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Round multi-digit numbers

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Difficulty: 1 – 5

5544634

Use the number line. Round point \(A\) to the nearest thousand.

Figure for problem 554463

Hints

- Read the value of point \(A\) from the labeled ticks. - Locate the two thousands between which the point lies. - Recall what happens when a value is exactly at the midpoint.

Solution

1. The midpoint between \(4000\) and \(5000\) is \(4500\). 2. A value exactly halfway between two thousands rounds to the greater thousand. 3. Therefore, \(4500\) rounds to \(5000\).

Answer

\(5000\)

5165864

The table shows the number of visitors to four amusement parks. <table> <tr><th>Amusement park</th><th>Visitors</th></tr> <tr><td>Skyline Adventure Park</td><td>\(342{,}189\)</td></tr> <tr><td>Riverstone Fun Park</td><td>\(567{,}802\)</td></tr> <tr><td>Maple Grove Park</td><td>\(12{,}455\)</td></tr> <tr><td>Summit Springs Park</td><td>\(99{,}501\)</td></tr> </table> Round each visitor count to the nearest thousand.

Hints

- To round to the nearest thousand, look at the hundreds digit. - Digits \(0\) through \(4\) round down. - Digits \(5\) through \(9\) round up. - Replace all digits to the right of the thousands place with zeros.

Solution

1. Skyline Adventure Park: The hundreds digit is \(1\), so \(342{,}189\) rounds to \(342{,}000\). 2. Riverstone Fun Park: The hundreds digit is \(8\), so \(567{,}802\) rounds to \(568{,}000\). 3. Maple Grove Park: The hundreds digit is \(4\), so \(12{,}455\) rounds to \(12{,}000\). 4. Summit Springs Park: The hundreds digit is \(5\), so \(99{,}501\) rounds to \(100{,}000\).

Answer

Skyline Adventure Park: \(342{,}000\) Riverstone Fun Park: \(568{,}000\) Maple Grove Park: \(12{,}000\) Summit Springs Park: \(100{,}000\)

5165884

Consider the number \(674{,}512\). Round it to the nearest: a) thousand b) ten thousand c) hundred thousand

Hints

- Identify the place to which you are rounding. - Look at the digit immediately to its right. - Replace all digits to the right of the rounding place with zeros.

Solution

1. For a), the hundreds digit is \(5\), so \(674{,}512\) rounds to \(675{,}000\). 2. For b), the thousands digit is \(4\), so \(674{,}512\) rounds to \(670{,}000\). 3. For c), the ten-thousands digit is \(7\), so \(674{,}512\) rounds to \(700{,}000\).

Answer

a) \(675{,}000\) b) \(670{,}000\) c) \(700{,}000\)

5173604

Round each number to the nearest ten and to the nearest hundred. a) \(432\) b) \(758\) c) \(945\)

Hints

- To round to the nearest ten, look at the ones digit. - To round to the nearest hundred, look at the tens digit. - Digits \(0\) through \(4\) round down; digits \(5\) through \(9\) round up.

Solution

1. For \(432\), the ones digit is \(2\), so the nearest ten is \(430\). The tens digit is \(3\), so the nearest hundred is \(400\). 2. For \(758\), the ones digit is \(8\), so the nearest ten is \(760\). The tens digit is \(5\), so the nearest hundred is \(800\). 3. For \(945\), the ones digit is \(5\), so the nearest ten is \(950\). The tens digit is \(4\), so the nearest hundred is \(900\).

Answer

a) Nearest ten: \(430\); nearest hundred: \(400\) b) Nearest ten: \(760\); nearest hundred: \(800\) c) Nearest ten: \(950\); nearest hundred: \(900\)

5173724

Round each number to the indicated places. a) \(34{,}274\): nearest ten and nearest thousand b) \(845{,}193\): nearest hundred, nearest ten-thousand, and nearest hundred-thousand c) \(99{,}996\): nearest ten, nearest hundred, and nearest thousand

Hints

- Look at the digit immediately to the right of the rounding place. - Round down for \(0\) through \(4\) and up for \(5\) through \(9\). - Watch for regrouping when several digits are \(9\).

Solution

1. a) \(34{,}274\) rounds to \(34{,}270\) to the nearest ten and \(34{,}000\) to the nearest thousand. 2. b) \(845{,}193\) rounds to \(845{,}200\) to the nearest hundred, \(850{,}000\) to the nearest ten-thousand, and \(800{,}000\) to the nearest hundred-thousand. 3. c) \(99{,}996\) rounds to \(100{,}000\) to the nearest ten, hundred, and thousand.

Answer

a) Nearest ten: \(34{,}270\); nearest thousand: \(34{,}000\) b) Nearest hundred: \(845{,}200\); nearest ten-thousand: \(850{,}000\); nearest hundred-thousand: \(800{,}000\) c) Nearest ten: \(100{,}000\); nearest hundred: \(100{,}000\); nearest thousand: \(100{,}000\)

5173754

Round each number to the nearest hundred and nearest thousand. a) \(4382\) b) \(12{,}549\) c) \(99{,}950\) d) \(456\)

Hints

- Look at the digit immediately to the right of the rounding place. - Digits \(0\) through \(4\) round down. - Digits \(5\) through \(9\) round up. - Rounding up a \(9\) may require regrouping.

Solution

1. a) \(4382\) rounds to \(4400\) to the nearest hundred and \(4000\) to the nearest thousand. 2. b) \(12{,}549\) rounds to \(12{,}500\) to the nearest hundred and \(13{,}000\) to the nearest thousand. 3. c) \(99{,}950\) rounds to \(100{,}000\) to both the nearest hundred and nearest thousand. 4. d) \(456\) rounds to \(500\) to the nearest hundred and \(0\) to the nearest thousand.

Answer

a) Nearest hundred: \(4400\); nearest thousand: \(4000\) b) Nearest hundred: \(12{,}500\); nearest thousand: \(13{,}000\) c) Nearest hundred: \(100{,}000\); nearest thousand: \(100{,}000\) d) Nearest hundred: \(500\); nearest thousand: \(0\)

5173784

First write each number name as a numeral. Then round each number to the nearest ten-thousand and nearest hundred-thousand. a) four hundred seventy-five thousand eight hundred thirty-two b) nine hundred ninety-nine thousand four hundred seventeen

Hints

- Write each number name in standard form first. - Look at the digit immediately to the right of the rounding place. - Regroup carefully when rounding up a place containing \(9\).

Solution

1. a) The numeral is \(475{,}832\). It rounds to \(480{,}000\) to the nearest ten-thousand and \(500{,}000\) to the nearest hundred-thousand. 2. b) The numeral is \(999{,}417\). It rounds to \(1{,}000{,}000\) to both the nearest ten-thousand and nearest hundred-thousand.

Answer

a) \(475{,}832\); nearest ten-thousand: \(480{,}000\); nearest hundred-thousand: \(500{,}000\) b) \(999{,}417\); nearest ten-thousand: \(1{,}000{,}000\); nearest hundred-thousand: \(1{,}000{,}000\)

5173944

A large city library has many items available to borrow. a) Round \(124{,}812\) books to the nearest ten-thousand. b) The library has \(9827\) novels and \(12{,}305\) nonfiction books. Round each number to the nearest hundred. c) The library subscribes to \(512\) magazines. Round this number to the nearest ten.

Hints

- Look at the digit immediately to the right of the place to which you are rounding. - Digits \(0\) through \(4\) tell you to round down. - Digits \(5\) through \(9\) tell you to round up. - When you round, what happens to the digits to the right of the rounding place?

Solution

1. a) In \(124{,}812\), the thousands digit is \(4\), so the number rounds down to \(120{,}000\). 2. b) In \(9827\), the tens digit is \(2\), so the number rounds down to \(9800\). In \(12{,}305\), the tens digit is \(0\), so the number rounds down to \(12{,}300\). 3. c) In \(512\), the ones digit is \(2\), so the number rounds down to \(510\).

Answer

a) \(120{,}000\) b) Novels: \(9800\); nonfiction books: \(12{,}300\) c) \(510\)

5173954

You are writing a short news report about a local soccer tournament. Round each number as directed. - Spectators: \(412\), nearest hundred - Hot dogs sold: \(187\), nearest ten - Total revenue: \(\$592\), nearest hundred dollars - Length of the final game: \(92\) minutes, nearest ten minutes

Hints

- Use the rounding place stated beside each quantity. - Look one place to the right of the place you are rounding to. - Check that your rounded result has zeros in all places to the right of the rounding place.

Solution

1. \(412\) rounds to \(400\) to the nearest hundred. 2. \(187\) rounds to \(190\) to the nearest ten. 3. \(\$592\) rounds to \(\$600\) to the nearest hundred dollars. 4. \(92\) minutes rounds to \(90\) minutes to the nearest ten minutes.

Answer

- about \(400\) spectators - about \(190\) hot dogs - about \(\$600\) in revenue - about \(90\) minutes

5544644

Use the number line to round point \(B\) to the nearest thousand.

Figure for problem 554464

Hints

- Read the marker value using the \(100\)-unit tick spacing. - Locate the halfway point between the two nearest thousands. - Decide which thousand is closer to the marker.

Solution

1. Point \(B\) is at \(62{,}400\). 2. The midpoint between \(62{,}000\) and \(63{,}000\) is \(62{,}500\). 3. Since \(62{,}400<62{,}500\), the number rounds to \(62{,}000\).

Answer

\(62{,}000\)

5544654

Use the number line. Round point \(C\) to the nearest ten-thousand.

Figure for problem 554465

Hints

- Read the value of point \(C\) from the labeled ticks. - Identify the two ten-thousands between which the point lies. - Apply the midpoint rounding rule at this place value.

Solution

1. The midpoint is \(345{,}000\). 2. A midpoint rounds to the greater ten-thousand. 3. Therefore, \(345{,}000\) rounds to \(350{,}000\).

Answer

\(350{,}000\)

5544674

Point \(E\) is placed on one of the unlabeled ticks of the number line. a) What value does point \(E\) represent? b) Explain why that value rounds to \(82{,}000\) to the nearest thousand.

Figure for problem 554467

Hints

- Use the tick spacing to count from the labeled left endpoint to \(E\). - Compare the marker value with the midpoint between the two labeled thousands.

Solution

1. The line runs from \(81{,}000\) to \(82{,}000\) in steps of \(100\). 2. Point \(E\) is five \(100\)-steps to the right of \(81{,}000\), so \(E=81{,}500\). 3. \(81{,}500\) is exactly halfway between \(81{,}000\) and \(82{,}000\), so it rounds up to \(82{,}000\).

Answer

a) \(81{,}500\) b) It is the midpoint between \(81{,}000\) and \(82{,}000\), so it rounds up to \(82{,}000\).

5159324

For each equation, round every number on the left to the nearest ten and write the resulting estimate. Use that estimate to decide whether the stated result is plausible. Then calculate an exact correction only for equations your estimate flags as implausible. a) \(423+371=794\) b) \(582+129=611\) c) \(864-452=412\) d) \(735-198=637\)

Hints

- Round the numbers to nearby tens or hundreds. - Compare each estimate with the stated result. - Calculate exactly when a stated result is far from the estimate.

Solution

1. a) Nearest-ten estimate: \(420+370=790\). The stated \(794\) is plausible, and the equation is correct. 2. b) Nearest-ten estimate: \(580+130=710\). The stated \(611\) is far from the estimate; exact calculation gives \(711\). 3. c) Nearest-ten estimate: \(860-450=410\). The stated \(412\) is plausible, and the equation is correct. 4. d) Nearest-ten estimate: \(740-200=540\). The stated \(637\) is far from the estimate; exact calculation gives \(537\).

Answer

a) Estimate: \(420+370=790\); plausible and correct. b) Estimate: \(580+130=710\); \(611\) is implausible. Exact correction: \(582+129=711\). c) Estimate: \(860-450=410\); plausible and correct. d) Estimate: \(740-200=540\); \(637\) is implausible. Exact correction: \(735-198=537\).

5159954

Use the numbers in this list: \(234\), \(156\), \(318\), \(89\), \(427\), \(112\), \(281\), \(65\) a) Choose three different pairs whose sums are between \(300\) and \(400\). For every pair, first write the nearest-ten rounded expression and its estimated sum, then write the exact equation. b) Choose two different pairs whose sums are greater than \(600\). Again give the nearest-ten rounded expression before the exact equation.

Hints

- Round each number to the nearest ten. - Compare the hundreds and tens to find pairs near the target range. - Check every selected pair with exact addition.

Solution

1. Round candidate pairs to the nearest ten before checking them exactly. 2. For part a, examples are \(230+160=390\), \(320+70=390\), and \(280+110=390\); the corresponding exact sums are \(390\), \(383\), and \(393\). 3. For part b, examples are \(430+230=660\) and \(430+320=750\); the corresponding exact sums are \(661\) and \(745\).

Answer

a) Possible answers: \(230+160=390\); \(234+156=390\) \(320+70=390\); \(318+65=383\) \(280+110=390\); \(281+112=393\) b) Possible answers: \(430+230=660\); \(427+234=661\) \(430+320=750\); \(427+318=745\)

5159964

Choose three different combinations of three numbers from the list whose exact sum is less than \(500\): \(145\), \(212\), \(88\), \(334\), \(176\), \(59\), \(287\). For each combination, first round every addend to the nearest ten and give the estimated sum. Then calculate the exact sum.

Hints

- Start with one of the smaller numbers. - Estimate how much room remains below \(500\) after choosing two numbers. - Consider which small numbers could be paired with \(334\).

Solution

1. Round each candidate combination to the nearest ten before calculating exactly. 2. \(145,212,88\) estimate to \(150+210+90=450\), and the exact sum is \(445\). 3. \(212,88,59\) estimate to \(210+90+60=360\), and the exact sum is \(359\). 4. \(145,176,59\) estimate to \(150+180+60=390\), and the exact sum is \(380\).

Answer

One possible set is: 1. Estimate: \(150+210+90=450\); exact: \(145+212+88=445\). 2. Estimate: \(210+90+60=360\); exact: \(212+88+59=359\). 3. Estimate: \(150+180+60=390\); exact: \(145+176+59=380\).

5159974

The number cards show \(367\), \(128\), \(452\), \(215\), \(94\), and \(503\). Use nearest-ten estimates to nominate the two most promising pairs for a sum closest to \(600\). Write both rounded expressions. Then calculate exact sums for only those two pairs and decide which is closer to \(600\).

Hints

- Round the numbers to nearby tens or hundreds. - Look for pairs with an estimated sum near \(600\). - Remember that \(94\) is close to \(100\).

Solution

1. Round to the nearest ten. Two strong candidates are \(503,94\), which estimate to \(500+90=590\), and \(367,215\), which estimate to \(370+220=590\). 2. Calculate only those candidate sums: \(503+94=597\) and \(367+215=582\). 3. Their distances from \(600\) are \(3\) and \(18\), respectively, so \(503\) and \(94\) are closest.

Answer

Promising estimates: \(500+90=590\) for \(503\) and \(94\). \(370+220=590\) for \(367\) and \(215\). Exact sums: \(503+94=597\) and \(367+215=582\). The pair \(503\) and \(94\) is closest; \(597\) is \(3\) from \(600\).

5164384

Consider the sum \(237 + 142 + 319\). a) Estimate the sum by rounding each addend to the nearest hundred. b) Estimate the sum by rounding each addend to the nearest ten. c) Find the exact sum. Which estimate is closer to the exact sum?

Hints

- Round each addend separately. - Find both estimates before calculating the exact sum. - Compare each estimate’s distance from the exact sum.

Solution

1. For part a, round to hundreds: \(200 + 100 + 300 = 600\). 2. For part b, round to tens: \(240 + 140 + 320 = 700\). 3. The exact sum is \(237 + 142 + 319 = 698\). 4. The hundreds estimate is \(98\) away from the exact sum, while the tens estimate is \(2\) away. Therefore, the estimate made by rounding to tens is closer.

Answer

a) \(600\) b) \(700\) c) The exact sum is \(698\). The estimate made by rounding to tens, \(700\), is closer.

5164424

Lara calculated \(342 + 189 + 211\) and got \(942\). a) Round each addend to the nearest hundred to decide whether Lara’s result is reasonable. b) Find the exact sum.

Hints

- Round each number to its nearest hundred. - A stated result far from the estimate is likely incorrect. - Add the original numbers to find the exact sum.

Solution

1. Round to hundreds: \(300 + 200 + 200 = 700\). 2. Since \(942\) is far from the estimate \(700\), Lara’s result is not reasonable. 3. The exact sum is \(342 + 189 + 211 = 742\).

Answer

a) The estimate is \(700\), so Lara’s result of \(942\) is not reasonable. b) The exact sum is \(742\).

5164434

Lucas rounds \(678\) and \(243\) to the nearest hundred. Mia rounds the same numbers to the nearest ten. a) Write Lucas’s rounded subtraction and estimate. b) Write Mia’s rounded subtraction and estimate. c) Find the exact difference. Which estimate is closer?

Hints

- Calculate the exact difference first. - Find the distance from each estimate to the exact result.

Solution

1. Nearest hundred: \(678\to700\) and \(243\to200\), so Lucas estimates \(500\). 2. Nearest ten: \(678\to680\) and \(243\to240\), so Mia estimates \(440\). 3. The exact difference is \(678-243=435\). 4. \(440\) is \(5\) away, while \(500\) is \(65\) away, so Mia’s estimate is closer.

Answer

a) \(700-200=500\) b) \(680-240=440\) c) Exact difference: \(435\). Mia’s estimate of \(440\) is closer.

5164444

Round each addend in \(214+187+392\) to the nearest hundred and write the estimated sum. Which answer choice is within \(50\) of that estimate? A: \(593\) B: \(793\) C: \(993\)

Hints

- Round each addend to the nearest hundred. - Compare the estimated sum with all three choices. - Choose the result closest to your estimate.

Solution

1. Round to the nearest hundred: \(214\to200\), \(187\to200\), and \(392\to400\). 2. The estimate is \(200+200+400=800\). 3. Of the choices, \(793\) is within \(50\) of \(800\).

Answer

Rounded expression: \(200+200+400=800\). Choice B, \(793\), is within \(50\) of the estimate.

5164754

For each equation, round every number on the left to the nearest ten and write an estimated result. Use the estimate to identify the two implausible stated results. Then calculate exact corrections only for those two equations. \(204+398+115=717\) \(145+236+412=593\) \(321+188+256=765\) \(478+52+311=941\)

Hints

- Round to nearby tens or hundreds to test whether each result is reasonable. - Compare the leading place values with the stated result. - Use the standard algorithm to calculate the suspected errors exactly.

Solution

1. Round each addend to the nearest ten and add the rounded values. 2. The four estimates are \(720\), \(800\), \(770\), and \(840\). 3. The stated results \(593\) and \(941\) are far from their estimates, so those equations are the two implausible ones. 4. Exact corrections are \(793\) and \(841\).

Answer

Estimates: \(200+400+120=720\): \(717\) is plausible. \(150+240+410=800\): \(593\) is implausible; exact correction \(145+236+412=793\). \(320+190+260=770\): \(765\) is plausible. \(480+50+310=840\): \(941\) is implausible; exact correction \(478+52+311=841\).

5164764

For each equation, round every number on the left to the nearest ten and write an estimated result. Use the estimate to identify the two implausible stated results. Then calculate exact corrections only for those two equations. \(987-456-123=408\) \(742-219-88=635\) \(654-321-145=188\) \(833-175-294=164\)

Hints

- Use the hundreds digits for a quick first estimate. - Round to nearby tens for a more precise estimate. - Calculate corrections from left to right.

Solution

1. Round all values to the nearest ten before computing exactly. 2. The estimates are \(410\), \(430\), \(180\), and \(360\). 3. The second and fourth stated results are inconsistent with those estimates. 4. Exact calculation gives corrected results \(435\) and \(364\).

Answer

Estimates: \(990-460-120=410\): \(408\) is plausible. \(740-220-90=430\): \(635\) is implausible; exact correction \(742-219-88=435\). \(650-320-150=180\): \(188\) is plausible. \(830-180-290=360\): \(164\) is implausible; exact correction \(833-175-294=364\).

5164774

For each equation, round every number on the left to the nearest ten and write an estimated result. Use the estimate to identify the two implausible stated results. Then calculate exact corrections only for those two equations. \(127+258+369=754\) \(890-455-212=423\) \(333+333+333=699\) \(421-105-99=217\)

Hints

- Describe what each equation is doing before estimating. - Round to nearby hundreds or tens to test whether the result is reasonable. - Compare each estimate with the stated result. - In the third equation, notice that there are three addends near \(300\).

Solution

1. Round each value to the nearest ten. 2. The estimates are \(760\), \(220\), \(990\), and \(210\). 3. The second and third stated results are inconsistent with those estimates. 4. Their exact corrected results are \(223\) and \(999\).

Answer

Estimates: \(130+260+370=760\): \(754\) is plausible. \(890-460-210=220\): \(423\) is implausible; exact correction \(890-455-212=223\). \(330+330+330=990\): \(699\) is implausible; exact correction \(333+333+333=999\). \(420-110-100=210\): \(217\) is plausible.

5165874

A news report says, “About \(45{,}000\) people attended the outdoor concert.” Which exact attendance numbers round to \(45{,}000\) to the nearest thousand? \(44{,}499\), \(45{,}499\), \(45{,}500\), \(44{,}500\), \(45{,}049\)

Hints

- Round each choice separately to the nearest thousand. - Look at the hundreds digit to decide whether to round down or up. - Remember that a number below \(45{,}000\) can still round to \(45{,}000\).

Solution

1. \(44{,}499\) rounds to \(44{,}000\) because the hundreds digit is \(4\). 2. \(45{,}499\) rounds to \(45{,}000\) because the hundreds digit is \(4\). 3. \(45{,}500\) rounds to \(46{,}000\) because the hundreds digit is \(5\). 4. \(44{,}500\) rounds to \(45{,}000\), and \(45{,}049\) rounds to \(45{,}000\). 5. Therefore, the matching numbers are \(45{,}499\), \(44{,}500\), and \(45{,}049\).

Answer

\(45{,}499\), \(44{,}500\), and \(45{,}049\)

5165894

A national swim organization reports its youth membership by age group and region. <table> <tr><th>Age group</th><th>East region</th><th>West region</th></tr> <tr><td>Ages 14 and under</td><td>\(124{,}389\)</td><td>\(135{,}612\)</td></tr> <tr><td>Ages 15–18</td><td>\(42{,}705\)</td><td>\(38{,}490\)</td></tr> </table> Round all four numbers to the nearest thousand. About how many members under age \(19\) are there in all?

Hints

- Use the hundreds digit to round each value to the nearest thousand. - Include all four values because both age groups are under \(19\). - Add only after rounding.

Solution

1. Round the four values: \(124{,}389 \approx 124{,}000\), \(135{,}612 \approx 136{,}000\), \(42{,}705 \approx 43{,}000\), and \(38{,}490 \approx 38{,}000\). 2. Add the rounded values for ages \(14\) and under: \(124{,}000 + 136{,}000 = 260{,}000\). 3. Add the rounded values for ages \(15\)–\(18\): \(43{,}000 + 38{,}000 = 81{,}000\). 4. Add the two age-group totals: \(260{,}000 + 81{,}000 = 341{,}000\).

Answer

There are about \(341{,}000\) members under age \(19\).

5165904

A national sports organization compares membership in two sports. <table> <tr><th>Sport</th><th>Ages 14 and under</th><th>Over age 14</th></tr> <tr><td>Table tennis</td><td>\(85{,}430\)</td><td>\(421{,}280\)</td></tr> <tr><td>Badminton</td><td>\(52{,}910\)</td><td>\(118{,}750\)</td></tr> </table> Round the membership numbers for people over age \(14\) to the nearest thousand. About how many more members are in table tennis than in badminton?

Hints

- Use only the “Over age 14” column. - Round each value before subtracting. - “How many more” asks for the difference.

Solution

1. The relevant values are \(421{,}280\) for table tennis and \(118{,}750\) for badminton. 2. Round to the nearest thousand: \(421{,}280 \approx 421{,}000\) and \(118{,}750 \approx 119{,}000\). 3. Find the difference: \(421{,}000 - 119{,}000 = 302{,}000\).

Answer

There are about \(302{,}000\) more table tennis members.

5165914

A national music program reports enrollment in three instruments. <table> <tr><th>Instrument</th><th>Ages 14 and under</th><th>Ages 15–18</th></tr> <tr><td>Piano</td><td>\(145{,}230\)</td><td>\(56{,}890\)</td></tr> <tr><td>Guitar</td><td>\(98{,}450\)</td><td>\(41{,}210\)</td></tr> <tr><td>Flute</td><td>\(112{,}780\)</td><td>\(23{,}450\)</td></tr> </table> Round every number to the nearest thousand. About how many students are enrolled in these three instruments altogether?

Hints

- Round all six values before adding. - Organize the rounded values by row or column so none are missed. - Use \(\approx\) because the total is based on rounded values.

Solution

1. Round the younger-group values: \(145{,}230 \approx 145{,}000\), \(98{,}450 \approx 98{,}000\), and \(112{,}780 \approx 113{,}000\). 2. Round the older-group values: \(56{,}890 \approx 57{,}000\), \(41{,}210 \approx 41{,}000\), and \(23{,}450 \approx 23{,}000\). 3. Add the rounded younger-group values: \(145{,}000 + 98{,}000 + 113{,}000 = 356{,}000\). 4. Add the rounded older-group values: \(57{,}000 + 41{,}000 + 23{,}000 = 121{,}000\). 5. Add the two totals: \(356{,}000 + 121{,}000 = 477{,}000\).

Answer

About \(477{,}000\) students are enrolled altogether.

5166194

A state measured its forest area in three years. <table> <tr><td>Year</td><td>Forest area in acres</td></tr> <tr><td>2000</td><td>\(45{,}287\)</td></tr> <tr><td>2010</td><td>\(52{,}814\)</td></tr> <tr><td>2020</td><td>\(58{,}490\)</td></tr> </table> a) Round each area to the nearest thousand acres. b) Using the rounded values, find the increase from 2000 to 2010 and from 2010 to 2020.

Hints

- Use the hundreds digit to round to the nearest thousand. - Find an increase by subtracting the earlier rounded value from the later rounded value. - Use only the rounded values in part b.

Solution

1. Round to the nearest thousand: \(45{,}287 \approx 45{,}000\), \(52{,}814 \approx 53{,}000\), and \(58{,}490 \approx 58{,}000\). 2. From 2000 to 2010, the increase is \(53{,}000 - 45{,}000 = 8000\) acres. 3. From 2010 to 2020, the increase is \(58{,}000 - 53{,}000 = 5000\) acres.

Answer

a) \(45{,}000\) acres, \(53{,}000\) acres, and \(58{,}000\) acres b) \(8000\) acres from 2000 to 2010; \(5000\) acres from 2010 to 2020

5166204

A public pool recorded its summer attendance. <table> <tr><td>Month</td><td>Visitors</td></tr> <tr><td>June</td><td>\(18{,}432\)</td></tr> <tr><td>July</td><td>\(25{,}918\)</td></tr> <tr><td>August</td><td>\(21{,}045\)</td></tr> </table> a) Round each attendance number to the nearest hundred. b) Using the rounded values, find the difference between June and July. c) Using the rounded values, how many fewer visitors came in August than in July?

Hints

- To round to the nearest hundred, look at the tens digit. - Use your rounded values for both subtraction problems. - “Difference” and “how many fewer” both call for subtraction.

Solution

1. Round to the nearest hundred: \(18{,}432 \approx 18{,}400\), \(25{,}918 \approx 25{,}900\), and \(21{,}045 \approx 21{,}000\). 2. The difference between June and July is \(25{,}900 - 18{,}400 = 7500\). 3. The difference between July and August is \(25{,}900 - 21{,}000 = 4900\).

Answer

a) June: \(18{,}400\); July: \(25{,}900\); August: \(21{,}000\) b) \(7500\) visitors c) \(4900\) fewer visitors

5166214

A large shipping center recorded the number of packages sent during three reporting periods. <table> <tr><td>Period</td><td>Packages</td></tr> <tr><td>Period 1</td><td>\(124{,}560\)</td></tr> <tr><td>Period 2</td><td>\(156{,}210\)</td></tr> <tr><td>Period 3</td><td>\(148{,}890\)</td></tr> </table> a) Round each package count to the nearest ten thousand. b) Using the rounded values, find the difference between Periods 1 and 2 and between Periods 2 and 3.

Hints

- To round to the nearest ten thousand, look at the thousands digit. - Identify the greater rounded value before subtracting. - Keep the rounded values organized by period.

Solution

1. Round to the nearest ten thousand: \(124{,}560 \approx 120{,}000\), \(156{,}210 \approx 160{,}000\), and \(148{,}890 \approx 150{,}000\). 2. The difference between Periods 1 and 2 is \(160{,}000 - 120{,}000 = 40{,}000\). 3. The difference between Periods 2 and 3 is \(160{,}000 - 150{,}000 = 10{,}000\).

Answer

a) \(120{,}000\), \(160{,}000\), and \(150{,}000\) b) \(40{,}000\) packages and \(10{,}000\) packages

5166384

Two cities compare their populations. Northfield has \(425{,}670\) residents, and Southbrook has \(421{,}340\) residents. 1) Round both populations to the nearest thousand. 2) Find the difference between the rounded populations. 3) Which city has the greater rounded population?

Hints

- Use the hundreds digit to round each population. - Find the difference by subtracting the lesser rounded value from the greater one. - Compare the rounded values for part 3.

Solution

1. \(425{,}670\) rounds to \(426{,}000\) because the hundreds digit is \(6\). \(421{,}340\) rounds to \(421{,}000\) because the hundreds digit is \(3\). 2. The difference is \(426{,}000 - 421{,}000 = 5000\). 3. Since \(426{,}000 > 421{,}000\), Northfield has the greater rounded population.

Answer

1) Northfield: \(426{,}000\); Southbrook: \(421{,}000\) 2) \(5000\) residents 3) Northfield

5166394

Three schools raised money in a charity run. Oak Valley Elementary: \(\$124{,}560\) Harbor View School: \(\$132{,}100\) Cedar Ridge Academy: \(\$127{,}890\) 1) Round each amount to the nearest thousand dollars. 2) Find the total of the three rounded amounts. 3) Is the total greater than or less than \(\$380{,}000\)?

Hints

- Round each amount before adding. - “Total” means add all three rounded amounts. - Compare the total with \(\$380{,}000\) from left to right.

Solution

1. The rounded amounts are \(\$125{,}000\), \(\$132{,}000\), and \(\$128{,}000\). 2. Add the rounded amounts: \(125{,}000 + 132{,}000 + 128{,}000 = 385{,}000\), so the total is \(\$385{,}000\). 3. Since \(385{,}000 > 380{,}000\), the total is greater than \(\$380{,}000\).

Answer

1) \(\$125{,}000\), \(\$132{,}000\), and \(\$128{,}000\) 2) \(\$385{,}000\) 3) Greater than \(\$380{,}000\)

5173614

Round each number to the indicated place. Pay attention to regrouping across place values. a) \(2996\) to the nearest ten b) \(8049\) to the nearest hundred c) \(14{,}995\) to the nearest ten

Hints

- Identify the digit immediately to the right of the rounding place. - When rounding up a digit of \(9\), regrouping may affect several places. - Check that all digits to the right of the rounding place become \(0\).

Solution

1. a) The ones digit is \(6\), so \(2996\) rounds up to \(3000\). 2. b) The tens digit is \(4\), so \(8049\) rounds down to \(8000\). 3. c) The ones digit is \(5\), so \(14{,}995\) rounds up to \(15{,}000\).

Answer

a) \(3000\) b) \(8000\) c) \(15{,}000\)

5173624

Consider the numbers \(645\), \(654\), and \(704\). a) Which numbers round to \(650\) to the nearest ten? b) Which numbers round to \(700\) to the nearest hundred? c) Which number has the same result when rounded to the nearest ten and nearest hundred?

Hints

- Round all three numbers to both places and record the results. - Compare the results with the target values. - For part c), find the row where the two rounded values match.

Solution

1. To the nearest ten: \(645\to650\), \(654\to650\), and \(704\to700\). Thus, \(645\) and \(654\) answer part a). 2. To the nearest hundred: \(645\to600\), \(654\to700\), and \(704\to700\). Thus, \(654\) and \(704\) answer part b). 3. Only \(704\) rounds to \(700\) both ways.

Answer

a) \(645\) and \(654\) b) \(654\) and \(704\) c) \(704\)

5173664

A news report about a music festival says: “On Friday, \(34{,}562\) fans attended. That was about \(2000\) more people than on Thursday. Therefore, exactly \(32{,}562\) people attended on Thursday.” Explain why the word “exactly” is not justified.

Hints

- Compare the meanings of “about” and “exactly.” - What happens to the precision of a result when one input is approximate?

Solution

1. “About \(2000\)” is an approximate difference, not an exact difference. 2. Subtracting an approximate amount from an exact count gives only an approximate result. 3. The report may estimate Thursday’s attendance as about \(32{,}562\), but it cannot claim that count is exact.

Answer

The difference is only approximate, so Thursday’s attendance can only be estimated, not stated as exactly \(32{,}562\).

5173734

Complete the table by rounding each number to the indicated place. <table> <tr> <th>Number</th> <th>Nearest hundred</th> <th>Nearest thousand</th> <th>Nearest ten-thousand</th> <th>Nearest hundred-thousand</th> </tr> <tr> <td>\(237{,}845\)</td> <td></td> <td></td> <td></td> <td></td> </tr> <tr> <td>\(845{,}298\)</td> <td></td> <td></td> <td></td> <td></td> </tr> </table>

Hints

- For each column, identify the digit immediately to the right of the rounding place. - Decide separately whether to round up or down for every place. - Larger rounding places produce less precise results.

Solution

1. For \(237{,}845\): nearest hundred \(237{,}800\); nearest thousand \(238{,}000\); nearest ten-thousand \(240{,}000\); nearest hundred-thousand \(200{,}000\). 2. For \(845{,}298\): nearest hundred \(845{,}300\); nearest thousand \(845{,}000\); nearest ten-thousand \(850{,}000\); nearest hundred-thousand \(800{,}000\).

Answer

First row: \(237{,}800\); \(238{,}000\); \(240{,}000\); \(200{,}000\) Second row: \(845{,}300\); \(845{,}000\); \(850{,}000\); \(800{,}000\)

5173744

Use the number \(79{,}995\). a) Round to the nearest ten. b) Round to the nearest hundred. c) Round to the nearest thousand. d) Round to the nearest ten-thousand. What do you notice about the results?

Hints

- Regrouping may continue through several places when rounding up. - Complete each rounding calculation separately and compare the results.

Solution

1. To the nearest ten, the ones digit is \(5\), so the value rounds up to \(80{,}000\). 2. To the nearest hundred, the tens digit is \(9\), so the value rounds up to \(80{,}000\). 3. To the nearest thousand, the hundreds digit is \(9\), so the value rounds up to \(80{,}000\). 4. To the nearest ten-thousand, the thousands digit is \(9\), so the value rounds up to \(80{,}000\). 5. All four rounded values are the same.

Answer

a) \(80{,}000\) b) \(80{,}000\) c) \(80{,}000\) d) \(80{,}000\) All results are equal.

5173764

A whole number rounds to \(45{,}000\) to the nearest thousand. Find the least possible number and the greatest possible number.

Hints

- Find the halfway point between \(44{,}000\) and \(45{,}000\). - Find the last number before the halfway point to \(46{,}000\). - A hundreds digit of \(5\) is the first that rounds up.

Solution

1. The least number that rounds up to \(45{,}000\) is \(44{,}500\). 2. The greatest number that rounds down to \(45{,}000\) is \(45{,}499\).

Answer

Least possible number: \(44{,}500\) Greatest possible number: \(45{,}499\)

5173774

Consider \(245{,}499\) and \(245{,}500\). a) Round both numbers to the nearest thousand. b) Round both numbers to the nearest ten-thousand.

Hints

- To round to the nearest thousand, look at the hundreds digit. - To round to the nearest ten-thousand, look at the thousands digit. - Compare the deciding digits in the two numbers.

Solution

1. a) The hundreds digit in \(245{,}499\) is \(4\), so it rounds to \(245{,}000\). The hundreds digit in \(245{,}500\) is \(5\), so it rounds to \(246{,}000\). 2. b) Both numbers have \(5\) in the thousands place, so both round to \(250{,}000\).

Answer

a) \(245{,}499\to245{,}000\); \(245{,}500\to246{,}000\) b) \(245{,}499\to250{,}000\); \(245{,}500\to250{,}000\)

5173804

Use the number \(849{,}502\). Round it to each place. a) nearest thousand b) nearest ten-thousand c) nearest hundred-thousand d) nearest million

Hints

- Identify the digit immediately to the right of each rounding place. - Round down for \(0\) through \(4\) and up for \(5\) through \(9\). - Regroup when rounding up changes a \(9\).

Solution

1. a) The hundreds digit is \(5\), so \(849{,}502\) rounds to \(850{,}000\). 2. b) The thousands digit is \(9\), so it rounds to \(850{,}000\). 3. c) The ten-thousands digit is \(4\), so it rounds to \(800{,}000\). 4. d) The hundred-thousands digit is \(8\), so it rounds to \(1{,}000{,}000\).

Answer

a) \(850{,}000\) b) \(850{,}000\) c) \(800{,}000\) d) \(1{,}000{,}000\)

5173824

A concert report says, “About \(40{,}000\) people attended.” The number was rounded to the nearest ten-thousand. What is the least possible exact attendance? What is the greatest possible exact attendance?

Hints

- To round to the nearest ten-thousand, look at the thousands digit. - What is the first number that rounds up to \(40{,}000\)? - What is the last number before values begin rounding to \(50{,}000\)?

Solution

1. The least whole number that rounds up to \(40{,}000\) is \(35{,}000\). 2. The greatest whole number that rounds down to \(40{,}000\) is \(44{,}999\).

Answer

Least possible attendance: \(35{,}000\) Greatest possible attendance: \(44{,}999\)

5173924

A city has \(45{,}812\) residents. 1. Round the population to the nearest thousand and to the nearest ten-thousand. 2. The city is planning a community festival. Officials estimate that about one out of every ten residents will attend. Use the population rounded to the nearest thousand to estimate the number of visitors.

Hints

- Which digit should you check when rounding to the nearest thousand? - What operation represents “one out of every ten”? - What happens to a whole number when you divide it by \(10\)?

Solution

1. To round to the nearest thousand, look at the hundreds digit. It is \(8\), so \(45{,}812\) rounds up to \(46{,}000\). To round to the nearest ten-thousand, look at the thousands digit. It is \(5\), so the number rounds up to \(50{,}000\). 2. Use the population rounded to the nearest thousand: \(46{,}000\div10=4600\). The city should expect about \(4600\) visitors.

Answer

1. Nearest thousand: \(46{,}000\); nearest ten-thousand: \(50{,}000\) 2. About \(4600\) visitors

5173974

The attendance at a school festival was rounded to the nearest ten and reported as \(1200\). What whole numbers could the actual attendance be? Give the least and greatest possible numbers.

Hints

- What is the first number below \(1200\) that rounds up to it? - What is the last number above \(1200\) that still rounds down to it? - Use the halfway point between consecutive tens.

Solution

1. The least whole number that rounds to \(1200\) is \(1195\), because a ones digit of \(5\) rounds up. 2. The greatest whole number that rounds to \(1200\) is \(1204\), because a ones digit of \(4\) still rounds down. 3. Therefore, every whole number from \(1195\) through \(1204\) is possible.

Answer

Least: \(1195\); greatest: \(1204\)

5173994

The exact cost of renovating a school athletic field was rounded to the nearest ten-thousand dollars. A news report listed the cost as about \(\$460{,}000\). Find the least and greatest whole-number costs that could round to \(\$460{,}000\) to the nearest ten-thousand dollars.

Hints

- Find the halfway point between \(450{,}000\) and \(460{,}000\). - Find the last whole-dollar amount before the halfway point between \(460{,}000\) and \(470{,}000\). - Include both endpoints that round to the reported amount.

Solution

1. The halfway point below \(460{,}000\) is \(455{,}000\), so \(455{,}000\) is the least amount that rounds up to \(460{,}000\). 2. The halfway point above \(460{,}000\) is \(465{,}000\). Therefore, \(464{,}999\) is the greatest amount that still rounds down to \(460{,}000\).

Answer

Least: \(\$455{,}000\); greatest: \(\$464{,}999\)

5174004

A school newspaper recorded the exact numbers of raffle tickets sold at three school events: - Fall festival: \(1244\) tickets - Winter fair: \(876\) tickets - Field day: \(1055\) tickets The headline will show each amount rounded to the nearest hundred. a) Round each number to the nearest hundred. b) Find the sum of the exact numbers and the sum of the rounded numbers. What is the difference between the two totals?

Hints

- To round to the nearest hundred, look at the tens digit. - Add the three exact amounts, then add the three rounded amounts. - Subtract the smaller total from the larger total to find the difference.

Solution

1. a) The rounded amounts are \(1244\approx1200\), \(876\approx900\), and \(1055\approx1100\). 2. b) The exact total is \(1244+876+1055=3175\). 3. The rounded total is \(1200+900+1100=3200\). 4. The difference is \(3200-3175=25\).

Answer

a) Fall festival: \(1200\); winter fair: \(900\); field day: \(1100\) b) Exact total: \(3175\); rounded total: \(3200\); difference: \(25\)

5174014

Exactly \(4837\) tickets were sold for an outdoor concert. Three newspapers report the ticket sales in different ways: - Metro Ledger: “Nearly \(5000\) tickets sold.” - Daily Beacon: “About \(4800\) tickets sold.” - City Chronicle: “About \(4840\) tickets sold.” a) To which place did each newspaper round? b) Which report is most precise? c) Why might the Metro Ledger choose \(5000\), even though it is farthest from the exact number?

Hints

- Look at the ending digits and zeros in each reported number. - Compare how far each reported number is from \(4837\). - Think about why a headline might favor a simple number over greater precision.

Solution

1. a) The Metro Ledger rounded to the nearest thousand, the Daily Beacon rounded to the nearest hundred, and the City Chronicle rounded to the nearest ten. 2. b) Rounding to the nearest ten changes the number the least, so the City Chronicle is most precise. 3. c) A large, rounded number such as \(5000\) is easy to remember and may make a headline sound more dramatic.

Answer

a) Metro Ledger: nearest thousand; Daily Beacon: nearest hundred; City Chronicle: nearest ten b) City Chronicle c) The rounded number \(5000\) is memorable and may make the headline sound more dramatic.

5174144

Lucas and Sarah are discussing how to round \(447\) to the nearest hundred. Lucas says, “I first round to the nearest ten to get \(450\), and then I round to the nearest hundred to get \(500\).” Sarah says, “I look at the tens digit of \(447\). It is \(4\), so I round down to \(400\).” a) Decide who gets the correct result. Explain using the rounding rule. b) Give another three-digit number for which Lucas’s method gives a different result from Sarah’s method.

Hints

- Which digit determines how to round directly to the nearest hundred? - Compare direct rounding with rounding in two stages. - For part b, think about when the first rounding step could move a number across a midpoint between hundreds.

Solution

1. a) Sarah is correct. To round to the nearest hundred, look directly at the tens digit. The tens digit in \(447\) is \(4\), so \(447\) rounds down to \(400\). 2. Lucas rounds in stages. The first rounding changes \(447\) to \(450\), which then rounds up to \(500\). This does not give the same result as rounding the original number directly to the nearest hundred. 3. b) Any three-digit number whose first rounding to the nearest ten moves it across a hundreds midpoint can work. For example, \(145\) rounds directly to \(100\), but rounding first to \(150\) and then to the nearest hundred gives \(200\).

Answer

a) Sarah is correct. \(447\) rounds directly to \(400\) because its tens digit is \(4\). b) One possible answer is \(145\).

5177174

A town has exactly \(4328\) residents. A local newspaper reports the population as “about \(4300\).” A regional information website reports it as “about \(4000\).” a) To which place did the newspaper round? To which place did the website round? b) Give one reason the website might use a less precise number than the newspaper.

Hints

- Look at how many zeros appear at the end of each rounded number. - Identify which digits in \(4328\) were replaced by zeros. - Think about the different audiences and purposes of a local newspaper and a regional website.

Solution

1. a) The newspaper rounded \(4328\) to the nearest hundred to get \(4300\). The website rounded \(4328\) to the nearest thousand to get \(4000\). 2. b) A regional website may only need a broad description that remains useful even when the town’s population changes slightly. A local newspaper may use newer, more precise information.

Answer

a) Newspaper: nearest hundred; website: nearest thousand b) The website may use a broad estimate that stays useful longer as the population changes.

5177184

A trail guide says, “The trail to the next shelter is about \(12{,}400\,\text{ft}\) long.” Could the actual trail length be \(12{,}452\,\text{ft}\) if the guide rounded to the nearest hundred feet? Explain.

Hints

- Which digit determines how to round to the nearest hundred? - Round \(12{,}452\) to the nearest hundred. - Compare your result with \(12{,}400\).

Solution

1. To round to the nearest hundred, look at the tens digit. 2. The tens digit in \(12{,}452\) is \(5\), so the number rounds up. 3. Therefore, \(12{,}452\,\text{ft}\) rounds to \(12{,}500\,\text{ft}\), not \(12{,}400\,\text{ft}\).

Answer

No. \(12{,}452\,\text{ft}\) rounds to \(12{,}500\,\text{ft}\) to the nearest hundred feet.

5177234

Use the number \(80{,}000\) to answer the questions. a) Find the least and greatest whole numbers that round to \(80{,}000\) to the nearest ten-thousand. b) Find the least and greatest whole numbers that round to \(80{,}000\) to the nearest thousand. c) How many whole numbers round to \(80{,}000\) to the nearest thousand?

Hints

- Find the halfway points between consecutive multiples of \(10{,}000\). - Then find the halfway points between consecutive multiples of \(1000\). - Count an inclusive range by subtracting the endpoints and adding \(1\).

Solution

1. a) The whole numbers from \(75{,}000\) through \(84{,}999\) round to \(80{,}000\) to the nearest ten-thousand. 2. b) The whole numbers from \(79{,}500\) through \(80{,}499\) round to \(80{,}000\) to the nearest thousand. 3. c) Count the second range inclusively: \(80{,}499-79{,}500+1=1000\).

Answer

a) Least: \(75{,}000\); greatest: \(84{,}999\) b) Least: \(79{,}500\); greatest: \(80{,}499\) c) \(1000\) whole numbers

5544664

Point \(D\) is shown on the number line. a) Round \(D\) to the nearest hundred. b) Round \(D\) to the nearest thousand.

Figure for problem 554466

Hints

- Determine the marker value from the tick spacing. - The midpoint depends on which place you are rounding to. - Treat the nearest-hundred and nearest-thousand questions separately.

Solution

1. The ticks show that \(D=47{,}650\). 2. To the nearest hundred, \(47{,}650\) is exactly halfway between \(47{,}600\) and \(47{,}700\), so it rounds to \(47{,}700\). 3. To the nearest thousand, \(47{,}650\) is above the midpoint \(47{,}500\), so it rounds to \(48{,}000\).

Answer

a) \(47{,}700\) b) \(48{,}000\)

5104704

A newspaper reports that about \(40{,}000\) people attended a concert. A blog reports about \(38{,}000\) people at the same concert. Explain how both statements could be true if they used different rounding places. Give one possible exact attendance.

Hints

- Find the rounding interval for \(40{,}000\) to the nearest ten thousand. - Find the rounding interval for \(38{,}000\) to the nearest thousand. - Choose a whole number in the overlap.

Solution

1. The newspaper may have rounded to the nearest ten thousand. Values from \(35{,}000\) through \(44{,}999\) round to \(40{,}000\). 2. The blog may have rounded to the nearest thousand. Values from \(37{,}500\) through \(38{,}499\) round to \(38{,}000\). 3. Every whole number from \(37{,}500\) through \(38{,}499\) satisfies both conditions. One possible exact attendance is \(38{,}200\).

Answer

The newspaper could have rounded to the nearest ten thousand, while the blog rounded to the nearest thousand. One possible exact attendance is \(38{,}200\).

5173794

Find every digit that can replace \(\square\) so that each rounding statement is correct. a) \(43\square{,}567\approx440{,}000\), rounded to the nearest ten-thousand b) \(\square40{,}890\approx500{,}000\), rounded to the nearest hundred-thousand

Hints

- Decide whether the digit in the rounding place increases or stays the same. - Which digits in the next smaller place cause rounding up? - Which digits cause rounding down?

Solution

1. a) The thousands digit determines the rounding. To round \(43\square{,}567\) up to \(440{,}000\), the missing digit can be \(5,6,7,8\), or \(9\). 2. b) The ten-thousands digit is \(4\), so the number rounds down. The hundred-thousands digit must therefore be \(5\).

Answer

a) \(\square\in\{5,6,7,8,9\}\) b) \(\square=5\)

5173834

Find all whole numbers that meet both conditions. 1. They round to \(150\) to the nearest ten. 2. They round to \(200\) to the nearest hundred.

Hints

- Write the full range of numbers satisfying the first condition. - Write the full range satisfying the second condition. - Find the numbers that appear in both ranges.

Solution

1. Whole numbers that round to \(150\) to the nearest ten are \(145\) through \(154\). 2. Whole numbers that round to \(200\) to the nearest hundred are \(150\) through \(249\). 3. The numbers in both ranges are \(150,151,152,153\), and \(154\).

Answer

\(150,151,152,153,154\)

5173874

a) Which whole numbers round to \(8000\) to the nearest thousand? Give the least and greatest numbers. b) How many whole numbers round to \(8000\) to the nearest thousand? c) Are there the same number of whole numbers that round to \(25{,}000\) to the nearest thousand? Explain.

Hints

- Find the first number with a hundreds digit that rounds up. - Find the last number that still rounds down. - To count an inclusive range, subtract the endpoints and add \(1\). - Compare the widths of the two rounding intervals.

Solution

1. a) The least number is \(7500\), and the greatest is \(8499\). 2. b) Count inclusively: \(8499-7500+1=1000\) numbers. 3. c) The numbers from \(24{,}500\) through \(25{,}499\) round to \(25{,}000\). There are \(25{,}499-24{,}500+1=1000\) numbers, so the counts are equal.

Answer

a) Least: \(7500\); greatest: \(8499\) b) \(1000\) numbers c) Yes. The range \(24{,}500\) through \(25{,}499\) also contains \(1000\) whole numbers.

5173894

a) A group of consecutive whole numbers all round to \(70\). The least number is \(65\), and the greatest is \(74\). To which place were the numbers rounded? b) How many numbers are in the group? c) How many whole numbers round to \(60{,}000\) to the nearest ten-thousand?

Hints

- Look at the final nonzero place in \(70\). - Count the integers inclusively from \(65\) through \(74\). - Find the least and greatest numbers that round to \(60{,}000\).

Solution

1. a) The numbers \(65\) through \(74\) round to \(70\) to the nearest ten. 2. b) The inclusive count is \(74-65+1=10\). 3. c) The numbers from \(55{,}000\) through \(64{,}999\) round to \(60{,}000\) to the nearest ten-thousand. The count is \(64{,}999-55{,}000+1=10{,}000\).

Answer

a) nearest ten b) \(10\) numbers c) \(10{,}000\) numbers

5173984

Investigate how many whole numbers can round to \(800\). a) How many whole numbers round to \(800\) to the nearest ten? b) How many whole numbers round to \(800\) to the nearest hundred? In which case are there more possible original numbers?

Hints

- Find the least and greatest possible numbers in each case. - To count all whole numbers in an inclusive range, subtract the endpoints and add \(1\). - Think about whether rounding to a larger place creates a wider or narrower interval.

Solution

1. a) The whole numbers from \(795\) through \(804\) round to \(800\) to the nearest ten. The inclusive count is \(804-795+1=10\). 2. b) The whole numbers from \(750\) through \(849\) round to \(800\) to the nearest hundred. The inclusive count is \(849-750+1=100\). 3. Since \(100>10\), rounding to the nearest hundred gives more possible original numbers.

Answer

a) \(10\) numbers, from \(795\) through \(804\) b) \(100\) numbers, from \(750\) through \(849\) There are more possibilities when rounding to the nearest hundred.

5175734

Give one whole number that makes all three statements true at the same time: - Rounded to the nearest ten-thousand: \(60{,}000\) - Rounded to the nearest thousand: \(62{,}000\) - Rounded to the nearest hundred: \(61{,}800\)

Hints

- Find the possible range for each rounded value separately. - Look for numbers that belong to all three ranges. - Start with the narrowest rounding range.

Solution

1. Numbers from \(55{,}000\) through \(64{,}999\) round to \(60{,}000\) to the nearest ten-thousand. 2. Numbers from \(61{,}500\) through \(62{,}499\) round to \(62{,}000\) to the nearest thousand. 3. Numbers from \(61{,}750\) through \(61{,}849\) round to \(61{,}800\) to the nearest hundred. 4. The numbers from \(61{,}750\) through \(61{,}849\) satisfy all three conditions. One example is \(61{,}800\).

Answer

One possible number is \(61{,}800\). Any whole number from \(61{,}750\) through \(61{,}849\) works.

5175744

Determine whether a whole number can satisfy all three conditions. If one exists, give an example. If none exists, explain why. - Rounded to the nearest thousand: \(5000\) - Rounded to the nearest hundred: \(5600\) - Rounded to the nearest ten: \(5580\)

Hints

- Find the least and greatest possible number for each rounded result. - Compare the ranges for the first two conditions. - If two required ranges do not overlap, no number can satisfy every condition.

Solution

1. A whole number that rounds to \(5000\) to the nearest thousand must be from \(4500\) through \(5499\). 2. A whole number that rounds to \(5600\) to the nearest hundred must be from \(5550\) through \(5649\). 3. These ranges do not overlap: the first condition requires a number no greater than \(5499\), while the second requires a number at least \(5550\). 4. Therefore, no whole number can satisfy all three conditions. The third condition does not need to be checked.

Answer

No such whole number exists. The first condition requires a number from \(4500\) through \(5499\), while the second requires a number from \(5550\) through \(5649\).

5175754

Find a whole number that rounds to \(10{,}000\) when rounded to each of these places: - nearest thousand - nearest hundred - nearest ten How many whole numbers satisfy all three conditions?

Hints

- First find the numbers that round to \(10{,}000\) to the nearest ten. - Check whether that narrow range also works for rounding to the nearest hundred and thousand. - Count the whole numbers in the final inclusive range.

Solution

1. Numbers from \(9500\) through \(10{,}499\) round to \(10{,}000\) to the nearest thousand. 2. Numbers from \(9950\) through \(10{,}049\) round to \(10{,}000\) to the nearest hundred. 3. Numbers from \(9995\) through \(10{,}004\) round to \(10{,}000\) to the nearest ten. 4. The narrowest range, \(9995\) through \(10{,}004\), lies inside the other two ranges, so every number in it satisfies all three conditions. 5. The inclusive count is \(10{,}004-9995+1=10\).

Answer

One possible number is \(9998\). There are \(10\) possible whole numbers: \(9995\) through \(10{,}004\).

5177254

A whole number \(n\) satisfies both conditions: - Rounded to the nearest thousand, \(n\) is \(3000\). - Rounded to the nearest hundred, \(n\) is \(3500\). a) Explain why \(3510\) does not satisfy the first condition. b) Find the least and greatest whole numbers that satisfy both conditions.

Hints

- Round \(3510\) directly to the nearest thousand. - Find the possible range for each condition separately. - Identify the numbers that appear in both ranges.

Solution

1. a) To round \(3510\) to the nearest thousand, look at the hundreds digit. It is \(5\), so \(3510\) rounds up to \(4000\), not \(3000\). 2. Numbers from \(2500\) through \(3499\) round to \(3000\) to the nearest thousand. 3. Numbers from \(3450\) through \(3549\) round to \(3500\) to the nearest hundred. 4. The overlap is \(3450\) through \(3499\).

Answer

a) \(3510\) rounds to \(4000\), not \(3000\), because its hundreds digit is \(5\). b) Least: \(3450\); greatest: \(3499\)

5197324

Investigate every whole number from \(340\) through \(350\). For which numbers does rounding first to the nearest ten and then rounding that result to the nearest hundred give a different result from rounding the original number directly to the nearest hundred?

Hints

- Compare direct rounding with two-stage rounding for each part of the interval. - Which numbers round to \(350\) to the nearest ten? - What happens when \(350\) is then rounded to the nearest hundred?

Solution

1. Direct rounding sends \(340\) through \(349\) to \(300\), while \(350\) rounds to \(400\). 2. When rounding in two stages, \(340\) through \(344\) first round to \(340\), which then rounds to \(300\). 3. The numbers \(345\) through \(349\) first round to \(350\), which then rounds to \(400\). Directly, each of these numbers rounds to \(300\). 4. The number \(350\) gives \(400\) by either method. 5. Therefore, only \(345, 346, 347, 348\), and \(349\) give different results.

Answer

\(345, 346, 347, 348\), and \(349\)

5544684

Point \(F\) is shown on the number line. Jordan first rounds \(F\) to the nearest hundred and then rounds that result to the nearest thousand. Casey rounds \(F\) directly to the nearest thousand. Do they get the same result? Explain.

Figure for problem 554468

Hints

- Read \(F\) from the tick spacing. - Follow Jordan's two rounding steps separately. - For Casey's method, compare the original number directly with the nearest-thousand midpoint.

Solution

1. The number line shows \(F=44{,}450\). 2. Jordan rounds \(44{,}450\) to \(44{,}500\) to the nearest hundred, then \(44{,}500\) to \(45{,}000\) to the nearest thousand. 3. Casey compares \(44{,}450\) directly with the midpoint \(44{,}500\), so \(44{,}450\) rounds to \(44{,}000\) to the nearest thousand. 4. The results are different; successive rounding can change the outcome.

Answer

No. Jordan gets \(45{,}000\), while Casey gets \(44{,}000\).

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