Free math worksheets

- How many days are in one week? - Recall the multiples of \(7\). - Multiply the number of weeks by the number of days in each week.

1. One week has \(7\) days. 2. Multiply: \(9 \times 7 = 63\).

The project lasts \(63\) days.

- One kilogram equals \(1000\) grams. - Find the number of complete thousands in \(4055\). - The remaining grams stay in the smaller unit.

1. One kilogram equals \(1000\) grams. 2. Split \(4055\,\text{g}\) into \(4000\,\text{g} + 55\,\text{g}\). 3. Since \(4000\,\text{g} = 4\,\text{kg}\), the mass is \(4\,\text{kg}\ 55\,\text{g}\).

The package weighs \(4\,\text{kg}\ 55\,\text{g}\).

- What smaller unit is used in the question? - Think about how many inches make one foot. - The number of inches should be larger than the number of feet for the same length.

1. One foot is \(12\) inches. 2. Multiply: \(3 \times 12 = 36\).

\(36\,\text{in}\)

- Read the point's meter value from the equally spaced tenth-meter ticks. - Connect one whole meter with centimeters. - Check that using the smaller unit gives a larger numerical value.

1. Point \(A\) is at \(0.7\,\text{m}\). 2. Since \(1\,\text{m} = 100\,\text{cm}\), \(0.7\,\text{m} = 70\,\text{cm}\).

\(70\,\text{cm}\)

- Convert both sides of each comparison to the same unit. - Pay close attention to place value when converting meters. - Check for zeros or reversed digits.

1. For a), \(3\,\text{m}\ 5\,\text{cm} = 305\,\text{cm}\), and \(305 < 350\). 2. For b), \(6\,\text{cm} = 60\,\text{mm}\), so the measurements are equal. 3. For c), \(4\,\text{cm}\ 2\,\text{mm} = 42\,\text{mm}\), and \(420 > 42\). 4. For d), \(1\,\text{m}\ 10\,\text{cm} = 110\,\text{cm}\), and \(110 > 101\).

a) \(<\) b) \(=\) c) \(>\) d) \(>\)

- Use \(1\,\text{m}=100\,\text{cm}\) and think of the decimal part of a meter in hundredths. - Separate any whole meters from the decimal part before expressing the total in centimeters. - Remember that tenths of a meter are groups of ten centimeters and hundredths of a meter are single centimeters.

1. Since \(1\,\text{m}=100\,\text{cm}\), each hundredth of a meter is \(1\,\text{cm}\). 2. \(1.25\,\text{m}\) is \(1\,\text{m}\) and \(25\) hundredths of a meter, so it is \(100\,\text{cm}+25\,\text{cm}=125\,\text{cm}\). 3. \(0.80\,\text{m}\) is \(80\) hundredths of a meter, so it is \(80\,\text{cm}\). 4. \(2.06\,\text{m}\) is \(2\,\text{m}\) and \(6\) hundredths of a meter, so it is \(200\,\text{cm}+6\,\text{cm}=206\,\text{cm}\). 5. \(0.09\,\text{m}\) is \(9\) hundredths of a meter, so it is \(9\,\text{cm}\).

a) \(125\,\text{cm}\) b) \(80\,\text{cm}\) c) \(206\,\text{cm}\) d) \(9\,\text{cm}\)

- Convert all cards to meters. - One kilometer equals \(1000\) meters. - Pay close attention to place value and zeros.

1. Convert each mixed measurement to meters. 2. \(3\,\text{km}\ 200\,\text{m} = 3000\,\text{m} + 200\,\text{m} = 3200\,\text{m}\). 3. \(3\,\text{km}\ 20\,\text{m} = 3000\,\text{m} + 20\,\text{m} = 3020\,\text{m}\). 4. \(30\,\text{km}\ 200\,\text{m} = 30{,}000\,\text{m} + 200\,\text{m} = 30{,}200\,\text{m}\).

\(3200\,\text{m} = 3\,\text{km}\ 200\,\text{m}\) \(3020\,\text{m} = 3\,\text{km}\ 20\,\text{m}\) \(30{,}200\,\text{m} = 30\,\text{km}\ 200\,\text{m}\)

- Convert \(1\,\text{km}\) to meters first. - Subtract the given length from the total. - Think about completing each number to \(1000\).

1. Convert the target length: \(1\,\text{km} = 1000\,\text{m}\). 2. Subtract each given length from \(1000\,\text{m}\). 3. The differences are \(1000 - 600 = 400\), \(1000 - 150 = 850\), \(1000 - 880 = 120\), and \(1000 - 405 = 595\).

a) \(400\,\text{m}\) b) \(850\,\text{m}\) c) \(120\,\text{m}\) d) \(595\,\text{m}\)

- Convert \(1\,\text{km}\) to meters. - Ask how many equal shorter distances make \(1000\,\text{m}\). - Look for a multiplication fact with the shorter distance as one factor and \(1000\) as the product.

1. Convert the total distance: \(1\,\text{km}=1000\,\text{m}\). 2. Use related multiplication facts to find how many equal shorter distances make \(1000\,\text{m}\): \(4\times250=1000\), \(10\times100=1000\), \(2\times500=1000\), and \(5\times200=1000\).

a) \(4\) times b) \(10\) times c) \(2\) times d) \(5\) times

- Convert both measurements to the same unit. - Change kilometers to meters before comparing. - One kilometer equals \(1000\) meters.

1. Convert the kilometer measurements to meters. 2. Compare \(1000\,\text{m}\) and \(900\,\text{m}\): \(1000 > 900\). 3. Compare \(1500\,\text{m}\) and \(2000\,\text{m}\): \(1500 < 2000\). 4. Compare \(1000\,\text{m}\) and \(1000\,\text{m}\): they are equal. 5. Compare \(750\,\text{m}\) and \(1000\,\text{m}\): \(750 < 1000\).

a) \(>\) b) \(<\) c) \(=\) d) \(<\)

- Use \(100\,\text{cm}=1\,\text{m}\) to separate whole meters from leftover centimeters. - Think of leftover centimeters as hundredths of a meter. - Use zeros as placeholders when the tenths or whole-meter place has no units.

1. Since \(100\,\text{cm}=1\,\text{m}\), each full group of \(100\,\text{cm}\) makes one whole meter, and leftover centimeters are hundredths of a meter. 2. \(408\,\text{cm}=4\,\text{m}\ 8\,\text{cm}=4.08\,\text{m}\). 3. \(560\,\text{cm}=5\,\text{m}\ 60\,\text{cm}=5.60\,\text{m}\). 4. \(90\,\text{cm}=0\,\text{m}\ 90\,\text{cm}=0.90\,\text{m}\). 5. \(7\,\text{cm}=0\,\text{m}\ 7\,\text{cm}=0.07\,\text{m}\).

a) \(4.08\,\text{m}\) b) \(5.60\,\text{m}\) c) \(0.90\,\text{m}\) d) \(0.07\,\text{m}\)

- How many meters are in \(1\) kilometer? - Convert the kilometers first. - Then add the extra \(450\) meters.

1. Convert the kilometers to meters: \(128 \times 1000 = 128{,}000\,\text{m}\). 2. Add the remaining distance. Adding \(450\,\text{m}\) gives \(128{,}450\,\text{m}\).

The road is \(128{,}450\,\text{m}\) long.

- Remember that \(1\,\text{km} = 1000\,\text{m}\). - After converting, where do the \(8\) meters belong in the place-value positions? - A place-value chart can help you keep the needed zeros.

1. Convert the kilometers to meters: \(14 \times 1000 = 14{,}000\,\text{m}\). 2. Add the remaining distance. Adding \(8\,\text{m}\) gives \(14{,}008\,\text{m}\).

The bike path will be \(14{,}008\,\text{m}\) long.

- Convert both quantities to the same unit before comparing them. - How many pounds are in \(1\,\text{ton}\)? - How many ounces are in \(1\,\text{lb}\)? - What is half of the number of pounds in \(1\,\text{ton}\)?

1. a) Since \(1\,\text{ton}=2000\,\text{lb}\), \(3 \times 2000=6000\,\text{lb}\). Therefore, \(3\,\text{tons}=6000\,\text{lb}\). 2. b) Since \(1\,\text{lb}=16\,\text{oz}\), \(300 \times 16=4800\,\text{oz}\). Because \(4500<4800\), \(4500\,\text{oz}<300\,\text{lb}\). 3. c) Half of \(2000\,\text{lb}\) is \(1000\,\text{lb}\). Because \(1000>800\), \(\frac{1}{2}\,\text{ton}>800\,\text{lb}\). 4. d) Since \(1\,\text{lb}=16\,\text{oz}\) and \(16<1000\), \(1\,\text{lb}<1000\,\text{oz}\).

a) \(=\) b) \(<\) c) \(>\) d) \(<\)

- One hour equals \(60\) minutes. - To convert minutes to hours and minutes, find how many groups of \(60\) fit. - The remainder is the number of extra minutes.

1. For a), \(60 + 25 = 85\) minutes. 2. For b), \(3 \times 60 + 5 = 185\) minutes. 3. For c), \(85 = 60 + 25\), so the time is \(1\) hour \(25\) minutes. 4. For d), \(190 = 3 \times 60 + 10\), so the time is \(3\) hours \(10\) minutes.

a) \(85\,\text{min}\) b) \(185\,\text{min}\) c) \(1\,\text{hr}\ 25\,\text{min}\) d) \(3\,\text{hr}\ 10\,\text{min}\)

- Convert both times in each pair to the same unit. - Use the smaller unit when it makes comparison easier. - One minute equals \(60\) seconds.

1. \(180\,\text{s} = 3\,\text{min}\), so the measurements are equal. 2. \(1\) hour \(15\) minutes is \(75\) minutes, and \(75 < 80\). 3. \(2\) minutes \(10\) seconds is \(130\) seconds, and \(130 > 120\). 4. \(3\) hours \(10\) minutes is \(190\) minutes, and \(200 > 190\).

a) \(=\) b) \(<\) c) \(>\) d) \(>\)

- Recall the number of minutes in one-fourth hour and one-half hour. - Convert whole hours and fractional hours separately. - For larger numbers of minutes or seconds, find how many groups of \(60\) there are.

1. Three-fourths of an hour is \(3 \times 15 = 45\) minutes. 2. Two hours is \(120\) minutes, and one-half hour is \(30\) minutes. The total is \(150\) minutes. 3. \(420 \div 60 = 7\) hours. 4. \(600 \div 60 = 10\) minutes.

a) \(45\,\text{min}\) b) \(150\,\text{min}\) c) \(7\,\text{hr}\) d) \(10\,\text{min}\)

- Convert both sides to the same unit. - One hour is \(60\) minutes. - One minute is \(60\) seconds. - One day is \(24\) hours.

1. \(1\) hour is \(60\) minutes, and \(60 < 75\). 2. \(2\) minutes is \(120\) seconds, so the measurements are equal. 3. \(1\) day is \(24\) hours, and \(24 > 20\). 4. \(1\) hour \(30\) minutes is \(90\) minutes, and \(90 < 95\). 5. Three-fourths of an hour is \(45\) minutes, so the measurements are equal.

a) \(<\) b) \(=\) c) \(>\) d) \(<\) e) \(=\)

- One hour equals \(60\) minutes. - For more than \(60\) minutes, separate out a full hour and find the remaining minutes. - Ask how many groups of \(60\) fit in the number of minutes.

1. \(1\) hour \(15\) minutes is \(60 + 15 = 75\) minutes. 2. \(90\) minutes is \(1\) hour \(30\) minutes. 3. \(1\) hour \(40\) minutes is \(60 + 40 = 100\) minutes. 4. \(110\) minutes is \(1\) hour \(50\) minutes. 5. \(2\) hours is \(2 \times 60 = 120\) minutes.

a) \(75\,\text{min}\) b) \(1\,\text{hr}\ 30\,\text{min}\) c) \(100\,\text{min}\) d) \(1\,\text{hr}\ 50\,\text{min}\) e) \(120\,\text{min}\)

- Recall the lengths of one-fourth of an hour, one-half of an hour, and three-fourths of an hour. - One day has \(24\) hours. - One minute has \(60\) seconds.

1. One-fourth of an hour is \(15\) minutes, and one-half of an hour is \(30\) minutes. Together, they are \(15 + 30 = 45\) minutes. 2. Three-fourths of an hour is \(45\) minutes, so \(60 - 45 = 15\) minutes are needed. 3. Two days have \(2 \times 24 = 48\) hours. 4. Since \(60\) seconds is \(1\) minute, \(80\) seconds is \(1\) minute \(20\) seconds.

a) \(45\) minutes b) \(15\) minutes c) \(48\) hours d) \(1\) minute \(20\) seconds

- Convert both amounts to the same unit before comparing them. - How many cups are in \(1\,\text{quart}\)? - How many cups are in \(1\,\text{gallon}\)?

1. Use \(1\,\text{quart}=4\,\text{cups}\) and \(1\,\text{gallon}=16\,\text{cups}\). 2. a) Since \(3<4\), \(3\,\text{cups}<1\,\text{quart}\). 3. b) \(2\,\text{quarts}=8\,\text{cups}\). Since \(10>8\), \(10\,\text{cups}>2\,\text{quarts}\). 4. c) Half of \(16\,\text{cups}\) is \(8\,\text{cups}\), so \(\frac{1}{2}\,\text{gallon}=8\,\text{cups}\). 5. d) \(3\,\text{quarts}=12\,\text{cups}\). Since \(12<14\), \(3\,\text{quarts}<14\,\text{cups}\).

a) \(<\) b) \(>\) c) \(=\) d) \(<\)

- How many fluid ounces are in \(1\,\text{cup}\)? - Multiply the number of cups by \(8\). - Think of each fraction of a cup as the same fraction of \(8\,\text{fl oz}\).

1. Use \(1\,\text{cup}=8\,\text{fl oz}\). 2. Multiply each number of cups by \(8\): \(3 \times 8=24\) \(\frac{1}{2} \times 8=4\) \(\frac{3}{4} \times 8=6\) \(1\frac{1}{2} \times 8=12\) \(\frac{1}{4} \times 8=2\)

\(24\,\text{fl oz}\); \(4\,\text{fl oz}\); \(6\,\text{fl oz}\); \(12\,\text{fl oz}\); \(2\,\text{fl oz}\)

- Count the days when the museum is open. - How many hours are in one day? - Multiply the hours in one day by the number of days in a week.

1. The museum is open on Thursday, Friday, Saturday, and Sunday, which is \(4\) days. 2. Find the weekly open hours: \(4 \times 10\,\text{hours} = 40\,\text{hours}\). 3. One day has \(24\) hours, so a full week has \(7 \times 24\,\text{hours} = 168\,\text{hours}\).

a) The museum is open \(40\) hours each week. b) A full week has \(168\) hours.

- How many hours are in \(1\) day? - Once you know the hours in one day, how can you find the hours in many days? - Which operation combines equal groups?

1. One day has \(24\) hours. 2. Multiply to find the total number of hours: \(42 \times 24=1008\).

The expedition lasts \(1008\,\text{hours}\).

- One dollar equals \(100\) cents. - One meter equals \(100\) centimeters. - In each decimal, use the digits before and after the decimal point to identify the two units.

1. In dollar notation, the whole-number part gives dollars and the two decimal places give cents. Therefore, \(\$6.02\) is \(6\) dollars \(2\) cents, and \(\$35.70\) is \(35\) dollars \(70\) cents. 2. Since \(100\) cents equals \(1\) dollar, \(915\) cents is \(9\) dollars \(15\) cents. 3. Since \(100\,\text{cm} = 1\,\text{m}\), the two decimal places in a meter measurement represent centimeters. Therefore, \(4.08\,\text{m} = 4\,\text{m}\ 8\,\text{cm}\) and \(12.55\,\text{m} = 12\,\text{m}\ 55\,\text{cm}\).

a) \(6\,\text{dollars}\ 2\,\text{cents}\) b) \(35\,\text{dollars}\ 70\,\text{cents}\) c) \(9\,\text{dollars}\ 15\,\text{cents}\) d) \(4\,\text{m}\ 8\,\text{cm}\) e) \(12\,\text{m}\ 55\,\text{cm}\)

- How many meters are in \(1\,\text{km}\)? - How many grams are in \(1\,\text{kg}\)? - Subtract each given amount from its target amount.

1. Convert the target measurements: \(1\,\text{km}=1000\,\text{m}\) and \(1\,\text{kg}=1000\,\text{g}\). 2. a) \(1000\,\text{m}-420\,\text{m}=580\,\text{m}\), and \(1000\,\text{m}-885\,\text{m}=115\,\text{m}\). 3. b) \(1000\,\text{g}-350\,\text{g}=650\,\text{g}\), and \(1000\,\text{g}-75\,\text{g}=925\,\text{g}\).

a) \(580\,\text{m}\) and \(115\,\text{m}\) b) \(650\,\text{g}\) and \(925\,\text{g}\)

- Remember that \(1\) hour has \(60\) minutes, not \(100\). - How many cups are in \(1\) gallon? - How many pounds are in \(1\) ton?

1. Use \(1\,\text{gallon}=16\,\text{cups}\), \(1\,\text{hour}=60\,\text{min}\), and \(1\,\text{ton}=2000\,\text{lb}\). 2. a) \(16-10=6\), and \(16-15=1\). The missing capacities are \(6\,\text{cups}\) and \(1\,\text{cup}\). 3. b) \(60-12=48\), and \(60-38=22\). The missing times are \(48\,\text{min}\) and \(22\,\text{min}\). 4. c) \(2000-1300=700\), and \(2000-30=1970\). The missing weights are \(700\,\text{lb}\) and \(1970\,\text{lb}\).

a) \(6\,\text{cups}\) and \(1\,\text{cup}\) b) \(48\,\text{min}\) and \(22\,\text{min}\) c) \(700\,\text{lb}\) and \(1970\,\text{lb}\)

- How many minutes are in \(1\) full hour? - Multiply the number of hours by \(60\), then add the remaining minutes.

1. Soccer: \(1 \times 60\,\text{min}+30\,\text{min}=90\,\text{min}\). 2. Swimming: \(1 \times 60\,\text{min}+15\,\text{min}=75\,\text{min}\). 3. Gymnastics: \(2 \times 60\,\text{min}+5\,\text{min}=125\,\text{min}\).

Soccer: \(90\,\text{min}\); swimming: \(75\,\text{min}\); gymnastics: \(125\,\text{min}\).

- Convert both time intervals to minutes before comparing. - One hour equals \(60\) minutes.

1. a) \(1\,\text{hr}\ 20\,\text{min} = 60\,\text{min} + 20\,\text{min} = 80\,\text{min}\), so the intervals are equal. 2. b) \(2\,\text{hr}\ 15\,\text{min} = 120\,\text{min} + 15\,\text{min} = 135\,\text{min}\). Since \(130 < 135\), the first interval is shorter. 3. c) \(1\,\text{hr}\ 55\,\text{min} = 60\,\text{min} + 55\,\text{min} = 115\,\text{min}\). Since \(115 > 110\), the first interval is longer.

a) \(80\,\text{min} = 1\,\text{hr}\ 20\,\text{min}\) b) \(130\,\text{min} < 2\,\text{hr}\ 15\,\text{min}\) c) \(1\,\text{hr}\ 55\,\text{min} > 110\,\text{min}\)

- One hour equals \(60\) minutes. - One minute equals \(60\) seconds. - To find a unit fraction of a duration, divide the whole duration by the denominator.

1. \(\frac{1}{2}\) hour is \(60\,\text{min} \div 2 = 30\,\text{min}\). 2. \(\frac{3}{4}\) hour is \((60\,\text{min} \div 4) \times 3 = 45\,\text{min}\). 3. One minute equals \(60\) seconds. 4. \(\frac{1}{4}\) minute is \(60\,\text{s} \div 4 = 15\,\text{s}\). 5. \(\frac{1}{6}\) minute is \(60\,\text{s} \div 6 = 10\,\text{s}\).

\(\frac{1}{2}\,\text{hr} = 30\,\text{min}\) \(\frac{3}{4}\,\text{hr} = 45\,\text{min}\) \(1\,\text{min} = 60\,\text{s}\) \(\frac{1}{4}\,\text{min} = 15\,\text{s}\) \(\frac{1}{6}\,\text{min} = 10\,\text{s}\)

- One pound equals \(16\) ounces. - One ton equals \(2000\) pounds. - Divide the whole-unit amount by the denominator of each unit fraction.

1. Since \(1\,\text{lb} = 16\,\text{oz}\), \(\frac{1}{2}\,\text{lb} = 16\,\text{oz} \div 2 = 8\,\text{oz}\). 2. \(\frac{1}{4}\,\text{lb} = 16\,\text{oz} \div 4 = 4\,\text{oz}\). 3. \(\frac{1}{8}\,\text{lb} = 16\,\text{oz} \div 8 = 2\,\text{oz}\). 4. Since \(1\,\text{ton} = 2000\,\text{lb}\), \(\frac{1}{10}\,\text{ton} = 2000\,\text{lb} \div 10 = 200\,\text{lb}\).

\(\frac{1}{2}\,\text{lb} = 8\,\text{oz}\) \(\frac{1}{4}\,\text{lb} = 4\,\text{oz}\) \(\frac{1}{8}\,\text{lb} = 2\,\text{oz}\) \(\frac{1}{10}\,\text{ton} = 200\,\text{lb}\)

- How many hours are in a full day? - Separate the time into one day and the additional hours. - Add the hours in the day to the remaining hours.

1. Convert the day to hours: \(1\,\text{day} = 24\,\text{hr}\). 2. Add the remaining hours: \(24\,\text{hr} + 9\,\text{hr} = 33\,\text{hr}\).

The trip lasts \(33\) hours altogether.

- Convert both measurements to the same unit. - One meter equals \(100\) centimeters. - Compare only after both sides are in centimeters.

1. Use \(1\,\text{m} = 100\,\text{cm}\). 2. \(3\,\text{m} = 300\,\text{cm}\), so the measurements are equal. 3. \(5\,\text{m} = 500\,\text{cm}\), and \(500 < 520\). 4. \(7\,\text{m} = 700\,\text{cm}\), and \(800 > 700\).

a) \(=\) b) \(<\) c) \(>\)

- How many centimeters are in one meter? - Converting meters to centimeters makes the number larger. - Converting centimeters to meters makes the number smaller. - For a mixed measurement, convert the meters first and then add the centimeters.

1. Use \(1\,\text{m} = 100\,\text{cm}\). 2. For a), \(6 \times 100 = 600\,\text{cm}\). 3. For b), \(900 \div 100 = 9\,\text{m}\). 4. For c), \(4 \times 100 + 20 = 420\,\text{cm}\). 5. For d), \(10 \times 100 = 1000\,\text{cm}\).

a) \(600\,\text{cm}\) b) \(9\,\text{m}\) c) \(420\,\text{cm}\) d) \(1000\,\text{cm}\)

- Convert both sides to the same unit. - Change meters to centimeters before comparing. - Pay close attention to the tens digit in part c).

1. For a), \(5\,\text{m} = 500\,\text{cm}\), and \(500 > 50\). 2. For b), \(3\,\text{m} = 300\,\text{cm}\), so the measurements are equal. 3. For c), \(2\,\text{m}\ 5\,\text{cm} = 205\,\text{cm}\), and \(205 < 250\). 4. For d), \(8\,\text{m} = 800\,\text{cm}\), and \(800 < 801\).

a) \(>\) b) \(=\) c) \(<\) d) \(<\)

- One ton equals \(2000\) pounds. - When converting to a smaller unit, the numerical value becomes larger. - Look for complete groups of \(2000\) pounds and any remainder.

1. a) Since \(2000\,\text{lb}=1\,\text{ton}\) and \(4\times2000=8000\), \(8000\,\text{lb}=4\,\text{tons}\). 2. b) \(9\times2000\,\text{lb}=18{,}000\,\text{lb}\). 3. c) \(5000\,\text{lb}\) contains \(4000\,\text{lb}=2\,\text{tons}\), with \(1000\,\text{lb}\) remaining. 4. d) \(6\times2000\,\text{lb}+80\,\text{lb}=12{,}080\,\text{lb}\).

a) \(4\,\text{tons}\) b) \(18{,}000\,\text{lb}\) c) \(2\,\text{tons}\ 1000\,\text{lb}\) d) \(12{,}080\,\text{lb}\)

- Convert both sides to the same unit before comparing. - One ton equals \(2000\) pounds. - One pound equals \(16\) ounces.

1. a) \(3\,\text{tons} = 6000\,\text{lb}\). Since \(6000 > 300\), \(3\,\text{tons} > 300\,\text{lb}\). 2. b) \(2\,\text{tons}\ 1200\,\text{lb} = 4000\,\text{lb} + 1200\,\text{lb} = 5200\,\text{lb}\), so the weights are equal. 3. c) \(1\,\text{ton}\ 5\,\text{lb} = 2005\,\text{lb}\). Since \(2005 < 2050\), the first weight is less. 4. d) Since \(1\,\text{lb} = 16\,\text{oz}\), \(500\,\text{lb} = 500 \times 16\,\text{oz} = 8000\,\text{oz}\), so the weights are equal.

a) \(>\) b) \(=\) c) \(<\) d) \(=\)

- Convert both measurements in a pair to the same unit. - Recall the conversion factors for yards, feet, inches, pounds, ounces, and tons. - Divide the larger numerical value by the smaller one.

1. a) Since \(1\,\text{yd}=3\,\text{ft}\), the first measurement is \(3\) times as large. 2. b) Since \(1\,\text{ft}=12\,\text{in.}\), the first measurement is \(12\) times as large. 3. c) Since \(1\,\text{lb}=16\,\text{oz}\), the first measurement is \(16\) times as large. 4. d) Since \(1\,\text{ton}=2000\,\text{lb}\), calculate \(2000 \div 200=10\). The first measurement is \(10\) times as large.

a) \(3\) times b) \(12\) times c) \(16\) times d) \(10\) times

- Can you compare the measurements directly, or must you first use the same unit? - It is often helpful to convert the larger unit to the smaller unit. - Recall how many centimeters are in a meter and how many millimeters are in a centimeter.

1. Convert the larger unit in each pair to the smaller unit. 2. a) \(5\,\text{km}=5000\,\text{m}\), and \(5000>500\), so \(5\,\text{km}>500\,\text{m}\). 3. b) \(2\,\text{km}=2000\,\text{m}\), so \(2000\,\text{m}=2\,\text{km}\). 4. c) \(60\,\text{m}=6000\,\text{cm}\), and \(6000>600\), so \(60\,\text{m}>600\,\text{cm}\). 5. d) \(10\,\text{cm}=100\,\text{mm}\), and \(100<1000\), so \(10\,\text{cm}<1000\,\text{mm}\). 6. e) \(4\,\text{km}=4000\,\text{m}\), and \(4000<4400\), so \(4\,\text{km}<4400\,\text{m}\).

a) \(>\) b) \(=\) c) \(>\) d) \(<\) e) \(<\)

- How many meters are in \(1\,\text{km}\)? - Does the numerical value become larger or smaller when you convert kilometers to meters? - Use place value and complete groups of \(1000\) to move between meters and kilometers.

1. Use \(1\,\text{km}=1000\,\text{m}\). For kilometers to meters: a) \(7\times1000=7000\) b) \(26\times1000=26{,}000\) c) \(400\times1000=400{,}000\) 2. For meters to kilometers, identify complete groups of \(1000\,\text{m}\): d) \(3\times1000=3000\), so \(3000\,\text{m}=3\,\text{km}\). e) \(80\times1000=80{,}000\), so \(80{,}000\,\text{m}=80\,\text{km}\).

a) \(7000\,\text{m}\) b) \(26{,}000\,\text{m}\) c) \(400{,}000\,\text{m}\) d) \(3\,\text{km}\) e) \(80\,\text{km}\)

- Use \(100\,\text{cm} = 1\,\text{m}\). - Convert each part of a mixed measurement separately. - Check whether the requested unit is larger or smaller than the given unit.

1. a) \(407\,\text{cm}\) contains \(4\) groups of \(100\,\text{cm}\) with \(7\,\text{cm}\) remaining, so \(407\,\text{cm} = 4\,\text{m}\ 7\,\text{cm}\). 2. b) \(2\,\text{m} = 200\,\text{cm}\). Adding \(50\,\text{cm}\) gives \(250\,\text{cm}\). 3. c) \(1\,\text{m} = 100\,\text{cm}\). Adding \(34\,\text{cm}\) gives \(134\,\text{cm}\). 4. d) Since \(100\,\text{cm} = 1\,\text{m}\), \(800\,\text{cm} = 8\,\text{m}\).

a) \(4\,\text{m}\ 7\,\text{cm}\) b) \(250\,\text{cm}\) c) \(134\,\text{cm}\) d) \(8\,\text{m}\)

- Use \(100\,\text{cm}=1\,\text{m}\) and \(1000\,\text{m}=1\,\text{km}\). - Find the number of complete groups of the conversion factor. - The remainder stays in the smaller unit.

1. a) Since \(100\,\text{cm}=1\,\text{m}\), \(504\,\text{cm}=5\,\text{m}\ 4\,\text{cm}\). 2. b) Since \(100\,\text{cm}=1\,\text{m}\), \(820\,\text{cm}=8\,\text{m}\ 20\,\text{cm}\). 3. c) Since \(1000\,\text{m}=1\,\text{km}\), \(1530\,\text{m}=1\,\text{km}\ 530\,\text{m}\). 4. d) Since \(1000\,\text{m}=1\,\text{km}\), \(2009\,\text{m}=2\,\text{km}\ 9\,\text{m}\).

a) \(5\,\text{m}\ 4\,\text{cm}\) b) \(8\,\text{m}\ 20\,\text{cm}\) c) \(1\,\text{km}\ 530\,\text{m}\) d) \(2\,\text{km}\ 9\,\text{m}\)

- One liter equals \(1000\) milliliters. - Divide the number of milliliters into groups of \(1000\). - The remainder stays in milliliters.

1. Use \(1000\,\text{mL} = 1\,\text{L}\). 2. a) \(3600\,\text{mL}\) has \(3\) groups of \(1000\,\text{mL}\) and \(600\,\text{mL}\) remaining, so it is \(3\,\text{L}\ 600\,\text{mL}\). 3. b) \(5900\,\text{mL} = 5\,\text{L}\ 900\,\text{mL}\). 4. c) \(1240\,\text{mL} = 1\,\text{L}\ 240\,\text{mL}\). 5. d) \(700\,\text{mL} = 0\,\text{L}\ 700\,\text{mL}\).

a) \(3\,\text{L}\ 600\,\text{mL}\) b) \(5\,\text{L}\ 900\,\text{mL}\) c) \(1\,\text{L}\ 240\,\text{mL}\) d) \(0\,\text{L}\ 700\,\text{mL}\)

- Convert both lengths to centimeters before comparing. - Use \(1\,\text{m} = 100\,\text{cm}\). - Convert every part of a mixed measurement.

1. a) \(4\,\text{m}\ 7\,\text{cm} = 400\,\text{cm} + 7\,\text{cm} = 407\,\text{cm}\). Since \(407 > 405\), the correct symbol is \(>\). 2. b) \(2\,\text{m}\ 30\,\text{cm} = 200\,\text{cm} + 30\,\text{cm} = 230\,\text{cm}\), so the lengths are equal. 3. c) \(15\,\text{m}\ 2\,\text{cm} = 1500\,\text{cm} + 2\,\text{cm} = 1502\,\text{cm}\). Since \(1502 > 1500\), the correct symbol is \(>\). 4. d) \(6\,\text{m} = 600\,\text{cm}\), so the lengths are equal.

a) \(>\) b) \(=\) c) \(>\) d) \(=\)

- One meter equals \(100\) centimeters. - Count the complete groups of \(100\) centimeters. - The remainder is the number of centimeters.

1. Use \(100\,\text{cm} = 1\,\text{m}\). 2. a) \(678\,\text{cm} = 6\,\text{m}\ 78\,\text{cm}\). 3. b) \(409\,\text{cm} = 4\,\text{m}\ 9\,\text{cm}\). 4. c) \(1050\,\text{cm} = 10\,\text{m}\ 50\,\text{cm}\). 5. d) Because \(82\,\text{cm}\) is less than \(1\) meter, it is \(0\,\text{m}\ 82\,\text{cm}\).

a) \(6\,\text{m}\ 78\,\text{cm}\) b) \(4\,\text{m}\ 9\,\text{cm}\) c) \(10\,\text{m}\ 50\,\text{cm}\) d) \(0\,\text{m}\ 82\,\text{cm}\)

- One kilometer equals \(1000\) meters. - Complete groups of \(1000\) meters become kilometers. - Keep zeros in their correct place values.

1. Use \(1\,\text{km} = 1000\,\text{m}\). 2. \(6842\,\text{m}\) contains \(6\) kilometers with \(842\) meters remaining, so it is \(6\,\text{km}\ 842\,\text{m}\). 3. \(4030\,\text{m} = 4\,\text{km}\ 30\,\text{m}\). 4. \(9009\,\text{m} = 9\,\text{km}\ 9\,\text{m}\). 5. \(15\,\text{km}\ 75\,\text{m} = 15 \times 1000\,\text{m} + 75\,\text{m} = 15{,}075\,\text{m}\).

a) \(6\,\text{km}\ 842\,\text{m}\) b) \(4\,\text{km}\ 30\,\text{m}\) c) \(9\,\text{km}\ 9\,\text{m}\) d) \(15{,}075\,\text{m}\)

- Identify whether each length already uses one unit or combines two units. - For a mixed-unit length, convert the larger unit and add the remaining smaller units.

1. \(7\,\text{km}\) already uses one unit, so it remains \(7\,\text{km}\). 2. \(3\,\text{m}\ 50\,\text{cm}=300\,\text{cm}+50\,\text{cm}=350\,\text{cm}\). 3. \(12\,\text{cm}\ 4\,\text{mm}=120\,\text{mm}+4\,\text{mm}=124\,\text{mm}\). 4. \(450\,\text{m}\) already uses one unit, so it remains \(450\,\text{m}\). 5. \(1\,\text{km}\ 5\,\text{m}=1000\,\text{m}+5\,\text{m}=1005\,\text{m}\).

\(7\,\text{km}\); \(350\,\text{cm}\); \(124\,\text{mm}\); \(450\,\text{m}\); \(1005\,\text{m}\)

- Convert both sides of each comparison to the smaller unit. - Recall the conversion factors for pounds and ounces, hours and minutes, feet and inches, and tons and pounds. - Pay close attention to place value when the smaller-unit amount has only one or two digits.

1. a) \(3\,\text{lb}\ 4\,\text{oz}=48\,\text{oz}+4\,\text{oz}=52\,\text{oz}\). Since \(52<56\), \(3\,\text{lb}\ 4\,\text{oz}<56\,\text{oz}\). 2. b) \(2\,\text{hr}\ 5\,\text{min}=120\,\text{min}+5\,\text{min}=125\,\text{min}\), so the measurements are equal. 3. c) \(5\,\text{ft}\ 7\,\text{in.}=60\,\text{in.}+7\,\text{in.}=67\,\text{in.}\), so the measurements are equal. 4. d) \(1\,\text{ton}\ 500\,\text{lb}=2000\,\text{lb}+500\,\text{lb}=2500\,\text{lb}\). Since \(2500>2050\), \(1\,\text{ton}\ 500\,\text{lb}>2050\,\text{lb}\).

a) \(3\,\text{lb}\ 4\,\text{oz}<56\,\text{oz}\) b) \(2\,\text{hr}\ 5\,\text{min}=125\,\text{min}\) c) \(5\,\text{ft}\ 7\,\text{in.}=67\,\text{in.}\) d) \(1\,\text{ton}\ 500\,\text{lb}>2050\,\text{lb}\)

- How many grams are in \(1\,\text{kg}\)? - For a mixed-unit mass, convert the kilograms first and then add the remaining grams. - What does one-fourth of a kilogram mean?

1. Use \(1\,\text{kg}=1000\,\text{g}\). 2. a) \(7 \times 1000\,\text{g}=7000\,\text{g}\). 3. b) \(15 \times 1000\,\text{g}=15{,}000\,\text{g}\). 4. c) \(3 \times 1000\,\text{g}+250\,\text{g}=3250\,\text{g}\). 5. d) \(20 \times 1000\,\text{g}+5\,\text{g}=20{,}005\,\text{g}\). 6. e) Since \(1000\,\text{g} \div 4=250\,\text{g}\), \(\frac{1}{4}\,\text{kg}=250\,\text{g}\).

a) \(7000\,\text{g}\) b) \(15{,}000\,\text{g}\) c) \(3250\,\text{g}\) d) \(20{,}005\,\text{g}\) e) \(250\,\text{g}\)

- How many pounds are in \(1\,\text{ton}\)? - How many ounces are in \(1\,\text{lb}\)? - For pounds to tons, look for the number of complete \(2000\)-pound groups. - For a mixed-unit weight, convert the larger unit first and then add.

1. Use \(1\,\text{ton}=2000\,\text{lb}\). 2. a) \(9\times2000=18{,}000\), so \(9\,\text{tons}=18{,}000\,\text{lb}\). 3. b) Since \(14\times2=28\), \(14\times2000=28{,}000\), so \(14\,\text{tons}=28{,}000\,\text{lb}\). 4. c) \(6\times2000=12{,}000\), so \(12{,}000\,\text{lb}=6\,\text{tons}\). 5. d) \(32\times2000=64{,}000\), so \(64{,}000\,\text{lb}=32\,\text{tons}\). 6. e) \(5\,\text{lb}=80\,\text{oz}\), and \(80+2=82\), so the weight is \(82\,\text{oz}\).

a) \(18{,}000\,\text{lb}\) b) \(28{,}000\,\text{lb}\) c) \(6\,\text{tons}\) d) \(32\,\text{tons}\) e) \(82\,\text{oz}\)

- How many pounds are in \(1\,\text{ton}\)? - Use zeros as place-value placeholders when you write each product. - Convert the tons first, then add the remaining pounds.

1. Use \(1\,\text{ton}=2000\,\text{lb}\). 2. Multiply the number of tons by \(2000\), then add the remaining pounds. 3. a) \(4 \times 2000\,\text{lb}+250\,\text{lb}=8250\,\text{lb}\). 4. b) \(6 \times 2000\,\text{lb}+80\,\text{lb}=12{,}080\,\text{lb}\). 5. c) \(2 \times 2000\,\text{lb}+5\,\text{lb}=4005\,\text{lb}\). 6. d) \(15 \times 2000\,\text{lb}+15\,\text{lb}=30{,}015\,\text{lb}\).

a) \(8250\,\text{lb}\) b) \(12{,}080\,\text{lb}\) c) \(4005\,\text{lb}\) d) \(30{,}015\,\text{lb}\)

- Each conversion uses a factor of \(1000\). - Find the number of complete groups of \(1000\). - The remainder stays in the smaller unit.

1. Use \(1000\,\text{m} = 1\,\text{km}\), \(1000\,\text{g} = 1\,\text{kg}\), and \(1000\,\text{mL} = 1\,\text{L}\). 2. a) \(3750\,\text{m} = 3\,\text{km}\ 750\,\text{m}\). 3. b) \(8005\,\text{g} = 8\,\text{kg}\ 5\,\text{g}\). 4. c) \(12{,}040\,\text{mL} = 12\,\text{L}\ 40\,\text{mL}\). 5. d) \(50{,}002\,\text{m} = 50\,\text{km}\ 2\,\text{m}\).

a) \(3\,\text{km}\ 750\,\text{m}\) b) \(8\,\text{kg}\ 5\,\text{g}\) c) \(12\,\text{L}\ 40\,\text{mL}\) d) \(50\,\text{km}\ 2\,\text{m}\)

- One ton equals \(2000\) pounds. - Convert tons to pounds before adding the remaining pounds. - For the last equation, think about how many pounds must be added to reach one ton.

1. Use \(1\,\text{ton} = 2000\,\text{lb}\). 2. \(3 \times 2000\,\text{lb} + 5\,\text{lb} = 6005\,\text{lb}\). 3. \(8070\,\text{lb}\) contains \(8000\,\text{lb} = 4\) tons with \(70\,\text{lb}\) remaining. 4. \(10 \times 2000\,\text{lb} + 10\,\text{lb} = 20{,}010\,\text{lb}\). 5. \(2000\,\text{lb} - 1200\,\text{lb} = 800\,\text{lb}\).

1. \(6005\,\text{lb}\) 2. \(4\,\text{tons}\ 70\,\text{lb}\) 3. \(20{,}010\,\text{lb}\) 4. \(800\,\text{lb}\)

- How many ounces are in \(1\,\text{lb}\)? - How many pounds are in \(1\,\text{ton}\)? - Find a multiplication fact whose product is the given smaller-unit amount.

1. a) Since \(9\times16=144\), \(144\,\text{oz}=9\,\text{lb}\). 2. b) Since \(24\times16=384\), \(384\,\text{oz}=24\,\text{lb}\). 3. c) Since \(5\times2000=10{,}000\), \(10{,}000\,\text{lb}=5\,\text{tons}\). 4. d) Since \(110\times2000=220{,}000\), \(220{,}000\,\text{lb}=110\,\text{tons}\).

a) \(9\,\text{lb}\) b) \(24\,\text{lb}\) c) \(5\,\text{tons}\) d) \(110\,\text{tons}\)

- One kilogram equals \(1000\) grams. - Separate each number into thousands and the remainder. - Pay close attention to zeros in the tens and hundreds places.

1. Use \(1000\,\text{g} = 1\,\text{kg}\). Complete groups of \(1000\) become kilograms, and the remainder stays in grams. 2. a) \(1405\,\text{g} = 1\,\text{kg}\ 405\,\text{g}\). 3. b) \(3060\,\text{g} = 3\,\text{kg}\ 60\,\text{g}\). 4. c) \(7008\,\text{g} = 7\,\text{kg}\ 8\,\text{g}\). 5. d) \(12{,}500\,\text{g} = 12\,\text{kg}\ 500\,\text{g}\). 6. e) \(20{,}030\,\text{g} = 20\,\text{kg}\ 30\,\text{g}\).

a) \(1\,\text{kg}\ 405\,\text{g}\) b) \(3\,\text{kg}\ 60\,\text{g}\) c) \(7\,\text{kg}\ 8\,\text{g}\) d) \(12\,\text{kg}\ 500\,\text{g}\) e) \(20\,\text{kg}\ 30\,\text{g}\)

- Convert all measurements to grams. - Use place value carefully, especially when zeros appear. - Compare the resulting whole numbers.

1. Convert each mixed measurement to grams. 2. a) \(3\,\text{kg}\ 50\,\text{g} = 3050\,\text{g}\), and \(3050 < 3500\). 3. b) \(6\,\text{kg}\ 40\,\text{g} = 6040\,\text{g}\), and \(6004 < 6040\). 4. c) \(2\,\text{kg}\ 7\,\text{g} = 2007\,\text{g}\), so the masses are equal. 5. d) \(10\,\text{kg}\ 200\,\text{g} = 10{,}200\,\text{g}\), and \(10{,}200 > 1200\). 6. e) \(4\,\text{kg}\ 8\,\text{g} = 4008\,\text{g}\), and \(4080 > 4008\).

a) \(<\) b) \(<\) c) \(=\) d) \(>\) e) \(>\)

- One meter equals \(100\) centimeters. - One kilometer equals \(1000\) meters. - Keep zeros in their correct place values.

1. a) \(408\,\text{cm}\) contains \(4\) complete meters and \(8\) centimeters, so \(408\,\text{cm} = 4\,\text{m}\ 8\,\text{cm}\). 2. b) \(1050\,\text{m}\) contains \(1\) kilometer and \(50\) meters, so \(1050\,\text{m} = 1\,\text{km}\ 50\,\text{m}\). 3. c) \(7\,\text{m}\ 3\,\text{cm} = 7 \times 100\,\text{cm} + 3\,\text{cm} = 703\,\text{cm}\). 4. d) \(5\,\text{km}\ 20\,\text{m} = 5 \times 1000\,\text{m} + 20\,\text{m} = 5020\,\text{m}\).

a) \(4\,\text{m}\ 8\,\text{cm}\) b) \(1\,\text{km}\ 50\,\text{m}\) c) \(703\,\text{cm}\) d) \(5020\,\text{m}\)

- One liter equals \(1000\) milliliters. - Separate the number into thousands and the remainder. - The complete thousands are liters.

1. Use \(1000\,\text{mL} = 1\,\text{L}\). 2. The amount \(15{,}700\,\text{mL}\) contains \(15\) complete liters with \(700\,\text{mL}\) remaining. 3. Therefore, \(15{,}700\,\text{mL} = 15\,\text{L}\ 700\,\text{mL}\).

The container holds \(15\,\text{L}\ 700\,\text{mL}\).

- One ton equals \(2000\) pounds. - Find the greatest multiple of \(2000\) that does not exceed \(24{,}040\). - The difference is the remaining number of pounds.

1. Use \(1\,\text{ton} = 2000\,\text{lb}\). 2. Twelve tons equals \(12 \times 2000\,\text{lb} = 24{,}000\,\text{lb}\). 3. The remaining weight is \(24{,}040\,\text{lb} - 24{,}000\,\text{lb} = 40\,\text{lb}\). 4. Therefore, the load weighs \(12\,\text{tons}\ 40\,\text{lb}\).

The load weighs \(12\,\text{tons}\ 40\,\text{lb}\).

- How many months are in one year? - Convert the full years to months first. - Add the extra months.

1. Anna: \(2 \times 12 + 3 = 24 + 3 = 27\) months. 2. Ben: \(4 \times 12 + 8 = 48 + 8 = 56\) months. 3. Clara: \(6 \times 12 + 1 = 72 + 1 = 73\) months.

Anna: \(27\) months Ben: \(56\) months Clara: \(73\) months

- How many years are in \(1\) century? - What number should you multiply by to convert centuries to years? - Compare your result with Maya's number.

1. One century is \(100\) years, so multiply: \(12 \times 100 = 1200\). 2. Therefore, \(12\) centuries is \(1200\) years. 3. Maya likely multiplied by \(10\) instead of \(100\), or omitted one zero.

\(12\) centuries is \(1200\,\text{years}\). Maya likely multiplied by \(10\) instead of \(100\).

- How many days are in one week? - Find the number of days in the full weeks. - Add the remaining days.

1. Convert the weeks to days: \(7 \times 7 = 49\) days. 2. Add the extra days: \(49 + 4 = 53\) days.

The circus stays for \(53\) days.

- How many seconds are in \(1\) minute? - First find the number of seconds in the full minutes. - Add the remaining seconds at the end. - You can break apart a product such as \(12 \times 60\) into easier partial products.

1. Use \(1\,\text{min}=60\,\text{s}\). 2. Convert the full minutes to seconds, then add the remaining seconds. 3. a) \(5 \times 60\,\text{s}+20\,\text{s}=300\,\text{s}+20\,\text{s}=320\,\text{s}\). 4. b) \(12 \times 60\,\text{s}+45\,\text{s}=720\,\text{s}+45\,\text{s}=765\,\text{s}\). 5. c) \(20 \times 60\,\text{s}+8\,\text{s}=1200\,\text{s}+8\,\text{s}=1208\,\text{s}\).

a) \(320\,\text{s}\) b) \(765\,\text{s}\) c) \(1208\,\text{s}\)

- One hour equals \(60\) minutes. - Find the greatest multiple of \(60\) that does not exceed each number. - The difference is the remaining number of minutes.

1. Find the greatest multiple of \(60\) that does not exceed each number of minutes. The number of groups gives the complete hours, and the difference gives the remaining minutes. 2. a) \(145=2\times60+25\), so \(145\,\text{min}=2\,\text{hr}\ 25\,\text{min}\). 3. b) \(310=5\times60+10\), so \(310\,\text{min}=5\,\text{hr}\ 10\,\text{min}\). 4. c) \(605=10\times60+5\), so \(605\,\text{min}=10\,\text{hr}\ 5\,\text{min}\).

a) \(2\,\text{hr}\ 25\,\text{min}\) b) \(5\,\text{hr}\ 10\,\text{min}\) c) \(10\,\text{hr}\ 5\,\text{min}\)

- Recall how many smaller time units are in each larger unit. - Write the conversion relationship for hours, minutes, seconds, months, or years. - Use related multiplication facts to check each statement.

1. a) Since \(3\times60=180\), \(180\) minutes is \(3\) hours. The conversion is true. 2. b) Since \(2\times12=24\), \(2\) years is \(24\) months. The conversion is false. 3. c) Since \(5\times60=300\), \(5\) minutes is \(300\) seconds. The conversion is false. 4. d) Since \(3\times24=72\), \(72\) hours is \(3\) days. The conversion is true.

a) True b) False c) False d) True

- What is the next larger unit after seconds? After minutes? - Look for the number of complete conversion-factor groups in each amount. - Use a related multiplication fact to check the conversion.

1. a) Since \(7\times60=420\), \(420\,\text{s}=7\,\text{min}\). 2. b) Since \(10\times60=600\), \(600\,\text{min}=10\,\text{hr}\). 3. c) Since \(4\times24=96\), \(96\,\text{hr}=4\,\text{days}\). 4. d) Since \(10\times12=120\), \(120\,\text{months}=10\,\text{years}\).

a) \(7\,\text{min}\) b) \(10\,\text{hr}\) c) \(4\,\text{days}\) d) \(10\,\text{years}\)

- One year has \(12\) months. - Find the number of complete groups of \(12\). - The remainder is the number of additional months.

1. Find the number of complete groups of \(12\) months. Each complete group is one year, and any amount left over is the additional number of months. 2. a) \(32=2\times12+8\), so \(32\) months is \(2\) years \(8\) months. 3. b) \(50=4\times12+2\), so \(50\) months is \(4\) years \(2\) months. 4. c) \(75=6\times12+3\), so \(75\) months is \(6\) years \(3\) months. 5. d) \(100=8\times12+4\), so \(100\) months is \(8\) years \(4\) months.

a) \(2\) years \(8\) months b) \(4\) years \(2\) months c) \(6\) years \(3\) months d) \(8\) years \(4\) months

- Convert both sides to seconds. - One minute equals \(60\) seconds. - Multiply the minutes by \(60\), then add the remaining seconds.

1. Convert each mixed time interval to seconds using \(1\,\text{min} = 60\,\text{s}\). 2. a) \(4 \times 60\,\text{s} + 15\,\text{s} = 255\,\text{s}\). Since \(260 > 255\), the correct symbol is \(>\). 3. b) \(8 \times 60\,\text{s} + 20\,\text{s} = 500\,\text{s}\), so the intervals are equal. 4. c) \(10 \times 60\,\text{s} + 5\,\text{s} = 605\,\text{s}\). Since \(605 > 600\), the correct symbol is \(>\). 5. d) \(12 \times 60\,\text{s} + 40\,\text{s} = 760\,\text{s}\), so the intervals are equal.

a) \(>\) b) \(=\) c) \(>\) d) \(=\)

- One meter equals \(100\) centimeters. - Convert the brother's claim back to centimeters. - Pay attention to the zero in \(405\).

1. Emma's brother's measurement equals \(4\,\text{m}\ 50\,\text{cm} = 400\,\text{cm} + 50\,\text{cm} = 450\,\text{cm}\). 2. Since \(450\,\text{cm} \ne 405\,\text{cm}\), he is not correct. 3. The measurement \(405\,\text{cm}\) contains \(4\) complete meters with \(5\) centimeters remaining, so \(405\,\text{cm} = 4\,\text{m}\ 5\,\text{cm}\).

He is not correct. \(405\,\text{cm} = 4\,\text{m}\ 5\,\text{cm}\), not \(4\,\text{m}\ 50\,\text{cm}\).

- How many seconds make \(1\) minute? - Add the minutes and seconds separately. - Regroup any total of at least \(60\) seconds as minutes and seconds.

1. Add \(18+37=55\) seconds and \(6+9=15\) minutes. The result is \(15\,\text{min}\ 55\,\text{s}\). 2. Add \(44+26=70\) seconds. Since \(70\) seconds is \(1\) minute \(10\) seconds, add \(12+5+1=18\) minutes. The result is \(18\,\text{min}\ 10\,\text{s}\). 3. Add \(52+39=91\) seconds. Since \(91\) seconds is \(1\) minute \(31\) seconds, add \(15+11+1=27\) minutes. The result is \(27\,\text{min}\ 31\,\text{s}\).

1. \(15\,\text{min}\ 55\,\text{s}\) 2. \(18\,\text{min}\ 10\,\text{s}\) 3. \(27\,\text{min}\ 31\,\text{s}\)

- One hour is \(60\) minutes, and one minute is \(60\) seconds. - Regroup any total of at least \(60\) minutes. - When subtracting, regroup one larger unit if the smaller-unit amount is not large enough.

1. a) Add \(35+45=80\) minutes. Regroup \(80\) minutes as \(1\) hour \(20\) minutes. Then \(14+8+1=23\) hours, so the result is \(23\,\text{hr}\ 20\,\text{min}\). 2. b) Regroup \(12\,\text{min}\ 15\,\text{s}\) as \(11\,\text{min}\ 75\,\text{s}\). Then subtract to get \(4\,\text{min}\ 35\,\text{s}\).

a) \(23\,\text{hr}\ 20\,\text{min}\) b) \(4\,\text{min}\ 35\,\text{s}\)

- First identify how many smaller units make one whole unit. - Divide the whole-unit amount by the fraction's denominator. - One dollar equals \(100\) cents.

1. Since \(1\,\text{m} = 100\,\text{cm}\), \(\frac{1}{2}\,\text{m} = 100\,\text{cm} \div 2 = 50\,\text{cm}\). 2. Since \(1\,\text{kg} = 1000\,\text{g}\), \(\frac{1}{4}\,\text{kg} = 1000\,\text{g} \div 4 = 250\,\text{g}\). 3. Since one dollar equals \(100\) cents, \(\frac{1}{10}\) of a dollar is \(100\,\text{cents} \div 10 = 10\,\text{cents}\).

a) \(50\,\text{cm}\) b) \(250\,\text{g}\) c) \(10\,\text{cents}\)

- Choose the unit that makes each measurement reasonable. - Then use the next smaller unit from the given list. - Millimeters are already the smallest unit in the list.

1. An elephant’s shoulder height is reasonably measured in meters: \(4\,\text{m} = 400\,\text{cm}\). 2. A wood ant’s length is reasonably measured in millimeters: \(6\,\text{mm}\). Millimeters are already the smallest unit in the list. 3. A blue whale’s length is reasonably measured in meters: \(33\,\text{m} = 3300\,\text{cm}\). 4. A long migration distance is reasonably measured in kilometers: more than \(10{,}000\,\text{km}\), which is more than \(10{,}000{,}000\,\text{m}\).

a) \(4\,\text{m} = 400\,\text{cm}\) b) \(6\,\text{mm}\) c) \(33\,\text{m} = 3300\,\text{cm}\) d) More than \(10{,}000\,\text{km}\), which is more than \(10{,}000{,}000\,\text{m}\)

- One liter equals \(1000\) milliliters. - One hour equals \(60\) minutes. - Divide by the denominator, then multiply by the numerator when needed.

1. Since \(1\,\text{L} = 1000\,\text{mL}\), \(\frac{1}{2}\,\text{L} = 500\,\text{mL}\), \(\frac{1}{4}\,\text{L} = 250\,\text{mL}\), and \(\frac{1}{10}\,\text{L} = 100\,\text{mL}\). 2. Since \(1\,\text{hr} = 60\,\text{min}\), \(\frac{1}{2}\,\text{hr} = 30\,\text{min}\), \(\frac{1}{4}\,\text{hr} = 15\,\text{min}\), and \(\frac{3}{4}\,\text{hr} = 3 \times 15\,\text{min} = 45\,\text{min}\).

a) \(500\,\text{mL}\); \(250\,\text{mL}\); \(100\,\text{mL}\) b) \(30\,\text{min}\); \(15\,\text{min}\); \(45\,\text{min}\)

- How many centimeters are in \(1\,\text{m}\)? - Convert the meters first, then add the remaining centimeters. - Pay close attention to place value when the centimeter amount is less than \(10\).

1. Use \(1\,\text{m}=100\,\text{cm}\). 2. a) \(3 \times 100\,\text{cm}+15\,\text{cm}=315\,\text{cm}\). 3. b) \(5 \times 100\,\text{cm}+4\,\text{cm}=504\,\text{cm}\). 4. c) \(10 \times 100\,\text{cm}+60\,\text{cm}=1060\,\text{cm}\). 5. d) \(2 \times 100\,\text{cm}+8\,\text{cm}=208\,\text{cm}\).

a) \(315\,\text{cm}\) b) \(504\,\text{cm}\) c) \(1060\,\text{cm}\) d) \(208\,\text{cm}\)

- Recall how many smaller units are in one larger unit. - Multiply the larger-unit amount by the conversion factor, then add the remaining smaller units. - Use place value carefully when adding a small remainder.

1. a) Since \(1\,\text{yd}=3\,\text{ft}\), \(12 \times 3\,\text{ft}+2\,\text{ft}=38\,\text{ft}\). 2. b) Since \(1\,\text{ft}=12\,\text{in.}\), \(40 \times 12\,\text{in.}+8\,\text{in.}=488\,\text{in.}\). 3. c) Since \(1\,\text{ton}=2000\,\text{lb}\), \(5 \times 2000\,\text{lb}+70\,\text{lb}=10{,}070\,\text{lb}\). 4. d) Since \(1\,\text{lb}=16\,\text{oz}\), \(10 \times 16\,\text{oz}+5\,\text{oz}=165\,\text{oz}\).

a) \(38\,\text{ft}\) b) \(488\,\text{in.}\) c) \(10{,}070\,\text{lb}\) d) \(165\,\text{oz}\)

- Convert each left-hand measurement to the unit on the right. - Use \(1\,\text{km} = 1000\,\text{m}\), \(1\,\text{m} = 100\,\text{cm}\), and \(1\,\text{ton} = 2000\,\text{lb}\). - Compare the resulting whole numbers.

1. a) \(20\,\text{km}\ 5\,\text{m} = 20{,}005\,\text{m}\). Since \(20{,}005 > 2500\), the correct symbol is \(>\). 2. b) \(8\,\text{m}\ 40\,\text{cm} = 840\,\text{cm}\). Since \(840 > 804\), the correct symbol is \(>\). 3. c) \(4\,\text{tons}\ 50\,\text{lb} = 8000\,\text{lb} + 50\,\text{lb} = 8050\,\text{lb}\). Since \(8050 < 8500\), the correct symbol is \(<\).

a) \(>\) b) \(>\) c) \(<\)

- Recall how many meters are in \(1\) kilometer. - Separate the full thousands of meters from the remaining meters. - What do the first two digits represent when the distance is split into kilometers and meters?

1. Use \(1000\,\text{m} = 1\,\text{km}\). 2. Separate the distance into \(42{,}000\,\text{m}\) and the remaining \(195\,\text{m}\). 3. Since \(42{,}000\,\text{m} = 42\,\text{km}\), the full distance is \(42\,\text{km}\,195\,\text{m}\).

\(42\,\text{km}\,195\,\text{m}\)

- Identify the conversion factor for each pair of units. - Keep zeros in their correct place values. - For part d), convert the mixed weight to pounds before comparing.

1. a) \(3050\,\text{mL}\) contains \(3\) liters with \(50\,\text{mL}\) remaining. 2. b) \(9\,\text{km}\ 4\,\text{m} = 9000\,\text{m} + 4\,\text{m} = 9004\,\text{m}\). 3. c) \(15{,}005\,\text{g}\) contains \(15\) kilograms with \(5\,\text{g}\) remaining. 4. d) \(4\,\text{tons}\ 20\,\text{lb} = 8020\,\text{lb}\). Since \(8020 < 8200\), the correct symbol is \(<\).

a) \(3\,\text{L}\ 50\,\text{mL}\) b) \(9004\,\text{m}\) c) \(15\,\text{kg}\ 5\,\text{g}\) d) \(<\)

- Find how many smaller units make one of the next larger units. - Find the number of complete groups of that conversion factor. - Write any remainder in the smaller unit.

1. a) Since \(100\,\text{cm}=1\,\text{m}\), \(940\,\text{cm}=9\,\text{m}\ 40\,\text{cm}\). 2. b) Since \(1000\,\text{m}=1\,\text{km}\), \(6200\,\text{m}=6\,\text{km}\ 200\,\text{m}\). 3. c) Since \(1000\,\text{g}=1\,\text{kg}\), \(5080\,\text{g}=5\,\text{kg}\ 80\,\text{g}\). 4. d) Since \(2000\,\text{lb}=1\,\text{ton}\), \(24{,}400\,\text{lb}=12\,\text{tons}\ 400\,\text{lb}\).

a) \(9\,\text{m}\ 40\,\text{cm}\) b) \(6\,\text{km}\ 200\,\text{m}\) c) \(5\,\text{kg}\ 80\,\text{g}\) d) \(12\,\text{tons}\ 400\,\text{lb}\)

- Convert each lettered measurement to the unit used by one of the numbered choices. - Keep zeros in their correct place values. - One ton equals \(2000\) pounds.

1. A: \(3\,\text{km}\ 5\,\text{m} = 3000\,\text{m} + 5\,\text{m} = 3005\,\text{m}\), so A matches 1. 2. B: \(350\,\text{cm} = 3\,\text{m}\ 50\,\text{cm}\), so B matches 3. 3. C: \(3\,\text{kg}\ 50\,\text{g} = 3050\,\text{g}\), so C matches 4. 4. D: \(35\,\text{tons} = 35 \times 2000\,\text{lb} = 70{,}000\,\text{lb}\), so D matches 2.

A-1, B-3, C-4, D-2

- Recall how many kilograms equal one metric ton. - Separate each number into complete thousands and leftover kilograms. - Do not treat missing place values as extra kilograms.

1. Use \(1000\,\text{kg} = 1\) metric ton. The number of complete thousands gives the metric tons, and the remaining kilograms stay as kilograms. 2. \(12{,}450\,\text{kg} = 12\) metric tons \(450\,\text{kg}\). 3. \(9070\,\text{kg} = 9\) metric tons \(70\,\text{kg}\). 4. \(25{,}004\,\text{kg} = 25\) metric tons \(4\,\text{kg}\).

\(12\) metric tons \(450\,\text{kg}\); \(9\) metric tons \(70\,\text{kg}\); \(25\) metric tons \(4\,\text{kg}\)

- Recall how many meters are in \(1\) kilometer. - Separate each number into full groups of \(1000\) meters and the remaining meters. - Pay close attention to zeros inside the numbers.

1. Use \(1000\,\text{m} = 1\,\text{km}\). 2. \(14{,}320\,\text{m} = 14\,\text{km}\,320\,\text{m}\). 3. \(5080\,\text{m} = 5\,\text{km}\,80\,\text{m}\). 4. \(40{,}009\,\text{m} = 40\,\text{km}\,9\,\text{m}\).

\(14\,\text{km}\,320\,\text{m}\) \(5\,\text{km}\,80\,\text{m}\) \(40\,\text{km}\,9\,\text{m}\)

- Identify whether the conversion factor is \(100\) or \(1000\). - Complete groups become the larger unit. - The remainder stays in the smaller unit.

1. Since \(1000\,\text{m} = 1\,\text{km}\), \(12{,}005\,\text{m} = 12\,\text{km}\ 5\,\text{m}\). 2. Since \(1000\,\text{mL} = 1\,\text{L}\), \(2080\,\text{mL} = 2\,\text{L}\ 80\,\text{mL}\). 3. Since \(100\) cents equals one dollar, \(505\) cents is \(5\) dollars \(5\) cents. 4. Since \(1000\,\text{g} = 1\,\text{kg}\), \(8070\,\text{g} = 8\,\text{kg}\ 70\,\text{g}\).

\(12\,\text{km}\ 5\,\text{m}\); \(2\,\text{L}\ 80\,\text{mL}\); \(5\,\text{dollars}\ 5\,\text{cents}\); \(8\,\text{kg}\ 70\,\text{g}\)

- How many ounces are in \(1\,\text{lb}\)? - How many pounds are in \(1\,\text{ton}\)? - Add the smaller units first and regroup when possible. - You can also convert everything to the smallest unit, add, and convert back.

1. a) \(12\,\text{oz}+8\,\text{oz}=20\,\text{oz}=1\,\text{lb}\ 4\,\text{oz}\). 2. b) Add \(12+3=15\) pounds and \(12+4=16\) ounces. Since \(16\,\text{oz}=1\,\text{lb}\), the result is \(16\,\text{lb}\). 3. c) Add \(4+5=9\) tons and \(1200+1100=2300\) pounds. Since \(2300\,\text{lb}=1\,\text{ton}\ 300\,\text{lb}\), the result is \(10\,\text{tons}\ 300\,\text{lb}\).

a) \(1\,\text{lb}\ 4\,\text{oz}\) b) \(16\,\text{lb}\) c) \(10\,\text{tons}\ 300\,\text{lb}\)

- Put the masses in one common unit before ordering them. - The two records already written only in pounds can stay in pounds. - Check neighboring values after you order them; two of the masses are fairly close.

1. Convert the mixed ton-pound masses to pounds: Alder is \(2000+800=2800\,\text{lb}\), and Cedar is \(4000+100=4100\,\text{lb}\). 2. Compare all four masses in pounds: \(2800<3500<3900<4100\). 3. Therefore, the order is Alder, Birch, Dogwood, Cedar.

Alder, Birch, Dogwood, Cedar

- Convert each record to pounds before matching it. - Keep the extra pounds when a record uses both tons and pounds. - Use each label exactly once and check every match by converting back mentally.

1. Juniper is \(2000+500=2500\,\text{lb}\), so 1 matches d. 2. Maple is \(3\times2000=6000\,\text{lb}\), so 2 matches c. 3. Oak is \(2\times2000+100=4100\,\text{lb}\), so 3 matches a. 4. Pine is \(2000+1500=3500\,\text{lb}\), so 4 matches b.

1) d 2) c 3) a 4) b

- How many grams are in \(1\,\text{kg}\)? - From kilograms to grams, use groups of \(1000\) grams. - From grams to kilograms, identify how many complete \(1000\)-gram groups are present.

1. Use \(1\,\text{kg}=1000\,\text{g}\). 2. \(8\,\text{kg}=8\times1000=8000\,\text{g}\). 3. Since \(15\times1000=15{,}000\), \(15{,}000\,\text{g}=15\,\text{kg}\). 4. \(250\,\text{kg}=250\times1000=250{,}000\,\text{g}\). 5. Since \(900\times1000=900{,}000\), \(900{,}000\,\text{g}=900\,\text{kg}\).

The missing values are \(8000\,\text{g}\), \(15\,\text{kg}\), \(250{,}000\,\text{g}\), and \(900\,\text{kg}\).

- One meter equals \(100\) centimeters. - Convert each part of a mixed measurement before adding. - When converting centimeters to meters, count complete groups of \(100\).

1. a) \(15 \times 100\,\text{cm} = 1500\,\text{cm}\). 2. b) \(3 \times 100\,\text{cm} + 7\,\text{cm} = 307\,\text{cm}\). 3. c) \(240\,\text{cm}\) contains \(2\) complete meters with \(40\) centimeters remaining, so it is \(2\,\text{m}\ 40\,\text{cm}\). 4. d) \(10 \times 100\,\text{cm} + 50\,\text{cm} = 1000\,\text{cm} + 50\,\text{cm} = 1050\,\text{cm}\).

a) \(1500\,\text{cm}\) b) \(307\,\text{cm}\) c) \(2\,\text{m}\ 40\,\text{cm}\) d) \(1050\,\text{cm}\)

- Convert each whole unit to the requested smaller unit. - Divide by the denominator. - Multiply by the numerator when it is greater than \(1\).

1. Use \(1\,\text{m} = 100\,\text{cm}\), \(1\,\text{km} = 1000\,\text{m}\), and \(1\,\text{L} = 1000\,\text{mL}\). 2. a) \(100\,\text{cm} \div 2 = 50\,\text{cm}\). 3. b) \(1000\,\text{m} \div 5 = 200\,\text{m}\). 4. c) \(1000\,\text{mL} \div 4 = 250\,\text{mL}\), and \(3 \times 250\,\text{mL} = 750\,\text{mL}\).

a) \(50\,\text{cm}\) b) \(200\,\text{m}\) c) \(750\,\text{mL}\)

- Identify how many smaller units make one whole unit. - Divide by the denominator. - Multiply the result by the numerator.

1. One hour equals \(60\) minutes, one day equals \(24\) hours, and one dollar equals \(100\) cents. 2. a) \(60\,\text{min} \div 3 = 20\,\text{min}\), and \(2 \times 20\,\text{min} = 40\,\text{min}\). 3. b) \(24\,\text{hr} \div 8 = 3\,\text{hr}\), and \(3 \times 3\,\text{hr} = 9\,\text{hr}\). 4. c) \(100\,\text{cents} \div 5 = 20\,\text{cents}\), and \(4 \times 20\,\text{cents} = 80\,\text{cents}\).

a) \(40\,\text{min}\) b) \(9\,\text{hr}\) c) \(80\,\text{cents}\)

- One dollar equals \(100\) cents. - The last two digits show the number of cents. - Use a zero in the tenths place when the remaining number of cents has only one digit.

1. One dollar equals \(100\) cents, so separate each amount into groups of \(100\) cents and the remaining cents. 2. a) \(804\) cents is \(8\) dollars and \(4\) cents, or \(\$8.04\). 3. b) \(2560\) cents is \(25\) dollars and \(60\) cents, or \(\$25.60\). 4. c) \(7003\) cents is \(70\) dollars and \(3\) cents, or \(\$70.03\). 5. d) \(15{,}020\) cents is \(150\) dollars and \(20\) cents, or \(\$150.20\).

a) \(\$8.04\) b) \(\$25.60\) c) \(\$70.03\) d) \(\$150.20\)

- Compare the sizes of a meter and a centimeter. - For one fixed length, what happens to the number of units when the units themselves get smaller? - Distinguish the size of the measurement unit from the numerical value written beside it.

1. The numerical values are \(4\) and \(400\), so \(400\) is greater. 2. Centimeters are smaller units than meters, so more centimeters are needed to describe the same length.

\(400\,\text{cm}\) has the greater numerical value. The same length uses a larger number when it is measured in the smaller unit.

- Which two units are actually being connected in the statement? - Recall the relationship between one yard and feet. - Check whether the proposed result is reasonable for only \(6\) yards.

1. Mila used the feet-to-inches relationship instead of the yards-to-feet relationship. 2. One yard is \(3\) feet, so \(6 \times 3 = 18\).

Mila used the wrong conversion relationship. \(6\,\text{yd} = 18\,\text{ft}\).

- Compare each option with familiar real-world lengths. - Eliminate units that would make the door impossibly tiny or enormous. - After choosing the reasonable value, convert meters to centimeters.

1. A classroom door is reasonably about \(2\,\text{m}\) tall; \(2\,\text{mm}\) is far too small and \(2\,\text{km}\) is far too large. 2. Convert: \(2\,\text{m} = 200\,\text{cm}\).

\(2\,\text{m}\), which is \(200\,\text{cm}\).

- Express the full mass and the amount already present in the same unit. - Decide whether the missing amount should be found by adding or subtracting. - Check that the known amount plus the missing amount gives the full mass.

1. Convert the full mass to grams: \(2\,\text{kg} = 2000\,\text{g}\). 2. Subtract the amount already in the container: \(2000 - 750 = 1250\).

\(1250\,\text{g}\)

- Convert all measurements to the smallest unit shown. - One centimeter equals \(10\) millimeters. - One meter equals \(100\) centimeters. - Compare the numerical values after the units match.

1. For a), convert to millimeters: \(8\), \(80\), \(18\), \(8000\), and \(880\). Order these values from least to greatest. 2. For b), convert to millimeters: \(25\), \(250\), \(52\), \(2000\), and \(50\). Order these values from least to greatest.

a) \(8\,\text{mm} < 18\,\text{mm} < 8\,\text{cm} < 88\,\text{cm} < 8\,\text{m}\) b) \(2\,\text{cm}\ 5\,\text{mm} < 5\,\text{cm} < 52\,\text{mm} < 25\,\text{cm} < 2\,\text{m}\)

- Make a list with all lengths in the same unit. - Use everyday experience to estimate the order before calculating. - Pay attention to the difference between centimeters and millimeters.

1. Convert all lengths to millimeters: paper clip \(32\,\text{mm}\), toothpick \(65\,\text{mm}\), paintbrush \(180\,\text{mm}\), shoelace \(800\,\text{mm}\), and belt \(1050\,\text{mm}\). 2. Compare the values: \(32 < 65 < 180 < 800 < 1050\).

Paper clip, toothpick, paintbrush, shoelace, belt

- Multiply the numbers while keeping the unit meters. - Every \(1000\) meters makes \(1\) kilometer. - Split each result into thousands and the remaining meters.

1. a) \(4 \times 800\,\text{m} = 3200\,\text{m}\). Since \(3200 = 3000 + 200\), this is \(3\,\text{km}\ 200\,\text{m}\). 2. b) \(3 \times 1200\,\text{m} = 3600\,\text{m}\). Since \(3600 = 3000 + 600\), this is \(3\,\text{km}\ 600\,\text{m}\). 3. c) \(5 \times 450\,\text{m} = 2250\,\text{m}\). Since \(2250 = 2000 + 250\), this is \(2\,\text{km}\ 250\,\text{m}\).

a) \(3200\,\text{m} = 3\,\text{km}\ 200\,\text{m}\) b) \(3600\,\text{m} = 3\,\text{km}\ 600\,\text{m}\) c) \(2250\,\text{m} = 2\,\text{km}\ 250\,\text{m}\)

- Convert each kilometer measurement to meters first. - Complete each multiplication before finding the missing addend. - Check whether the equation requires subtraction or addition.

1. Use \(1\,\text{km} = 1000\,\text{m}\). 2. For a), \(1000 - 350 = 650\). 3. For b), \(2 \times 400 = 800\), then \(1000 - 800 = 200\). 4. For c), \(1000 - 720 = 280\). 5. For d), \(3 \times 200 = 600\), then \(1000 - 600 = 400\).

a) \(650\,\text{m}\) b) \(200\,\text{m}\) c) \(280\,\text{m}\) d) \(400\,\text{m}\)

- Convert both lengths to the same unit before comparing. - Write each meter measurement through the hundredths place. - Compare the ones, then tenths, then hundredths.

1. a) \(160\,\text{cm} = 1.60\,\text{m}\). Since \(1.60 > 1.06\), \(160\,\text{cm} > 1.06\,\text{m}\). 2. b) \(203\,\text{cm} = 2.03\,\text{m}\). Since \(2.03 < 2.30\), \(203\,\text{cm} < 2.30\,\text{m}\). 3. c) \(80\,\text{cm} = 0.80\,\text{m}\), so the lengths are equal. 4. d) \(405\,\text{cm} = 4.05\,\text{m}\). Since \(4.50 > 4.05\), \(4.50\,\text{m} > 405\,\text{cm}\).

a) \(>\) b) \(<\) c) \(=\) d) \(>\)

- How many centimeters are in \(1\) meter? - Convert each measurement to the requested unit before multiplying. - Keep the unit with each product.

1. Convert the meter measurements to centimeters: \(1.20\,\text{m} = 120\,\text{cm}\), \(0.75\,\text{m} = 75\,\text{cm}\), and \(2.15\,\text{m} = 215\,\text{cm}\). 2. a) \(4 \times 120\,\text{cm} = 480\,\text{cm}\). 3. b) \(5 \times 75\,\text{cm} = 375\,\text{cm}\). 4. c) \(3 \times 215\,\text{cm} = 645\,\text{cm}\).

a) \(480\,\text{cm}\) b) \(375\,\text{cm}\) c) \(645\,\text{cm}\)

- Measurements must use the same unit before you add or subtract them. - Decide whether each expression requires addition or subtraction. - Use \(1\,\text{m} = 100\,\text{cm}\).

1. Convert the meter measurements: \(2.40\,\text{m} = 240\,\text{cm}\), \(3.05\,\text{m} = 305\,\text{cm}\), and \(1.12\,\text{m} = 112\,\text{cm}\). 2. a) \(135\,\text{cm} + 240\,\text{cm} = 375\,\text{cm}\). 3. b) \(305\,\text{cm} - 80\,\text{cm} = 225\,\text{cm}\). 4. c) \(112\,\text{cm} + 88\,\text{cm} = 200\,\text{cm}\).

a) \(375\,\text{cm}\) b) \(225\,\text{cm}\) c) \(200\,\text{cm}\)

- Multiply before adding or subtracting. - Express both terms in the same unit before combining them. - Converting the meter measurements to centimeters lets you use whole-number arithmetic.

1. a) Multiply first: \(8\times25\,\text{cm}=200\,\text{cm}\). Convert \(1.75\,\text{m}=175\,\text{cm}\). Then \(200\,\text{cm}+175\,\text{cm}=375\,\text{cm}=3.75\,\text{m}\). 2. b) Convert \(1.20\,\text{m}=120\,\text{cm}\). Then \(5\times120\,\text{cm}=600\,\text{cm}\), and \(600\,\text{cm}-350\,\text{cm}=250\,\text{cm}=2.50\,\text{m}\).

a) \(3.75\,\text{m}\) b) \(2.50\,\text{m}\)

- Multiply the mass of one crate by \(5\). - Recall that \(1000\,\text{kg}\) equals \(1\) metric ton. - Recall that \(1\,\text{kg} = 1000\,\text{g}\).

1. Find the total mass in kilograms: \(5 \times 200\,\text{kg} = 1000\,\text{kg}\). 2. Since \(1000\,\text{kg} = 1\) metric ton, the total is \(1\) metric ton. 3. Each kilogram is \(1000\,\text{g}\). Converting \(1000\,\text{kg}\) gives \(1{,}000{,}000\,\text{g}\).

a) \(1000\,\text{kg}\) b) \(1\) metric ton c) \(1{,}000{,}000\,\text{g}\)

- How many seconds are in \(1\) minute? - Express both times in seconds before comparing. - Subtract to find the difference.

1. Convert Lucas's time to seconds: \(1\,\text{min} = 60\,\text{s}\), so \(60\,\text{s} + 15\,\text{s} = 75\,\text{s}\). 2. Compare the times: \(85\,\text{s} > 75\,\text{s}\), so Emma jumps longer. 3. Find the difference: \(85\,\text{s} - 75\,\text{s} = 10\,\text{s}\).

Emma jumps longer by \(10\,\text{s}\).

- How many seconds are in one minute? - Convert both lap times to the same unit before comparing them. - For a race time, what does “faster” mean about the number of seconds?

1. Convert the second-lap time to seconds: \(1\,\text{minute} = 60\,\text{seconds}\), so \(60\,\text{seconds} + 15\,\text{seconds} = 75\,\text{seconds}\). 2. Compare the times. Since \(75\,\text{seconds} < 85\,\text{seconds}\), the second lap is faster. 3. Find the difference: \(85\,\text{seconds} - 75\,\text{seconds} = 10\,\text{seconds}\).

a) The second lap takes \(75\) seconds. b) The second lap is faster by \(10\) seconds.

- Find complete groups of \(60\) minutes inside \(230\) minutes. - Use the difference from the greatest multiple of \(60\) for the leftover minutes. - To compare the two times, express them in the same unit.

1. Find the greatest multiple of \(60\) below \(230\): \(3\times60=180\) and \(4\times60=240\). So there are \(3\) full hours, with \(230-180=50\) minutes remaining. Thus \(230\) minutes is \(3\) hours \(50\) minutes. 2. Convert the second train's time to minutes: \(4\times60+10=250\) minutes. 3. Compare the times: \(230\,\text{minutes}<250\,\text{minutes}\), so the express train is faster.

a) \(230\) minutes is \(3\) hours \(50\) minutes. b) The express train is faster.

- How many minutes are in \(1\) hour? Multiply that number by the number of hours in a day. - After you find the number of minutes in a full day, how can you find half as many?

1. a) Multiply \(24\) hours by \(60\) minutes per hour. Using partial products, \(20 \times 60=1200\) and \(4 \times 60=240\). Then \(1200+240=1440\). One full day has \(1440\) minutes. 2. b) Half of \(1440\) is \(720\). Equivalently, half a day is \(12\) hours, and \(12 \times 60=720\).

a) \(1440\,\text{minutes}\) b) \(720\,\text{minutes}\)

- To convert hours to seconds, use two conversion steps: hours to minutes, then minutes to seconds. - Find the number of seconds in \(1\) hour and use that result in the later parts. - For part c), convert the hours and minutes separately, then add the results.

1. a) One hour has \(60\) minutes, and each minute has \(60\) seconds. Calculate \(60 \times 60=3600\). Therefore, \(1\) hour is \(3600\) seconds. 2. b) Multiply the number of seconds in \(1\) hour by \(5\): \(5 \times 3600=18{,}000\). Therefore, \(5\) hours is \(18{,}000\) seconds. 3. c) Convert each part separately. Two hours is \(2 \times 3600=7200\) seconds, and \(15\) minutes is \(15 \times 60=900\) seconds. Then \(7200+900=8100\).

a) \(3600\,\text{seconds}\) b) \(18{,}000\,\text{seconds}\) c) \(8100\,\text{seconds}\)

- Check whether the measurements use the same unit before adding or subtracting. - Recall how many smaller units make one larger unit. - What operations do “more than” and “less than” indicate? - Converting everything to the smaller unit can make the arithmetic easier.

1. For parts a) and b), convert \(10\,\text{km}=10{,}000\,\text{m}\). 2. a) \(10{,}000\,\text{m}+400\,\text{m}=10{,}400\,\text{m}=10.4\,\text{km}\). 3. b) \(10{,}000\,\text{m}-400\,\text{m}=9600\,\text{m}=9.6\,\text{km}\). 4. For parts c) and d), convert \(5\,\text{m}=500\,\text{cm}\). 5. c) \(500\,\text{cm}+80\,\text{cm}=580\,\text{cm}=5.8\,\text{m}\). 6. d) \(500\,\text{cm}-80\,\text{cm}=420\,\text{cm}=4.2\,\text{m}\).

a) \(10.4\,\text{km}\), or \(10{,}400\,\text{m}\) b) \(9.6\,\text{km}\), or \(9600\,\text{m}\) c) \(5.8\,\text{m}\), or \(580\,\text{cm}\) d) \(4.2\,\text{m}\), or \(420\,\text{cm}\)

- How many seconds are in \(1\) minute? - How many pounds are in \(1\) ton? - Convert to the smaller unit before calculating. - What operations do “more than” and “less than” indicate? - How many pounds are in half a ton?

1. Use \(1\,\text{min}=60\,\text{s}\) and \(1\,\text{ton}=2000\,\text{lb}\). 2. a) \(3\,\text{min}=180\,\text{s}\). Then \(180\,\text{s}+15\,\text{s}=195\,\text{s}=3\,\text{min}\ 15\,\text{s}\). 3. b) \(180\,\text{s}-15\,\text{s}=165\,\text{s}=2\,\text{min}\ 45\,\text{s}\). 4. c) Half a ton is \(1000\,\text{lb}\). Then \(1000\,\text{lb}+75\,\text{lb}=1075\,\text{lb}\). 5. d) \(1000\,\text{lb}-75\,\text{lb}=925\,\text{lb}\).

a) \(195\,\text{s}\), or \(3\,\text{min}\ 15\,\text{s}\) b) \(165\,\text{s}\), or \(2\,\text{min}\ 45\,\text{s}\) c) \(1075\,\text{lb}\) d) \(925\,\text{lb}\)

- First calculate each total in the smaller unit. - Use \(1000\,\text{g} = 1\,\text{kg}\), \(1000\,\text{mL} = 1\,\text{L}\), and \(100\,\text{cm} = 1\,\text{m}\). - Separate each total into the greatest possible number of larger units and the remainder.

1. a) \(5 \times 300\,\text{g} = 1500\,\text{g} = 1\,\text{kg}\ 500\,\text{g}\). 2. b) \(8 \times 125\,\text{g} = 1000\,\text{g} = 1\,\text{kg}\ 0\,\text{g}\). 3. c) \(4 \times 450\,\text{mL} = 1800\,\text{mL} = 1\,\text{L}\ 800\,\text{mL}\). 4. d) \(6 \times 50\,\text{cm} = 300\,\text{cm} = 3\,\text{m}\ 0\,\text{cm}\).

a) \(1\,\text{kg}\ 500\,\text{g}\) b) \(1\,\text{kg}\ 0\,\text{g}\) c) \(1\,\text{L}\ 800\,\text{mL}\) d) \(3\,\text{m}\ 0\,\text{cm}\)

- For parts b) and c), convert both amounts to the smaller unit first. - For the money amount, you can count up to the next whole dollar and continue from there. - Include the correct unit with each answer.

1. a) Subtract the money amounts: \(\$20.00-\$12.50=\$7.50\). 2. b) Convert the target distance: \(2\,\text{km}=2000\,\text{m}\). Then \(2000\,\text{m}-850\,\text{m}=1150\,\text{m}\), or \(1\,\text{km}\ 150\,\text{m}\). 3. c) Convert both capacities to milliliters: \(3\,\text{L}=3000\,\text{mL}\), and \(1\,\text{L}\ 200\,\text{mL}=1200\,\text{mL}\). Then \(3000\,\text{mL}-1200\,\text{mL}=1800\,\text{mL}\), or \(1\,\text{L}\ 800\,\text{mL}\).

a) \(\$7.50\) b) \(1150\,\text{m}\), or \(1\,\text{km}\ 150\,\text{m}\) c) \(1800\,\text{mL}\), or \(1\,\text{L}\ 800\,\text{mL}\)

- Use \(1\,\text{m} = 100\,\text{cm}\). - Convert the fractional and decimal meter measurements to centimeters. - Put all measurements in one unit before sorting them.

1. Since \(1\,\text{m} = 100\,\text{cm}\), \(\frac{1}{2}\,\text{m} = 0.50\,\text{m} = 50\,\text{cm}\). 2. \(\frac{1}{10}\,\text{m} = 0.10\,\text{m} = 10\,\text{cm}\). 3. \(\frac{1}{5}\,\text{m} = 0.20\,\text{m} = 20\,\text{cm}\).

Group 1: \(\frac{1}{2}\,\text{m} = 0.50\,\text{m} = 50\,\text{cm}\) Group 2: \(\frac{1}{10}\,\text{m} = 0.10\,\text{m} = 10\,\text{cm}\) Group 3: \(\frac{1}{5}\,\text{m} = 0.20\,\text{m} = 20\,\text{cm}\)

- Try starting with the largest available weight. - For \(300\,\text{g}\), look for one combination with two weights and another with more weights. - Remember how many grams are in \(1\) kilogram. - In part b, use each weight no more times than it appears in the list.

1. Two combinations for \(300\,\text{g}\) are \(200\,\text{g} + 100\,\text{g} = 300\,\text{g}\) and \(200\,\text{g} + 50\,\text{g} + 50\,\text{g} = 300\,\text{g}\). 2. One combination for \(850\,\text{g}\) is \(500\,\text{g} + 200\,\text{g} + 100\,\text{g} + 50\,\text{g} = 850\,\text{g}\). 3. The total of all six weights is \(500\,\text{g} + 200\,\text{g} + 100\,\text{g} + 100\,\text{g} + 50\,\text{g} + 50\,\text{g} = 1000\,\text{g}\). Since \(1000\,\text{g} = 1\,\text{kg}\), they weigh \(1\,\text{kg}\) altogether.

a) For example, \(200\,\text{g} + 100\,\text{g}\) and \(200\,\text{g} + 50\,\text{g} + 50\,\text{g}\) b) \(500\,\text{g}\), \(200\,\text{g}\), \(100\,\text{g}\), and \(50\,\text{g}\) c) \(1\,\text{kg}\)

- Break \(387\) into hundreds, tens, and ones to find suitable weights. - Look closely at the ones digit of \(387\). - For part b, add all the weights carefully. - How many grams would the total need to equal \(1\,\text{kg}\)?

1. Decompose \(387\,\text{g}\) as \(300\,\text{g} + 80\,\text{g} + 7\,\text{g}\). Use \(200\,\text{g} + 100\,\text{g}\), \(50\,\text{g} + 20\,\text{g} + 10\,\text{g}\), and \(5\,\text{g} + 2\,\text{g}\). 2. Add all ten weights: \(1 + 2 + 2 + 5 + 10 + 20 + 50 + 100 + 200 + 500 = 890\), so the total is \(890\,\text{g}\). 3. Since \(1\,\text{kg} = 1000\,\text{g}\) and \(890\,\text{g} < 1000\,\text{g}\), Lucas is not correct. The set is \(110\,\text{g}\) short of \(1\,\text{kg}\).

a) Use \(200\,\text{g}\), \(100\,\text{g}\), \(50\,\text{g}\), \(20\,\text{g}\), \(10\,\text{g}\), \(5\,\text{g}\), and one \(2\,\text{g}\) weight. b) No. All ten weights total \(890\,\text{g}\), which is \(110\,\text{g}\) less than \(1\,\text{kg}\).

- Convert every weight to pounds. - One ton equals \(2000\) pounds. - Pay attention to the difference between \(1\) ton \(100\) pounds and \(3000\) pounds.

1. Convert every weight to pounds using \(1\,\text{ton} = 2000\,\text{lb}\). 2. \(2\,\text{tons} = 4000\,\text{lb}\). 3. \(1\,\text{ton}\ 100\,\text{lb} = 2100\,\text{lb}\). 4. \(2\,\text{tons}\ 5\,\text{lb} = 4005\,\text{lb}\). 5. The pound values are \(1800 < 2100 < 3000 < 4000 < 4005\), so place the original measurements in that order.

\(1800\,\text{lb} < 1\,\text{ton}\ 100\,\text{lb} < 3000\,\text{lb} < 2\,\text{tons} < 2\,\text{tons}\ 5\,\text{lb}\)

- First convert both measurements to the same smaller unit. - Divide to determine how many equal parts fit into the whole. - For numbers ending in zeros, use place value to simplify the division.

1. a) Convert \(1\,\text{m}\) to \(100\,\text{cm}\). Then \(100 \div 20=5\), so \(20\,\text{cm}\) fits \(5\) times. 2. b) Convert \(1\,\text{kg}\) to \(1000\,\text{g}\). Then \(1000 \div 250=4\), so \(250\,\text{g}\) fits \(4\) times. 3. c) Convert \(1\,\text{km}\) to \(1000\,\text{m}\). Then \(1000 \div 50=20\), so \(50\,\text{m}\) fits \(20\) times.

a) \(5\) times b) \(4\) times c) \(20\) times

- One kilogram equals \(1000\) grams. - Check whether the kilograms match the thousands in the gram measurement. - Convert each mixed-unit result back to grams as a check.

1. Conversion 2 is incorrect. Since \(1000\,\text{g} = 1\,\text{kg}\), \(2500\,\text{g} = 2\,\text{kg}\ 500\,\text{g}\), not \(25\,\text{kg}\). 2. Conversion 4 is incorrect. Since \(406\,\text{g}\) is less than \(1000\,\text{g}\), it is \(0\,\text{kg}\ 406\,\text{g}\), not \(4\,\text{kg}\ 6\,\text{g}\). 3. Conversions 1 and 3 are correct.

The incorrect conversions are 2 and 4. 2. \(2500\,\text{g} = 2\,\text{kg}\ 500\,\text{g}\) 4. \(406\,\text{g} = 0\,\text{kg}\ 406\,\text{g}\)

- Convert all lengths to meters before ordering them. - One kilometer equals \(1000\) meters. - Pay special attention to the zeros in \(4\) kilometers \(5\) meters.

1. Convert every length to meters. 2. \(4\,\text{km}\ 5\,\text{m} = 4005\,\text{m}\). 3. \(4\,\text{km}\ 500\,\text{m} = 4500\,\text{m}\). 4. The meter values are \(450 < 4005 < 4050 < 4500\). 5. Write the original measurements in that order.

\(450\,\text{m} < 4\,\text{km}\ 5\,\text{m} < 4050\,\text{m} < 4\,\text{km}\ 500\,\text{m}\)

- How many ounces are in \(1\,\text{lb}\)? - How many pounds are in \(1\,\text{ton}\)? - Convert all measurements to the same unit. - Then compare the numerical values.

1. Convert the weights to pounds: \(16\,\text{oz}=1\,\text{lb}\) \(160\,\text{oz}=10\,\text{lb}\) \(30\,\text{lb}=30\,\text{lb}\) \(300\,\text{lb}=300\,\text{lb}\) \(3\,\text{tons}=6000\,\text{lb}\) 2. Compare the values: \(1<10<30<300<6000\). 3. Write the original measurements in order: \(16\,\text{oz}<160\,\text{oz}<30\,\text{lb}<300\,\text{lb}<3\,\text{tons}\).

\(16\,\text{oz}\), \(160\,\text{oz}\), \(30\,\text{lb}\), \(300\,\text{lb}\), \(3\,\text{tons}\)

- Convert mixed units to grams before calculating. - One kilogram equals \(1000\) grams. - Convert the final gram total back to kilograms and grams.

1. a) \(4650\,\text{g} + 2350\,\text{g} = 7000\,\text{g} = 7\,\text{kg}\ 0\,\text{g}\). 2. b) Convert \(10\,\text{kg}\) to \(10{,}000\,\text{g}\). Then \(10{,}000\,\text{g} - 4200\,\text{g} = 5800\,\text{g} = 5\,\text{kg}\ 800\,\text{g}\). 3. c) \(6 \times 500\,\text{g} = 3000\,\text{g} = 3\,\text{kg}\ 0\,\text{g}\). 4. d) Convert \(1\,\text{kg}\) to \(1000\,\text{g}\). Then \(1000\,\text{g} \div 8 = 125\,\text{g} = 0\,\text{kg}\ 125\,\text{g}\).

a) \(7\,\text{kg}\ 0\,\text{g}\) b) \(5\,\text{kg}\ 800\,\text{g}\) c) \(3\,\text{kg}\ 0\,\text{g}\) d) \(0\,\text{kg}\ 125\,\text{g}\)

- Convert all the weights to the same unit before ordering them. - Compare the converted numbers by place value. - Recall the conversion factors between pounds and ounces and between tons and pounds. - Subtract each given amount from its target amount.

1. Convert the weights in part a) to pounds: \(4\,\text{tons}=8000\,\text{lb}\), \(400\,\text{lb}=400\,\text{lb}\), \(400\,\text{oz}=25\,\text{lb}\), and \(40\,\text{lb}=40\,\text{lb}\). 2. Order the values: \(25<40<400<8000\). Therefore, \(400\,\text{oz}<40\,\text{lb}<400\,\text{lb}<4\,\text{tons}\). 3. Since \(1\,\text{lb}=16\,\text{oz}\), \(16-12=4\). The first missing amount is \(4\,\text{oz}\). 4. Since \(1\,\text{ton}=2000\,\text{lb}\), \(2000-1300=700\). The second missing amount is \(700\,\text{lb}\).

a) \(400\,\text{oz}<40\,\text{lb}<400\,\text{lb}<4\,\text{tons}\) b) \(4\,\text{oz}\) and \(700\,\text{lb}\)

- One ton equals \(2000\) pounds. - Convert the whole tons to pounds first. - Subtract the pounds in the whole tons from the total.

1. Use \(1\,\text{ton} = 2000\,\text{lb}\). 2. a) Four tons is \(8000\,\text{lb}\), leaving \(8025 - 8000 = 25\,\text{lb}\). 3. b) \(14{,}105\,\text{lb}\) contains \(14{,}000\,\text{lb} = 7\) tons and \(105\,\text{lb}\). 4. c) Fifteen tons is \(30{,}000\,\text{lb}\), leaving \(8\,\text{lb}\).

a) \(25\) b) \(7\) c) \(8\)

- Convert both weights in each statement to the same unit. - How many pounds are in half a ton? - Pay close attention to comparison words such as “heavier,” “lighter,” and “the same weight.”

1. a) Since \(48\,\text{oz}=3\,\text{lb}\), the statement is false. The weights are equal. 2. b) Since \(1\,\text{ton}=2000\,\text{lb}\), \(5 \times 2000=10{,}000\). The statement is true. 3. c) \(192\,\text{oz}=12\,\text{lb}\). Since \(12<100\), the statement is true. 4. d) Half a ton is \(1000\,\text{lb}\). Since \(400<1000\), the statement is false. \(400\,\text{lb}\) is lighter than half a ton.

a) False; \(48\,\text{oz}=3\,\text{lb}\). b) True c) True d) False; \(400\,\text{lb}\) is lighter than half a ton.

- Convert both sides to the same unit. - One ton equals \(2000\) pounds, and one pound equals \(16\) ounces. - Use subtraction for a missing addend or subtrahend, and a related multiplication fact for a whole-number conversion.

1. Use \(1\,\text{ton}=2000\,\text{lb}\) and \(1\,\text{lb}=16\,\text{oz}\). 2. a) \(2000\,\text{lb}-900\,\text{lb}=1100\,\text{lb}\). 3. b) \(12\,\text{lb}=192\,\text{oz}\), so \(192\,\text{oz}-176\,\text{oz}=16\,\text{oz}\). 4. c) \(117\,\text{oz}\) contains \(112\,\text{oz}=7\,\text{lb}\), with \(5\,\text{oz}\) remaining. 5. d) \(2\,\text{tons}=4000\,\text{lb}\), and \(4000\,\text{lb}-3000\,\text{lb}=1000\,\text{lb}\). 6. e) Since \(40\times16=640\), \(640\,\text{oz}=40\,\text{lb}\).

a) \(1100\,\text{lb}\) b) \(16\,\text{oz}\) c) \(7\,\text{lb}\ 5\,\text{oz}\) d) \(1000\,\text{lb}\) e) \(40\,\text{lb}\)

- Convert both quantities to years before comparing them. - The phrase “how many are needed” indicates that you should find a difference. - One century is \(100\) years.

1. a) Convert \(8\) centuries to years: \(8 \times 100=800\) years. Then \(800-650=150\), so \(150\) years are needed. 2. b) Convert \(14\) centuries to years: \(14 \times 100=1400\) years. Then \(1400-1050=350\), so \(350\) years are needed.

a) \(150\,\text{years}\) b) \(350\,\text{years}\)

- One year has \(12\) months. - You can often calculate the years and months separately. - If an addition produces at least \(12\) months, regroup \(12\) months as \(1\) year. - If there are not enough months to subtract, regroup \(1\) year as \(12\) months.

1. a) Add years and months separately: \(6+4=10\) years and \(5+3=8\) months. The result is \(10\) years \(8\) months. 2. b) \(8+2=10\) years and \(9+11=20\) months. Since \(20\) months is \(1\) year \(8\) months, the result is \(11\) years \(8\) months. 3. c) Regroup \(1\) year as \(12\) months: \(15\) years \(2\) months becomes \(14\) years \(14\) months. Then \(14-6=8\) years and \(14-7=7\) months, so the result is \(8\) years \(7\) months. 4. d) One year has \(12\) months, and \(12-8=4\). Four months remain.

a) \(10\,\text{years}\ 8\,\text{months}\) b) \(11\,\text{years}\ 8\,\text{months}\) c) \(8\,\text{years}\ 7\,\text{months}\) d) \(4\,\text{months}\)

- How many minutes are in \(1\) hour? - When subtracting, regroup \(1\) hour as \(60\) minutes if needed. - Calculate the hours and minutes separately. - At the end, regroup any set of \(60\) minutes as \(1\) hour.

1. a) Add the hours and minutes: \(4+2=6\) hours and \(35+45=80\) minutes. Since \(80\) minutes is \(1\) hour \(20\) minutes, the result is \(7\) hours \(20\) minutes. 2. b) Regroup \(1\) hour as \(60\) minutes: \(15\) hours \(12\) minutes becomes \(14\) hours \(72\) minutes. Then \(14-6=8\) hours and \(72-45=27\) minutes. The result is \(8\) hours \(27\) minutes. 3. c) Add \(9+4=13\) hours and \(58+7=65\) minutes. Since \(65\) minutes is \(1\) hour \(5\) minutes, the result is \(14\) hours \(5\) minutes.

a) \(7\,\text{hr}\ 20\,\text{min}\) b) \(8\,\text{hr}\ 27\,\text{min}\) c) \(14\,\text{hr}\ 5\,\text{min}\)

- Calculate \(A\) and \(B\) separately. - Remember that \(1\,\text{hr}=60\,\text{min}\). - Subtract the shorter time span from the longer time span.

1. Calculate \(A\): \(5+3=8\) hours, and \(45+25=70\) minutes \(=1\) hour \(10\) minutes. Therefore, \(A=9\,\text{hr}\ 10\,\text{min}\). 2. Calculate \(B\): Regroup \(12\,\text{hr}\ 10\,\text{min}\) as \(11\,\text{hr}\ 70\,\text{min}\). Then \(11-2=9\) hours and \(70-50=20\) minutes, so \(B=9\,\text{hr}\ 20\,\text{min}\). 3. Compare and subtract: \(B\) is longer, and \(9\,\text{hr}\ 20\,\text{min}-9\,\text{hr}\ 10\,\text{min}=10\,\text{min}\).

Time span \(B\) is longer by \(10\,\text{min}\).

- Add like units separately. - How many minutes make \(1\) hour? - How many hours make \(1\) day? - Regroup any total of at least \(60\) minutes or \(24\) hours.

1. a) Add \(4+3=7\) hours and \(40+35=75\) minutes. Since \(75\) minutes is \(1\) hour \(15\) minutes, the result is \(8\,\text{hr}\ 15\,\text{min}\). 2. b) Add \(12+11=23\) hours and \(55+25=80\) minutes. Since \(80\) minutes is \(1\) hour \(20\) minutes, this becomes \(24\) hours \(20\) minutes. Since \(24\) hours is \(1\) day, the result is \(1\,\text{day}\ 20\,\text{min}\). 3. c) Add \(2+1=3\) days and \(15+18=33\) hours. Since \(33\) hours is \(1\) day \(9\) hours, the result is \(4\,\text{days}\ 9\,\text{hr}\).

a) \(8\,\text{hr}\ 15\,\text{min}\) b) \(1\,\text{day}\ 20\,\text{min}\) c) \(4\,\text{days}\ 9\,\text{hr}\)

- How many months are in \(1\) year? - If there are not enough months to subtract, regroup \(1\) year as \(12\) months. - Rewrite each time span in a form that makes subtraction possible.

1. a) Subtract like units: \(9-3=6\) years and \(8-5=3\) months. The result is \(6\) years \(3\) months. 2. b) Regroup \(1\) year as \(12\) months: \(14\) years \(2\) months becomes \(13\) years \(14\) months. Then \(13-6=7\) years and \(14-11=3\) months. The result is \(7\) years \(3\) months. 3. c) Rewrite \(7\) years as \(6\) years \(12\) months. Then \(6-4=2\) years and \(12-9=3\) months. The result is \(2\) years \(3\) months.

a) \(6\,\text{years}\ 3\,\text{months}\) b) \(7\,\text{years}\ 3\,\text{months}\) c) \(2\,\text{years}\ 3\,\text{months}\)

- Calculate each side separately. - Regroup any total of at least \(60\) seconds. - Compare the minutes first, then the seconds.

1. Calculate \(A\): \(9+4=13\) minutes, and \(35+40=75\) seconds \(=1\) minute \(15\) seconds. Therefore, \(A=14\,\text{min}\ 15\,\text{s}\). 2. Calculate \(B\): \(6+7=13\) minutes, and \(55+15=70\) seconds \(=1\) minute \(10\) seconds. Therefore, \(B=14\,\text{min}\ 10\,\text{s}\). 3. Since \(14\,\text{min}\ 15\,\text{s}>14\,\text{min}\ 10\,\text{s}\), \(A>B\).

\(A>B\), because \(14\,\text{min}\ 15\,\text{s}>14\,\text{min}\ 10\,\text{s}\).

- What can you regroup when the smaller-unit amount is not large enough to subtract? - How many minutes are in \(1\) hour? - How many hours are in \(1\) day? - How many seconds are in \(1\) minute? - Subtract like units after regrouping.

1. a) Regroup \(14\,\text{hr}\ 20\,\text{min}\) as \(13\,\text{hr}\ 80\,\text{min}\). Then subtract to get \(7\,\text{hr}\ 35\,\text{min}\). 2. b) Regroup \(5\,\text{days}\ 6\,\text{hr}\) as \(4\,\text{days}\ 30\,\text{hr}\). Then subtract to get \(2\,\text{days}\ 15\,\text{hr}\). 3. c) Regroup \(8\,\text{min}\ 12\,\text{s}\) as \(7\,\text{min}\ 72\,\text{s}\). Then subtract to get \(4\,\text{min}\ 32\,\text{s}\).

a) \(7\,\text{hr}\ 35\,\text{min}\) b) \(2\,\text{days}\ 15\,\text{hr}\) c) \(4\,\text{min}\ 32\,\text{s}\)

- Can an inverse operation help you find the missing value? - Decide whether to add or subtract to fill each blank. - Remember that \(1\) day is \(24\) hours and \(1\) minute is \(60\) seconds.

1. a) Find the missing subtrahend by subtracting: \(10\,\text{hr}\ 15\,\text{min}-4\,\text{hr}\ 50\,\text{min}\). Regroup as \(9\,\text{hr}\ 75\,\text{min}-4\,\text{hr}\ 50\,\text{min}=5\,\text{hr}\ 25\,\text{min}\). 2. b) Find the missing minuend by adding: \(3\,\text{days}\ 12\,\text{hr}+1\,\text{day}\ 18\,\text{hr}=4\,\text{days}\ 30\,\text{hr}=5\,\text{days}\ 6\,\text{hr}\). 3. c) Find the missing subtrahend by subtracting: \(20\,\text{min}-12\,\text{min}\ 25\,\text{s}\). Regroup as \(19\,\text{min}\ 60\,\text{s}-12\,\text{min}\ 25\,\text{s}=7\,\text{min}\ 35\,\text{s}\).

a) \(5\,\text{hr}\ 25\,\text{min}\) b) \(5\,\text{days}\ 6\,\text{hr}\) c) \(7\,\text{min}\ 35\,\text{s}\)

- Use an inverse operation to fill each blank. - Count up to the next full hour when that helps. - One hour is \(60\) minutes.

1. a) Subtract to find the missing addend: \(6\,\text{hr}\ 10\,\text{min}-3\,\text{hr}\ 40\,\text{min}=5\,\text{hr}\ 70\,\text{min}-3\,\text{hr}\ 40\,\text{min}=2\,\text{hr}\ 30\,\text{min}\). 2. b) Subtract the difference from the minuend: \(7\,\text{hr}\ 25\,\text{min}-4\,\text{hr}\ 50\,\text{min}=6\,\text{hr}\ 85\,\text{min}-4\,\text{hr}\ 50\,\text{min}=2\,\text{hr}\ 35\,\text{min}\). 3. c) Subtract: \(11\,\text{hr}-8\,\text{hr}\ 12\,\text{min}=10\,\text{hr}\ 60\,\text{min}-8\,\text{hr}\ 12\,\text{min}=2\,\text{hr}\ 48\,\text{min}\). 4. d) Subtract to find the missing addend: \(5\,\text{hr}\ 20\,\text{min}-2\,\text{hr}\ 55\,\text{min}=4\,\text{hr}\ 80\,\text{min}-2\,\text{hr}\ 55\,\text{min}=2\,\text{hr}\ 25\,\text{min}\).

a) \(2\,\text{hr}\ 30\,\text{min}\) b) \(2\,\text{hr}\ 35\,\text{min}\) c) \(2\,\text{hr}\ 48\,\text{min}\) d) \(2\,\text{hr}\ 25\,\text{min}\)

- How many seconds are in \(1\) minute? - If the top number of seconds is smaller, regroup \(1\) minute as \(60\) seconds. - Subtract the minutes and seconds after regrouping.

1. Regroup \(8\,\text{min}\ 15\,\text{s}\) as \(7\,\text{min}\ 75\,\text{s}\). Then \(7\,\text{min}\ 75\,\text{s}-3\,\text{min}\ 45\,\text{s}=4\,\text{min}\ 30\,\text{s}\). 2. Regroup \(20\,\text{min}\ 5\,\text{s}\) as \(19\,\text{min}\ 65\,\text{s}\). Then \(19\,\text{min}\ 65\,\text{s}-12\,\text{min}\ 18\,\text{s}=7\,\text{min}\ 47\,\text{s}\). 3. Regroup \(5\,\text{min}\) as \(4\,\text{min}\ 60\,\text{s}\). Then \(4\,\text{min}\ 60\,\text{s}-3\,\text{min}\ 52\,\text{s}=1\,\text{min}\ 8\,\text{s}\).

1. \(4\,\text{min}\ 30\,\text{s}\) 2. \(7\,\text{min}\ 47\,\text{s}\) 3. \(1\,\text{min}\ 8\,\text{s}\)

- Calculate the subtraction on the left side first. - Convert both time amounts to the same unit before comparing. - How many seconds are in \(5\) or \(6\) minutes?

1. a) Regroup and subtract: \(3\,\text{min}\ 70\,\text{s}-1\,\text{min}\ 50\,\text{s}=2\,\text{min}\ 20\,\text{s}\). Therefore, the two sides are equal. 2. b) The left side is \(9\,\text{min}\ 60\,\text{s}-4\,\text{min}\ 15\,\text{s}=5\,\text{min}\ 45\,\text{s}\). 3. Convert the right side: \(350\,\text{s}=5\,\text{min}\ 50\,\text{s}\), because \(5 \times 60=300\). 4. Since \(5\,\text{min}\ 45\,\text{s}<5\,\text{min}\ 50\,\text{s}\), the left side is less.

a) \(=\) b) \(<\)

- How many hours are in \(1\) day? - Convert to smaller units when that makes subtraction easier. - You can count up from the shorter time to the longer time as a check.

1. a) Add \(18+9=27\) hours. Since \(27\) hours is \(1\) day \(3\) hours, add \(4+2+1=7\) days. The result is \(7\,\text{days}\ 3\,\text{hr}\). 2. b) Convert \(1\) hour \(5\) minutes to \(65\) minutes. Regroup as \(64\,\text{min}\ 60\,\text{s}\). Then subtract \(35\,\text{min}\ 20\,\text{s}\) to get \(29\,\text{min}\ 40\,\text{s}\).

a) \(7\,\text{days}\ 3\,\text{hr}\) b) \(29\,\text{min}\ 40\,\text{s}\)

- Convert both quantities to grams. - Find each fraction of its whole amount. - Compare the two results.

1. Find half of \(600\,\text{g}\): \(600\,\text{g} \div 2 = 300\,\text{g}\). 2. Convert \(1\,\text{kg}\) to \(1000\,\text{g}\). 3. Find one-fourth of \(1000\,\text{g}\): \(1000\,\text{g} \div 4 = 250\,\text{g}\). 4. Since \(300\,\text{g} > 250\,\text{g}\), half of \(600\,\text{g}\) is heavier.

\(\frac{1}{2}\) of \(600\,\text{g}\) is heavier because \(300\,\text{g} > 250\,\text{g}\).

- Convert each expression to the unit on the right. - Use place value to separate complete larger units from the remainder. - Use \(1\,\text{ton} = 2000\,\text{lb}\) for part d).

1. a) Six meters equals \(600\,\text{cm}\), so the missing amount is \(609 - 600 = 9\,\text{cm}\). 2. b) \(5012\,\text{m}\) contains \(5000\,\text{m} = 5\,\text{km}\) and \(12\,\text{m}\), so the missing number is \(5\). 3. c) \(4\,\text{kg}\ 5\,\text{g} = 4000\,\text{g} + 5\,\text{g} = 4005\,\text{g}\). 4. d) \(3\,\text{tons} = 6000\,\text{lb}\). Adding \(20\,\text{lb}\) gives \(6020\,\text{lb}\).

a) \(9\) b) \(5\) c) \(4005\) d) \(6020\)

- How many cents equal one dollar? - First find how many cents are in \(\$15\). - Then add the \(6\) cents and compare your result with \(156\).

1. One dollar equals \(100\) cents. 2. Convert the whole dollars: \(15 \times 100 = 1500\) cents. 3. Add the remaining \(6\) cents: \(1500 + 6 = 1506\) cents. 4. Since \(1506 \ne 156\), Lucas is not correct. He likely treated \(\$15\) as \(150\) cents or left out a zero when writing the amount.

Lucas is not correct. \(\$15.06\) equals \(1506\) cents. He probably forgot that \(\$15\) alone equals \(1500\) cents.

- Add the kilometers and meters separately. - How many meters are in \(1\,\text{km}\)? - Check whether the meter total can be regrouped as kilometers and meters.

1. Add the kilometers: \(26+41+13=80\,\text{km}\). 2. Add the meters: \(380+920+705=2005\,\text{m}\). 3. Regroup \(2005\,\text{m}=2\,\text{km}\ 5\,\text{m}\). 4. Add the results: \(80\,\text{km}+2\,\text{km}\ 5\,\text{m}=82\,\text{km}\ 5\,\text{m}\).

\(82\,\text{km}\ 5\,\text{m}\)

- Use \(1\,\text{L}=1000\,\text{mL}\), \(1\,\text{m}=100\,\text{cm}\), and \(1\,\text{cm}=10\,\text{mm}\). - Regroup one larger unit when the smaller-unit amount is not large enough to subtract. - You can check by converting everything to the smallest unit first.

1. a) Regroup \(10\,\text{L}\ 200\,\text{mL}\) as \(9\,\text{L}\ 1200\,\text{mL}\). Then subtract to get \(5\,\text{L}\ 350\,\text{mL}\). 2. b) Regroup \(6\,\text{m}\ 12\,\text{cm}\) as \(5\,\text{m}\ 112\,\text{cm}\). Then subtract to get \(3\,\text{m}\ 67\,\text{cm}\). 3. c) Regroup \(14\,\text{cm}\ 3\,\text{mm}\) as \(13\,\text{cm}\ 13\,\text{mm}\). Then subtract to get \(5\,\text{cm}\ 4\,\text{mm}\).

a) \(5\,\text{L}\ 350\,\text{mL}\) b) \(3\,\text{m}\ 67\,\text{cm}\) c) \(5\,\text{cm}\ 4\,\text{mm}\)

- Calculate the expression on the left before comparing. - Regroup or convert to the smaller unit when needed. - Estimate whether each difference is a little more or less than a whole pound or ton. - Compare both sides using the same unit.

1. a) Convert to ounces: \(197\,\text{oz}-71\,\text{oz}=126\,\text{oz}=7\,\text{lb}\ 14\,\text{oz}\). Since this is less than \(8\,\text{lb}\), use \(<\). 2. b) Convert to pounds: \(10{,}020\,\text{lb}-2030\,\text{lb}=7990\,\text{lb}\). Also, \(3\,\text{tons}\ 1990\,\text{lb}=7990\,\text{lb}\), so use \(=\). 3. c) Convert to ounces: \(132\,\text{oz}-56\,\text{oz}=76\,\text{oz}=4\,\text{lb}\ 12\,\text{oz}\). Since \(4\,\text{lb}\ 12\,\text{oz}>4\,\text{lb}\ 8\,\text{oz}\), use \(>\).

a) \(<\) b) \(=\) c) \(>\)

- Convert everything to the smallest unit shown. - How many centimeters are in \(1\,\text{m}\)? - How many millimeters are in \(1\,\text{cm}\)? - Convert the final difference back to mixed units.

1. a) \(500\,\text{cm}-72\,\text{cm}=428\,\text{cm}=4\,\text{m}\ 28\,\text{cm}\). 2. b) \(1215\,\text{cm}-840\,\text{cm}=375\,\text{cm}=3\,\text{m}\ 75\,\text{cm}\). 3. c) \(32\,\text{mm}-18\,\text{mm}=14\,\text{mm}=1\,\text{cm}\ 4\,\text{mm}\). 4. d) \(10{,}000\,\text{cm}-4505\,\text{cm}=5495\,\text{cm}=54\,\text{m}\ 95\,\text{cm}\).

a) \(4\,\text{m}\ 28\,\text{cm}\) b) \(3\,\text{m}\ 75\,\text{cm}\) c) \(1\,\text{cm}\ 4\,\text{mm}\) d) \(54\,\text{m}\ 95\,\text{cm}\)

- How many ounces are in \(1\,\text{lb}\)? - How many pounds are in \(1\,\text{ton}\)? - Pay close attention to place value in mixed-unit weights. - Convert both weights to the same smaller unit before subtracting.

1. a) \(16\,\text{oz}-6\,\text{oz}=10\,\text{oz}\). 2. b) \(8000\,\text{lb}-2500\,\text{lb}=5500\,\text{lb}=2\,\text{tons}\ 1500\,\text{lb}\). 3. c) \(165\,\text{oz}-56\,\text{oz}=109\,\text{oz}=6\,\text{lb}\ 13\,\text{oz}\). 4. d) \(5000\,\text{lb}-1600\,\text{lb}=3400\,\text{lb}=1\,\text{ton}\ 1400\,\text{lb}\).

a) \(10\,\text{oz}\) b) \(2\,\text{tons}\ 1500\,\text{lb}\) c) \(6\,\text{lb}\ 13\,\text{oz}\) d) \(1\,\text{ton}\ 1400\,\text{lb}\)

- Recall how many inches are in a foot and how many feet are in a yard. - Regroup when a smaller-unit amount is not large enough to subtract. - Converting each full length to inches can simplify the subtraction.

1. a) Convert to inches: \(7\,\text{yd}\ 1\,\text{ft}\ 2\,\text{in.}=266\,\text{in.}\), and \(3\,\text{yd}\ 2\,\text{ft}\ 5\,\text{in.}=137\,\text{in.}\). Then \(266-137=129\,\text{in.}=3\,\text{yd}\ 1\,\text{ft}\ 9\,\text{in.}\). 2. b) Convert to inches: \(15\,\text{yd}\ 1\,\text{ft}\ 6\,\text{in.}=558\,\text{in.}\), and \(6\,\text{yd}\ 2\,\text{ft}\ 9\,\text{in.}=249\,\text{in.}\). Then \(558-249=309\,\text{in.}=8\,\text{yd}\ 1\,\text{ft}\ 9\,\text{in.}\).

a) \(3\,\text{yd}\ 1\,\text{ft}\ 9\,\text{in.}\) b) \(8\,\text{yd}\ 1\,\text{ft}\ 9\,\text{in.}\)

- Use \(1\,\text{lb}=16\,\text{oz}\) and \(1\,\text{gal}=4\,\text{qt}\). - For part c), convert both lengths to inches before subtracting.

1. a) Convert to ounces: \(197\,\text{oz}-134\,\text{oz}=63\,\text{oz}=3\,\text{lb}\ 15\,\text{oz}\). 2. b) Regroup \(14\,\text{gal}\ 2\,\text{qt}\) as \(13\,\text{gal}\ 6\,\text{qt}\). Then subtract to get \(8\,\text{gal}\ 3\,\text{qt}\). 3. c) Convert to inches: \(20\,\text{yd}=720\,\text{in.}\), and \(12\,\text{yd}\ 1\,\text{ft}\ 8\,\text{in.}=452\,\text{in.}\). Then \(720-452=268\,\text{in.}=7\,\text{yd}\ 1\,\text{ft}\ 4\,\text{in.}\).

a) \(3\,\text{lb}\ 15\,\text{oz}\) b) \(8\,\text{gal}\ 3\,\text{qt}\) c) \(7\,\text{yd}\ 1\,\text{ft}\ 4\,\text{in.}\)

- What must be subtracted from the first measurement to reach the result? - Subtract the result from the starting measurement. - Use the appropriate conversion factor before subtracting. - Think of each equation as an inverse-operation problem.

1. a) Convert to meters and subtract: \(5000-4250=750\). The missing length is \(750\,\text{m}\). 2. b) Convert to ounces: \(197-145=52\,\text{oz}=3\,\text{lb}\ 4\,\text{oz}\). 3. c) Convert to centimeters: \(300-185=115\,\text{cm}=1\,\text{m}\ 15\,\text{cm}\). 4. d) Convert to pounds: \(20{,}000-14{,}800=5200\,\text{lb}=2\,\text{tons}\ 1200\,\text{lb}\).

a) \(750\,\text{m}\) b) \(3\,\text{lb}\ 4\,\text{oz}\) c) \(1\,\text{m}\ 15\,\text{cm}\) d) \(2\,\text{tons}\ 1200\,\text{lb}\)

- One ton is \(2000\,\text{lb}\). - You may convert everything to pounds and convert back at the end. - When subtracting, regroup one ton as \(2000\) pounds if needed. - Calculate tons and pounds separately while tracking regrouping.

1. a) Add \(14+5=19\) tons and \(640+1780=2420\) pounds. Since \(2420\,\text{lb}=1\,\text{ton}\ 420\,\text{lb}\), the result is \(20\,\text{tons}\ 420\,\text{lb}\). 2. b) Regroup \(25\,\text{tons}\ 300\,\text{lb}\) as \(24\,\text{tons}\ 2300\,\text{lb}\). Subtract to get \(12\,\text{tons}\ 1500\,\text{lb}\). 3. c) Regroup \(8\,\text{tons}\ 100\,\text{lb}\) as \(7\,\text{tons}\ 2100\,\text{lb}\). Subtract to get \(4\,\text{tons}\ 600\,\text{lb}\).

a) \(20\,\text{tons}\ 420\,\text{lb}\) b) \(12\,\text{tons}\ 1500\,\text{lb}\) c) \(4\,\text{tons}\ 600\,\text{lb}\)

- How many pounds are in \(1\) ton? - Use inverse operations to solve missing-value equations. - Convert all values to pounds when that makes the calculation easier. - Each full group of \(2000\) pounds is one ton.

1. a) \(3\,\text{tons}=6000\,\text{lb}\), and \(2\,\text{tons}\ 800\,\text{lb}=4800\,\text{lb}\). The difference is \(6000-4800=1200\,\text{lb}\). 2. b) \(6 \times 600\,\text{lb}=3600\,\text{lb}=1\,\text{ton}\ 1600\,\text{lb}\). 3. c) \(10\,\text{tons}=20{,}000\,\text{lb}\), and \(7\,\text{tons}\ 500\,\text{lb}=14{,}500\,\text{lb}\). The difference is \(5500\,\text{lb}=2\,\text{tons}\ 1500\,\text{lb}\).

a) \(1200\,\text{lb}\) b) \(1\,\text{ton}\ 1600\,\text{lb}\) c) \(2\,\text{tons}\ 1500\,\text{lb}\)

- Convert both masses to grams, or regroup kilograms and grams when needed. - “Sum” means add, and “difference” means subtract. - Complete the three operations in the order stated.

1. Find the sum: \(12\,\text{kg}\,450\,\text{g} + 8\,\text{kg}\,700\,\text{g} = 21\,\text{kg}\,150\,\text{g}\). 2. Find the difference: \(12\,\text{kg}\,450\,\text{g} - 8\,\text{kg}\,700\,\text{g} = 3\,\text{kg}\,750\,\text{g}\). 3. Subtract the difference from the sum: \(21\,\text{kg}\,150\,\text{g} - 3\,\text{kg}\,750\,\text{g} = 17\,\text{kg}\,400\,\text{g}\).

The result is \(17\,\text{kg}\,400\,\text{g}\).

- Recall the conversion from liters to milliliters. - For part b), find how many groups of \(25\,\text{g}\) make \(1000\,\text{g}\). - If the comparison unit doubles in length, what happens to the number of times it fits into the same whole?

1. a) Since \(1\,\text{L}=1000\,\text{mL}\), calculate \(1000 \div 200=5\). 2. b) Since \(1\,\text{kg}=1000\,\text{g}\), calculate \(1000 \div 25=40\). 3. c) Since \(1\,\text{m}=100\,\text{cm}\), calculate \(100\,\text{cm} \div 2\,\text{cm}=50\). A \(2\,\text{cm}\) segment is twice as long as a \(1\,\text{cm}\) segment, so it fits half as many times.

a) \(5\) portions b) \(40\) objects c) \(50\) times; doubling the smaller segment halves the number of segments that fit.

- Convert both sides of each comparison to the same unit. - Evaluate each addition expression before comparing. - How many grams are in half a kilogram?

1. Convert all masses to grams. 2. a) \(4\,\text{kg}=4000\,\text{g}\). Since \(4000>400\), use \(>\). 3. b) \(12\,\text{kg}=12{,}000\,\text{g}\), so use \(=\). 4. c) \(3\,\text{kg}+50\,\text{g}=3000\,\text{g}+50\,\text{g}=3050\,\text{g}\). Since \(3050<3500\), use \(<\). 5. d) \(750\,\text{g}+250\,\text{g}=1000\,\text{g}=1\,\text{kg}\), so use \(=\). 6. e) \(\frac{1}{2}\,\text{kg}=500\,\text{g}\). Since \(600>500\), use \(>\).

a) \(>\) b) \(=\) c) \(<\) d) \(=\) e) \(>\)

- Put all three times in the same form before comparing them. - Use multiples of \(60\) to find complete hours. - Compare the hours first and then the minutes.

1. For Leo, \(3\times60=180\) and \(4\times60=240\). So \(215\) minutes is \(3\) hours with \(215-180=35\) minutes remaining: \(3\) hours \(35\) minutes. 2. For Noah, \(3\times60=180\), so \(220\) minutes is \(3\) hours with \(220-180=40\) minutes remaining: \(3\) hours \(40\) minutes. 3. Compare: \(3\,\text{hours}\ 35\,\text{minutes}<3\,\text{hours}\ 40\,\text{minutes}<3\,\text{hours}\ 45\,\text{minutes}\). 4. The order is Leo, Noah, Mia.

Leo: \(3\) hours \(35\) minutes Noah: \(3\) hours \(40\) minutes Mia: \(3\) hours \(45\) minutes Order: Leo, Noah, Mia.

- Which two smaller-unit amounts make exactly \(1000\,\text{g}\) or \(100\,\text{cm}\)? - Add the smaller units first and regroup one whole larger unit when possible. - Sorting or scanning the smaller-unit amounts can help you find useful pairs.

1. a) Pair \(16\,\text{kg}\ 350\,\text{g}\) with \(13\,\text{kg}\ 650\,\text{g}\): \(29\,\text{kg}\ 1000\,\text{g}=30\,\text{kg}\). Pair \(24\,\text{kg}\ 120\,\text{g}\) with \(15\,\text{kg}\ 880\,\text{g}\): \(39\,\text{kg}\ 1000\,\text{g}=40\,\text{kg}\). Then \(30\,\text{kg}+40\,\text{kg}=70\,\text{kg}\). 2. b) Pair \(7\,\text{m}\ 42\,\text{cm}\) with \(12\,\text{m}\ 58\,\text{cm}\): \(19\,\text{m}\ 100\,\text{cm}=20\,\text{m}\). Pair \(12\,\text{m}\ 15\,\text{cm}\) with \(7\,\text{m}\ 85\,\text{cm}\): \(19\,\text{m}\ 100\,\text{cm}=20\,\text{m}\). Then \(20\,\text{m}+20\,\text{m}=40\,\text{m}\).

a) \(70\,\text{kg}\) b) \(40\,\text{m}\)

- How many milliliters make \(1\) liter? - How many minutes make \(1\) hour? - Look for pairs that add to \(1000\) milliliters or \(60\) minutes.

1. a) Pair \(14\,\text{L}\ 250\,\text{mL}\) with \(5\,\text{L}\ 750\,\text{mL}\): \(19\,\text{L}\ 1000\,\text{mL}=20\,\text{L}\). Pair \(22\,\text{L}\ 680\,\text{mL}\) with \(17\,\text{L}\ 320\,\text{mL}\): \(39\,\text{L}\ 1000\,\text{mL}=40\,\text{L}\). Then \(20\,\text{L}+40\,\text{L}=60\,\text{L}\). 2. b) Pair \(1\,\text{hr}\ 15\,\text{min}\) with \(3\,\text{hr}\ 45\,\text{min}\): \(4\,\text{hr}\ 60\,\text{min}=5\,\text{hr}\). Pair \(2\,\text{hr}\ 50\,\text{min}\) with \(1\,\text{hr}\ 10\,\text{min}\): \(3\,\text{hr}\ 60\,\text{min}=4\,\text{hr}\). Then \(5\,\text{hr}+4\,\text{hr}=9\,\text{hr}\).

a) \(60\,\text{L}\) b) \(9\,\text{hr}\)

- Convert unfamiliar measurements to units you can picture. - Check both the numerical value and its unit. - Compare each measurement with a familiar benchmark; more than one replacement estimate may be reasonable.

1. A height of \(6\,\text{ft}\) is reasonable for an adult. 2. A standard stick of butter is about \(4\,\text{oz}\), or \(0.25\,\text{lb}\), so \(2.5\,\text{lb}\) is not reasonable. One reasonable replacement is \(4\,\text{oz}\). 3. Since \(45\times60=2700\), \(2700\) seconds is \(45\) minutes. A \(45\)-minute class period is reasonable. 4. Six hundred feet is far too short for a long trip between cities. One reasonable replacement is about \(600\,\text{mi}\). Other estimates can also be reasonable if they fit a long city-to-city trip.

a) Reasonable. b) Not reasonable. Example: about \(4\,\text{oz}\), or \(0.25\,\text{lb}\). Other reasonable estimates are acceptable. c) Reasonable. d) Not reasonable. Example: about \(600\,\text{mi}\). Other reasonable long-trip estimates are acceptable.

- Convert both movie lengths to the same unit. - How many minutes are in \(1\) hour? - Subtract the shorter time from the longer time.

1. Convert Movie A's length to minutes: \(1\,\text{hour}\,35\,\text{minutes} = 60 + 35 = 95\) minutes. 2. Compare the lengths: \(110 > 95\), so Movie B is longer. 3. Find the difference: \(110 - 95 = 15\) minutes.

Movie B is longer by \(15\) minutes.