- 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.

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 \(5000\).

\(5000\)

- Look at the two tens on either side of the number. - Compare how far the number is from each ten.

1. \(4386\) is \(6\) more than \(4380\) and \(4\) less than \(4390\). 2. It is closer to \(4390\), so \(4386\) rounds to \(4390\) to the nearest ten.

Closer to \(4390\); rounded value \(4390\).

- Find the nearest hundred below and above the number. - Compare the distances to those two hundreds. - Think about the usual rule for a halfway case.

1. The two neighboring hundreds are \(7400\) and \(7500\). 2. \(7450\) is halfway between them, so the usual whole-number rounding rule selects the higher hundred.

\(7500\). A halfway value rounds up to the next hundred.

- Identify the thousands place first. - Which digit is immediately to its right? - That next digit helps decide which thousand is nearer.

1. Look at the hundreds digit when rounding to the nearest thousand. 2. The hundreds digit is \(0\), so \(61{,}020\) is nearer to \(61{,}000\) than to \(62{,}000\).

\(61{,}000\). Use the hundreds digit, \(0\).

- 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.

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 \(100{,}000\).

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

- 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.

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 \(700{,}000\).

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

- 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.

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 \(900\).

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

- 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 \(9\).

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.

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: \(100{,}000\)

- 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.

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.

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: \(0\)

- 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 \(9\).

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.

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: \(1{,}000{,}000\)

- 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?

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 \(510\).

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

- 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.

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.

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

- 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.

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 \(62{,}000\).

\(62{,}000\)

- 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.

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

\(350{,}000\)

- 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.

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 \(82{,}000\).

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

- Read the tick spacing between the labeled values. - Find the value at P before rounding. - Compare P's distance from the two neighboring hundreds.

1. The small ticks are \(10\) apart, so P is at \(4560\). 2. P is \(60\) above \(4500\) and \(40\) below \(4600\), so it is closer to \(4600\).

P is \(4560\). It rounds to \(4600\) because it is \(40\) away from \(4600\) but \(60\) away from \(4500\).

- Each rounding place uses the digit immediately to its right. - Round separately for each requested place; do not round an already rounded answer again. - Compare which digit controls the nearest-ten and nearest-hundred choices.

1. The ones digit is \(7\), so to the nearest ten the number rounds to \(58{,}430\). 2. The tens digit is \(2\), so to the nearest hundred the number rounds to \(58{,}400\). 3. The hundreds digit is \(4\), so to the nearest thousand the number rounds to \(58{,}000\). 4. The ones digit, \(7\), makes the tens round up. The tens digit, \(2\), makes the hundreds round down.

Nearest ten: \(58{,}430\). Nearest hundred: \(58{,}400\). Nearest thousand: \(58{,}000\). The nearest-ten result is greater because the ones digit is \(7\). The nearest-hundred result is smaller because the tens digit is \(2\).

- 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.

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 \(537\).

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: \(735-198=537\).

- 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.

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 \(745\).

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

- 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 \(334\).

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 \(380\).

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: \(145+176+59=380\).

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

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.

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 \(600\).

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

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.

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

- 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.

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 \(342 + 189 + 211 = 742\).

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

- Round the two numbers to the place Lucas and Mia each chose before doing the exact subtraction. - Subtract the rounded values to get the two estimates. - Then find the exact difference and compare how far each estimate is from it.

1. To the nearest hundred, \(678\) rounds to \(700\) and \(243\) rounds to \(200\). Lucas estimates \(700-200=500\). 2. To the nearest ten, \(678\) rounds to \(680\) and \(243\) rounds to \(240\). Mia estimates \(680-240=440\). 3. The exact difference is \(678-243=435\). 4. Lucas's estimate is \(65\) away, while Mia's is \(5\) away, so Mia's estimate is closer.

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

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

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 \(800\).

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

- 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.

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 \(841\).

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 \(478+52+311=841\).

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

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 \(364\).

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 \(833-175-294=364\).

- 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 \(300\).

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 \(999\).

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.

- 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 \(45{,}000\).

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 \(45{,}049\).

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

- 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.

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: \(260{,}000 + 81{,}000 = 341{,}000\).

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

- Which digit decides whether each number rounds up or down to the nearest thousand? - Round both counts before finding their difference. - Is the result an exact difference or an estimate?

1. Round each count: \(421{,}280\approx421{,}000\) and \(118{,}750\approx119{,}000\). 2. Subtract the rounded counts: \(421{,}000-119{,}000=302{,}000\).

About \(302{,}000\) more players are listed for table tennis.

- 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.

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: \(356{,}000 + 121{,}000 = 477{,}000\).

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

- To round to the nearest thousand, look at the hundreds digit. - Estimate the change using the rounded areas, not the original figures. - Subtract the earlier rounded value from the later one.

1. Round to the nearest thousand: \(45{,}287\approx45{,}000\), \(52{,}814\approx53{,}000\), and \(58{,}490\approx58{,}000\). 2. The estimated increase from 2000 to 2010 is \(53{,}000-45{,}000=8000\) acres. 3. The estimated increase from 2010 to 2020 is \(58{,}000-53{,}000=5000\) acres.

a) \(45{,}000\), \(53{,}000\), and \(58{,}000\) acres. b) About \(8000\) acres, then about \(5000\) acres.

- 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.

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 \(25{,}900 - 21{,}000 = 4900\).

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

- 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.

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 \(160{,}000 - 150{,}000 = 10{,}000\).

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

- 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.

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.

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

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

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 \(\$380{,}000\).

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

- 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 \(0\).

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 \(15{,}000\).

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

- 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.

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.

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

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

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.

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

- 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.

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 \(800{,}000\).

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

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

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.

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

- 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.

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

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

- 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.

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 \(250{,}000\).

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

- 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 \(9\).

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 \(1{,}000{,}000\).

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

- 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 \(50{,}000\)?

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 \(44{,}999\).

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

- Multiply the number of equal groups by the size of each group. - Decide whether the report needs a broad estimate or a more precise one. - Name the rounding place you choose.

1. Multiply the number of classes by the students per class: \(22\times26=572\). 2. A reasonable broad estimate is \(600\), rounded to the nearest hundred. A more precise report could use \(570\), rounded to the nearest ten. Either choice is reasonable when its intended precision is explained.

1. \(572\) students 2. One reasonable answer is about \(600\) students, rounded to the nearest hundred. About \(570\) students, rounded to the nearest ten, is also reasonable with an appropriate explanation.

- 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 \(10\)?

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.

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

- 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.

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.

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

- 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.

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 \(460{,}000\).

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

- 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.

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 \(3200-3175=25\).

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

- 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.

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.

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.

- 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.

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 \(200\).

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

- 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.

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.

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

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

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 \(12{,}400\,\text{ft}\).

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

- 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 \(1\).

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: \(80{,}499-79{,}500+1=1000\).

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

- 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.

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 \(48{,}000\).

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

- Which whole numbers are close enough to \(450\) to round there to the nearest ten? - Where is the halfway point between \(400\) and \(500\)? - Check both rounding rules for the numbers near that point.

1. The whole numbers that round to \(450\) to the nearest ten are \(445\) through \(454\). 2. Those below \(450\) round to \(400\) to the nearest hundred, while \(450\) through \(454\) round to \(500\). 3. The numbers satisfying both conditions are \(445,446,447,448,449\). Numbers outside this list fail at least one of the two rounding conditions.

\(445,446,447,448,449\). These all round to \(450\) to the nearest ten and \(400\) to the nearest hundred. Smaller or larger numbers fail at least one condition.

- Think about the two halfway points on either side of \(26{,}000\). - The rounding-up rule treats the two halfway points differently for this target. - Because the answers must be whole numbers, check the integers next to each boundary.

1. Halfway between \(25{,}000\) and \(26{,}000\) is \(25{,}500\). That number rounds up to \(26{,}000\), so it is the least possible. 2. Halfway between \(26{,}000\) and \(27{,}000\) is \(26{,}500\). That number rounds up to \(27{,}000\), so the greatest whole number still rounding to \(26{,}000\) is \(26{,}499\). 3. The neighboring whole numbers \(25{,}499\) and \(26{,}500\) both round away from \(26{,}000\), confirming the endpoints.

Smallest: \(25{,}500\); greatest: \(26{,}499\). Just below the range, \(25{,}499\) rounds to \(25{,}000\). Just above it, \(26{,}500\) rounds to \(27{,}000\).

- 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.

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 \(38{,}200\).

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

- 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?

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 \(5\).

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

- 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.

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 \(154\).

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

- 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.

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.

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

- 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 \(60{,}000\).

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 \(64{,}999-55{,}000+1=10{,}000\).

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

- 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.

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.

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

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

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 \(61{,}800\).

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

- 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.

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.

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

- 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.

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 \(10{,}004-9995+1=10\).

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

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

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 \(3499\).

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

- 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?

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.

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

- 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.

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.

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