Free math worksheets

- Compare equal place values from left to right. - A trailing zero can make the decimal places easier to line up without changing the value. - Stop at the first place where the digits differ.

1. Write \(0.54\) as \(0.540\) so both decimals show thousandths. 2. The tenths digits are equal, but the hundredths digits are \(4\) and \(0\). 3. Therefore, \(0.540 > 0.504\).

\(0.54 > 0.504\)

- Identify the place value of each nonzero digit. - Tenths, hundredths, and thousandths correspond to denominators \(10\), \(100\), and \(1000\). - The zero hundredths digit contributes nothing.

1. The \(3\) is in the ones place. 2. The \(4\) is in the tenths place, contributing \(\frac{4}{10}\). 3. The hundredths digit is \(0\), so that term can be omitted. 4. The \(7\) is in the thousandths place, contributing \(\frac{7}{1000}\).

\(3 + \frac{4}{10} + \frac{7}{1000}\)

- Match each fraction denominator to its decimal place. - Include a placeholder when a place has value \(0\).

1. The whole-number part is \(5\). 2. \(\frac{2}{10}\) gives \(2\) tenths, \(\frac{9}{1000}\) gives \(9\) thousandths, and there are \(0\) hundredths. 3. The decimal is \(5.209\).

\(5.209\)

- Write the digits for four hundred five thousandths in place-value order. - Check that the zero remains in the hundredths place.

1. Four hundred five thousandths is written as \(0.405\). 2. The digits in the tenths, hundredths, and thousandths places are \(4\), \(0\), and \(5\). 3. The completed decimal is \(8.405\).

The boxes contain \(4\), \(0\), and \(5\); the decimal is \(8.405\).

- Identify the whole-number part. - Place the \(8\) in the thousandths place and use zeros as placeholders.

1. The whole-number part is \(103\). 2. Eight thousandths requires \(8\) in the thousandths place, with zeros in the tenths and hundredths places. 3. The decimal is \(103.008\).

\(103.008\)

- Identify the thousandths place. - Compare the position of each digit in the answer choices.

1. Seventy-five thousandths is \(\frac{75}{1000}\). 2. Writing seventy-five thousandths as a decimal gives \(0.075\). 3. Choice B is correct.

B: \(0.075\)

- First write the value using tenths. - Add trailing zeros until there are three digits after the decimal point.

1. Forty-nine and three tenths is \(49.3\). 2. Adding trailing zeros does not change the value. 3. With exactly three digits after the decimal point, the entry is \(49.300\).

\(49.300\)

- Write the whole-number part first. - Place the three decimal digits in tenths-to-thousandths order.

1. Write the whole-number part as \(10{,}002\). 2. Place \(1\), \(0\), and \(4\) in the tenths, hundredths, and thousandths places. 3. The decimal is \(10{,}002.104\).

\(10{,}002.104\)

- Place the \(5\) in the thousandths place. - Count all visible digits to the right of the decimal point.

1. The whole-number part is \(6\). 2. Five thousandths requires zeros in the tenths and hundredths places and \(5\) in the thousandths place. 3. The decimal is \(6.005\), and it has three digits after the decimal point.

\(6.005\); it has three digits after the decimal point.

- Match each digit to its place from tenths through thousandths. - Keep the zero in the tenths place.

1. The whole-number part is \(720\). 2. The digits after the decimal point are \(0\), \(4\), and \(2\), in that order. 3. The decimal is \(720.042\).

\(720.042\)

- Identify the whole-number part first. - Determine the place value of the digit \(7\).

1. The whole-number part is \(400\). 2. The digit \(7\) is in the thousandths place. 3. Therefore, the correct reading is “four hundred and seven thousandths,” which is choice B.

B: “four hundred and seven thousandths”

- Read the digits from left to right after the decimal point. - Match them to tenths, hundredths, and thousandths in order.

1. The digits to the right of the decimal point are \(0\), \(5\), and \(0\). 2. These digits are in the tenths, hundredths, and thousandths places, respectively.

tenths digit: \(0\) hundredths digit: \(5\) thousandths digit: \(0\)

- Count all digits to the right of the decimal point. - Use the place of the last digit to name the fractional part.

1. There are three digits after the decimal point. 2. The fractional part is \(25\) thousandths. 3. The number is read as “one and twenty-five thousandths.”

Three digits; “one and twenty-five thousandths”

- Identify the whole-number part. - Use the thousandths place to name the fractional part.

1. The whole-number part is \(1234\). 2. The fractional part is \(90\) thousandths. 3. The decimal is read as “one thousand two hundred thirty-four and ninety thousandths.”

“one thousand two hundred thirty-four and ninety thousandths”

- Use the place of the last digit to name the fractional part. - Read the digits to the right of the decimal point as one whole number.

1. The decimal has three digits after the decimal point, so the fractional part is measured in thousandths. 2. The digits form \(608\) thousandths. 3. Choice C is correct.

C: “six hundred eight thousandths”

- Read the three decimal digits from left to right. - Match them to tenths, hundredths, and thousandths in order.

1. The digits to the right of the decimal point are \(5\), \(0\), and \(0\). 2. Therefore, the tenths digit is \(5\), the hundredths digit is \(0\), and the thousandths digit is \(0\).

tenths digit: \(5\) hundredths digit: \(0\) thousandths digit: \(0\)

- Identify the whole-number part first. - The digit \(6\) is in the thousandths place.

1. The whole-number part is \(8000\). 2. The fractional part is \(6\) thousandths. 3. The decimal is read as “eight thousand and six thousandths.”

“eight thousand and six thousandths”

- Locate the digit \(1\) on a place-value chart. - Use the place of the last visible digit to read the decimal exactly as written.

1. The digits after the decimal point are \(0\), \(1\), and \(0\). 2. The digit \(1\) is in the hundredths place, not the tenths place. 3. As written to the thousandths place, the decimal is “three hundred five and ten thousandths.” It is also equivalent to \(305.01\), or “three hundred five and one hundredth.”

The \(1\) is in the hundredths place, not the tenths place. A correct reading as written is “three hundred five and ten thousandths.”

- Identify the place named by “thousandths.” - Use zeros as placeholders in the tenths and hundredths places.

1. The digit \(3\) must be in the thousandths place. 2. Zeros are needed in the tenths and hundredths places. 3. The correct decimal is \(7.003\).

The decimal is incorrect. The correct decimal is \(7.003\).

- Match each amount to its place value. - Keep the given zero in the hundredths place.

1. Place \(9\) in the tenths place. 2. The hundredths digit is \(0\), so place \(0\) in the hundredths place. 3. Place \(4\) in the thousandths place. 4. The boxes contain \(9\), \(0\), and \(4\), and the decimal is \(60.904\).

The boxes contain \(9\), \(0\), and \(4\); the decimal is \(60.904\).

- Identify the whole-number part. - Write two hundred five thousandths using three decimal places.

1. The whole-number part is \(18\). 2. Two hundred five thousandths is written as \(0.205\). 3. The stored value is \(18.205\).

\(18.205\)

- Identify the third place to the right of the decimal point. - Use zeros as placeholders before the \(9\).

1. The digit \(9\) belongs in the thousandths place. 2. Zeros are needed in the tenths and hundredths places. 3. The boxes contain \(0\), \(0\), and \(9\), and the decimal is \(0.009\).

The boxes contain \(0\), \(0\), and \(9\); the decimal is \(0.009\).

- Match each decimal digit to its place from left to right. - Use the thousandths place to name the fractional part.

1. The digits after the decimal point are \(3\), \(0\), and \(8\). 2. Enter \(3\) under tenths, \(0\) under hundredths, and \(8\) under thousandths. 3. The decimal is read as “three hundred eight thousandths.”

Table entries: \(3\), \(0\), \(8\). Reading: “three hundred eight thousandths.”

- Compare each description with the decimal one place at a time. - Pay special attention to the zero in the tenths place.

1. The digits after the decimal point are \(0\), \(7\), and \(2\). 2. These digits are in the tenths, hundredths, and thousandths places. 3. Therefore, choice B is correct.

B: one hundred forty-five, with \(0\) tenths, \(7\) hundredths, and \(2\) thousandths

- Match the digits to their places from left to right. - Use the place of the last digit to name the fractional part.

1. The digits in the tenths, hundredths, and thousandths places are \(0\), \(9\), and \(5\). 2. The fractional part is \(95\) thousandths. 3. The decimal is read as “ninety-five thousandths.”

tenths digit: \(0\); hundredths digit: \(9\); thousandths digit: \(5\); reading: “ninety-five thousandths”

- Locate the digit \(8\) relative to the decimal point. - Name each visible decimal place in order.

1. The digit \(8\) is the first digit after the decimal point, so it is in the tenths place. 2. The hundredths and thousandths digits are both \(0\). 3. As written, the decimal has \(8\) tenths, \(0\) hundredths, and \(0\) thousandths. It is read as “eighty and eight hundred thousandths.”

Paul confused tenths with hundredths. The decimal has \(8\) tenths, \(0\) hundredths, and \(0\) thousandths.

- Separate the whole-number part from the decimal digits. - Match the decimal digits to tenths, hundredths, and thousandths in order.

1. The whole-number part is \(600\). 2. The digits after the decimal point are \(0\), \(6\), and \(4\). 3. The table entries are \(600\), \(0\), \(6\), and \(4\).

Whole-number part: \(600\); tenths: \(0\); hundredths: \(6\); thousandths: \(4\)

- Count the decimal places before the digit \(7\). - Compare each description with the decimal place by place.

1. The digits after the decimal point are \(0\), \(0\), and \(7\). 2. The digit \(7\) is in the thousandths place. 3. Choice B is correct.

B: three, with \(0\) tenths, \(0\) hundredths, and \(7\) thousandths

- Count the digits to the right of the decimal point. - Use the place of the last digit to name the fractional part.

1. The decimal has three digits after the decimal point. 2. Therefore, the fractional part is measured in thousandths, not hundredths. 3. The correct reading is “forty and four hundred seven thousandths.”

“forty and four hundred seven thousandths”

- Match each decimal digit to its place from left to right. - Pay attention to the zero in the hundredths place.

1. The digits after the decimal point are \(3\), \(0\), and \(5\). 2. The digit \(5\) is in the thousandths place, not the hundredths place. 3. The decimal has \(3\) tenths, \(0\) hundredths, and \(5\) thousandths.

The \(5\) is in the thousandths place. The correct description is \(3\) tenths, \(0\) hundredths, and \(5\) thousandths.

- Write ninety-two thousandths using three decimal places. - Compare the decimal digits with the entered value.

1. Ninety-two thousandths is written as \(0.092\). 2. The tenths digit must be \(0\), the hundredths digit \(9\), and the thousandths digit \(2\). 3. The correct decimal is \(200.092\).

\(200.092\)

- Identify the place occupied by the digit \(5\). - Use zeros as placeholders in the two earlier decimal places.

1. The digit \(5\) is already in the thousandths place. 2. The tenths and hundredths places must both contain \(0\). 3. The completed decimal is \(2.005\).

Both boxes contain \(0\); the decimal is \(2.005\).

- Match the decimal digits to tenths, hundredths, and thousandths. - Identify the place occupied by the box.

1. The digits after the decimal point must be \(4\), \(0\), and \(0\). 2. The box is in the hundredths place. 3. The missing digit is \(0\), and the completed decimal is \(70.400\).

The missing digit is \(0\); the decimal is \(70.400\).

- Write six hundred two thousandths using three decimal places. - Match the first and third decimal digits to the boxes.

1. Six hundred two thousandths is written as \(0.602\). 2. The first box is the tenths digit, \(6\). 3. The second box is the thousandths digit, \(2\).

The boxes contain \(6\) and \(2\), in that order; the decimal is \(0.602\).

- Identify the whole-number part first. - The digit \(8\) is in the thousandths place.

1. The whole-number part is \(125\). 2. The fractional part is \(8\) thousandths. 3. The measurement is read as “one hundred twenty-five and eight thousandths meters.”

“one hundred twenty-five and eight thousandths meters”

- Identify the whole-number part first. - Use the thousandths place to name the fractional part.

1. The whole-number part is \(12\). 2. The fractional part is \(305\) thousandths. 3. The distance is read as “twelve and three hundred five thousandths miles.”

“twelve and three hundred five thousandths miles”

- Write seventy thousandths using three decimal places. - Include the trailing zero and the unit.

1. Seventy thousandths is written as \(0.070\). 2. Combine this with the whole-number part \(5\). 3. The display value is \(5.070\,\text{kg}\).

\(5.070\,\text{kg}\)

- Write one hundred twenty-five thousandths using three decimal places. - Include the metric unit in your answer.

1. One hundred twenty-five thousandths is \(\frac{125}{1000}\). 2. As a decimal, \(\frac{125}{1000} = 0.125\). 3. The mass is \(0.125\,\text{kg}\).

\(0.125\,\text{kg}\)

- Work on each part separately. - Use the named place value and preserve any required trailing zeros.

1. Four and six hundredths is \(4.06\). 2. Ninety and five tenths is \(90.5\); with two decimal places, it is \(90.50\). 3. Three thousandths is \(0.003\).

a) \(4.06\) b) \(90.50\) c) \(0.003\)

- Check the whole-number part and the decimal place separately. - Distinguish hundredths from thousandths.

1. In \(6.008\), the digit \(8\) is in the thousandths place, so A matches 3. 2. In \(60.08\), the whole-number part is \(60\) and the fractional part is eight hundredths, so B matches 2. 3. In \(6.08\), the fractional part is eight hundredths, so C matches 1.

\(\text{A} \to 3\), \(\text{B} \to 2\), \(\text{C} \to 1\)

- Check the whole-number part first. - Then compare the tenths, hundredths, and thousandths digits.

1. The whole-number part must be \(300\). 2. The decimal digits must be \(0\), \(7\), and \(0\), in that order. 3. Only choice A, \(300.070\), satisfies both conditions.

A: \(300.070\)

- Use three decimal places for thousandths. - Keep the zero in the hundredths place.

1. Five hundred two thousandths is \(\frac{502}{1000}\). 2. In decimal notation, \(\frac{502}{1000} = 0.502\). 3. Choice A is correct.

A: \(0.502\)

- Place each given digit in its named decimal place. - Use the thousandths place to name the fractional part.

1. Write the whole-number part as \(304\). 2. Place \(0\), \(6\), and \(2\) in the tenths, hundredths, and thousandths places. 3. The decimal is \(304.062\), read as “three hundred four and sixty-two thousandths.”

\(304.062\); “three hundred four and sixty-two thousandths”

- Keep the three decimal digits in the given order. - Use the thousandths place to name the fractional part.

1. Write the whole-number part as \(9\). 2. Place \(4\), \(0\), and \(7\) in the tenths, hundredths, and thousandths places. 3. The decimal is \(9.407\), read as “nine and four hundred seven thousandths.”

\(9.407\); “nine and four hundred seven thousandths”

- Use the place of the last digit to read the decimal. - Count three places to the right of the decimal point.

1. The decimal has three digits after the decimal point. 2. It is read as “two hundred nine thousandths.” 3. The digit \(9\) is in the thousandths place.

“two hundred nine thousandths”; the \(9\) is in the thousandths place.

- Place the digits in tenths, hundredths, and thousandths order. - Use the thousandths place to read the decimal exactly as written.

1. The whole-number part is \(7\). 2. The decimal digits are \(0\), \(2\), and \(0\). 3. The decimal is \(7.020\), read as “seven and twenty thousandths.”

\(7.020\); “seven and twenty thousandths”

- Keep the digits \(4\) and \(2\) in the given order. - Use the zeros as placeholders before the thousandths digit.

1. The whole-number part is \(4\). 2. Two thousandths requires \(0\) in the tenths place, \(0\) in the hundredths place, and \(2\) in the thousandths place. 3. The decimal is \(4.002\).

\(4.002\)

- Use the labeled tenths to determine the value of each small interval. - Count hundredths from the nearest labeled tenth. - Compare the two marker positions after identifying their values.

1. Each small tick represents one hundredth. 2. Marker \(A\) is at \(0.47\), and marker \(B\) is at \(0.53\). 3. Since \(47\) hundredths is less than \(53\) hundredths, \(0.47 < 0.53\).

a) \(A=0.47\), \(B=0.53\) b) \(0.47 < 0.53\)

- Line up the decimal places before comparing. - Adding trailing zeros does not change a decimal's value. - Compare the first place where two numbers differ.

1. Write equivalent forms to thousandths when helpful: \(0.706\), \(0.670\), \(0.700\), and \(0.076\). 2. Compare from the tenths place to the right. 3. The order is \(0.076 < 0.67 < 0.7 < 0.706\).

\(0.076 < 0.67 < 0.7 < 0.706\)

- Rewrite both decimals with the same number of displayed decimal places. - Compare the tenths digits first. - Stop as soon as one place determines the comparison.

1. Decimal digits must be compared by place value, not as separate whole-number strings. 2. Write \(0.4\) as \(0.400\). 3. At the tenths place, \(3 < 4\), so \(0.392 < 0.400\).

Ava's reasoning ignores place value. The correct comparison is \(0.392 < 0.4\).

- Think about equivalent fractions such as hundredths and thousandths. - Does adding a zero at the far right create a new nonzero place value? - Compare the amounts represented, not the number of written digits.

1. \(2.50\) means \(2\) ones and \(50\) hundredths. 2. \(2.500\) means \(2\) ones and \(500\) thousandths. 3. Since \(50\) hundredths equals \(500\) thousandths, the decimals are equal. The trailing zero does not change the value.

\(2.50 = 2.500\). The extra trailing zero does not change the value.

- Count chips by place-value column in each panel. - Write each represented decimal before comparing. - Compare from the largest place value toward the thousandths.

1. Panel a) shows \(1\) one, \(0\) tenths, \(2\) hundredths, and \(5\) thousandths, so it represents \(1.025\). 2. Panel b) shows \(1\) one, \(2\) tenths, \(0\) hundredths, and \(5\) thousandths, so it represents \(1.205\). 3. The ones digits are equal, but at the tenths place \(0 < 2\), so \(1.025 < 1.205\).

a) Panel a): \(1.025\); panel b): \(1.205\) b) \(1.025 < 1.205\)

- Name the place of each digit in \(0.604\). - Check the denominator used for the \(4\).

1. In \(0.604\), the \(6\) is in the tenths place, the hundredths digit is \(0\), and the \(4\) is in the thousandths place. 2. Jordan treated the \(4\) as hundredths instead of thousandths. 3. The correct expanded form is \(\frac{6}{10}+\frac{4}{1000}\).

Jordan placed the \(4\) in the hundredths place, but it represents \(4\) thousandths. The correct expanded form is \(\frac{6}{10}+\frac{4}{1000}\).

- Rewrite both endpoints with three decimal places. - Consider the thousandths between \(120\) thousandths and \(130\) thousandths. - Arrange your chosen values in increasing order.

1. Write the endpoints to the thousandths place: \(4.12 = 4.120\) and \(4.13 = 4.130\). 2. Choose any three thousandths between \(4.120\) and \(4.130\), such as \(4.121\), \(4.125\), and \(4.129\). 3. Order the five values.

Answers will vary. One possible answer is \(4.12 < 4.121 < 4.125 < 4.129 < 4.13\).

- Treat the two inequality signs as two separate conditions first. - For the left condition, compare the tenths digits and then the hundredths digits. - For the right condition, test what changes when the repeated digit reaches \(8\).

1. From \(4.35<4.\square2\), the box digit must be at least \(4\). A digit of \(3\) gives \(4.32\), which is too small. 2. From \(4.\square2<4.7\square\), digits \(8\) and \(9\) fail because the left number would be greater than \(4.7\square\). A digit of \(7\) gives \(4.72<4.77\), so digits through \(7\) can work. 3. Combining both conditions leaves \(4,5,6,7\).

The possible digits are \(4,5,6,7\).

- Focus first on \(0.\square7<0.\square\). - Adding a trailing zero does not change the value of a decimal. - Compare the digits from left to right by place value.

1. Consider only the first inequality: \(0.\square7<0.\square\). 2. Write the number on the right with a trailing zero: \(0.\square7<0.\square0\). 3. The tenths digits are identical, but the hundredths digit on the left is \(7\) and the hundredths digit on the right is \(0\). Therefore, \(0.\square7\) is always greater than \(0.\square0\). 4. The first inequality is impossible for every digit, so the entire compound inequality has no solution.

No digit works. The first inequality is impossible because \(0.\square7\) is always greater than \(0.\square=0.\square0\).

- Separate the whole-number part from the digits after the decimal point. - Use the thousandths place to name the fractional part.

1. The whole-number part is \(72\). 2. The tenths, hundredths, and thousandths digits are \(0\), \(4\), and \(3\). 3. The fractional part is \(43\) thousandths, so the decimal is read as “seventy-two and forty-three thousandths.”

Whole-number part: \(72\); tenths digit: \(0\); hundredths digit: \(4\); thousandths digit: \(3\); reading: “seventy-two and forty-three thousandths.”

- Determine how many decimal places are needed for thousandths. - Compare both entries place by place.

1. Thirty-six thousandths requires three digits after the decimal point. 2. The digits in the tenths, hundredths, and thousandths places are \(0\), \(3\), and \(6\). 3. Therefore, \(41.036\) is the correct entry.

\(41.036\) is correct because its decimal digits are \(0\), \(3\), and \(6\), representing thirty-six thousandths.

- Check each row separately. - Use all three decimal digits when reading a number to the thousandths place.

1. Row A is correct because \(0.04\) is four hundredths. 2. Row C is correct because \(0.70\) is seventy hundredths. 3. Row B is incorrect because \(0.305\) is three hundred five thousandths, not thirty-five thousandths.

Row B is incorrect. The correct reading is “twelve and three hundred five thousandths.”

- Work on each row separately. - Place the decimal point between the ones and tenths columns.

1. In row a), place \(4\) in the ones place and \(0\), \(3\), and \(7\) in the decimal places to get \(4.037\). 2. In row b), the digits form \(0.802\). 3. In row c), the digits form \(9.150\).

a) \(4.037\) b) \(0.802\) c) \(9.150\)

- Form the whole-number part from the hundreds, tens, and ones columns. - Then place the tenths, hundredths, and thousandths digits after the decimal point.

1. In row a), the whole-number digits form \(205\), and the decimal digits form \(048\), so the decimal is \(205.048\). 2. In row b), the whole-number digits form \(70\), and the decimal digits form \(603\), so the decimal is \(70.603\).

a) \(205.048\) b) \(70.603\)

- Work on each row separately. - Match the decimal digits to tenths, hundredths, and thousandths in order.

1. For row A, the decimal digits are \(0\), \(2\), and \(4\). The place-value description is fifteen, with \(0\) tenths, \(2\) hundredths, and \(4\) thousandths. 2. For row B, the decimal digits are \(4\), \(0\), and \(6\). 3. Combining those digits with the whole-number part \(2\) gives \(2.406\).

A: decimal digits \(0, 2, 4\); fifteen, with \(0\) tenths, \(2\) hundredths, and \(4\) thousandths B: \(2.406\); decimal digits \(4, 0, 6\)

- Work on each part separately. - Keep the decimal digits in the order shown.

1. Combining \(27\) with decimal digits \(0\) and \(9\) gives \(27.09\). 2. Combining \(4\) with \(3\), \(0\), and \(5\) gives \(4.305\). 3. Combining \(800\) with \(0\), \(0\), and \(4\) gives \(800.004\).

a) \(27.09\) b) \(4.305\) c) \(800.004\)

- Place the decimal point between the whole-number part and the tenths digit. - Use the thousandths place to name each fractional part.

1. In row a), the digits form \(31.026\). The decimal is read as “thirty-one and twenty-six thousandths.” 2. In row b), the digits form \(0.704\). The decimal is read as “seven hundred four thousandths.”

a) \(31.026\); “thirty-one and twenty-six thousandths” b) \(0.704\); “seven hundred four thousandths”

- Keep the digits in their original order. - Use the place of the last digit to name each fractional part.

1. After the first digit, the decimal is \(1.234\), read as “one and two hundred thirty-four thousandths.” 2. After the second digit, the decimal is \(12.34\), read as “twelve and thirty-four hundredths.” 3. After the third digit, the decimal is \(123.4\), read as “one hundred twenty-three and four tenths.”

\(1.234\): “one and two hundred thirty-four thousandths” \(12.34\): “twelve and thirty-four hundredths” \(123.4\): “one hundred twenty-three and four tenths”

- Check each item separately. - Use the named place value to position every digit.

1. Four hundredths is \(0.04\), so the first decimal should be \(3.04\). 2. Two hundred five thousandths is \(0.205\), so the second decimal should be \(70.205\).

a) \(3.04\) b) \(70.205\)

- Identify the three decimal digits and their places. - Use the thousandths place to name the fractional part.

1. The whole-number part is \(6007\). 2. The fractional part is \(85\) thousandths, so the decimal is read as “six thousand seven and eighty-five thousandths.” 3. The digit \(8\) is in the hundredths place, and the digit \(5\) is in the thousandths place.

“six thousand seven and eighty-five thousandths”; \(8\) is in the hundredths place, and \(5\) is in the thousandths place.

- Determine the value of one small interval. - Count thousandths from the labeled hundredths. - Use the markers' left-to-right positions to check your ordering.

1. Each small tick represents one thousandth. 2. The markers are \(P=0.345\), \(Q=0.348\), and \(R=0.352\). 3. Therefore, \(Q\) represents \(0.348\), and \(0.345 < 0.348 < 0.352\).

a) \(Q\) b) \(0.345 < 0.348 < 0.352\)

- The tenths digit decides whether the number can stay below \(2.4\). - After choosing the tenths digit, maximize the remaining places from left to right. - Check your number against \(2.400\).

1. To stay below \(2.4\), the tenths digit must be less than \(4\), so it must be \(3\). 2. To make the number as large as possible, put the larger remaining digit \(9\) in the hundredths place and \(8\) in the thousandths place. 3. The number is \(2.398\), and \(2.398 < 2.4\).

\(2.398\)