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Round to nearest 10

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5503763
Round \(263\) to the nearest \(10\). The ones digit is \(3\). Should the tens digit stay \(6\) or increase to \(7\)? Explain briefly.

Hints

- Focus on the ones digit when rounding to the nearest \(10\). - Decide whether the number is closer to the lower ten or the next ten. - Check that your rounded number is a multiple of \(10\).

Solution

1. The ones digit is \(3\), which is less than \(5\). 2. The tens digit stays \(6\), and the ones digit becomes \(0\). 3. Therefore, \(263\) rounds to \(260\).

Answer

The tens digit stays \(6\), so \(263\) rounds to \(260\).
5157323
Use the number lines. Each lettered marker lies exactly on a tick. First write the marked number, then round it to the nearest \(10\).
Figure for problem 515732

Hints

- Use the labeled endpoints and the tick spacing to determine each marker's value. - Compare each marker with the halfway tick between the neighboring tens. - For a marker exactly halfway, use the rounding convention.

Solution

1. Marker A is two 1-unit ticks past \(450\), so it represents \(452\). It is closer to \(450\) than to \(460\), so it rounds to \(450\). 2. Marker B is at \(708\). It is closer to \(710\) than to \(700\), so it rounds to \(710\). 3. Marker C is at \(195\), exactly halfway between \(190\) and \(200\). A halfway value rounds to the greater multiple of \(10\), so it rounds to \(200\).

Answer

a) A: \(452\), rounded to \(450\) b) B: \(708\), rounded to \(710\) c) C: \(195\), rounded to \(200\)
5158453
Use each number line. The lettered marker lies exactly on a tick. State the multiple of \(10\) immediately below the marked number, the multiple of \(10\) immediately above it, and the nearer multiple of \(10\).
Figure for problem 515845

Hints

- Use the endpoint labels and 1-unit tick spacing to locate the marker. - Identify the two multiples of \(10\) on either side. - Compare the marker's distance from those two endpoints.

Solution

1. A is at \(524\), between \(520\) and \(530\), and is closer to \(520\). 2. B is at \(607\), between \(600\) and \(610\), and is closer to \(610\). 3. C is at \(889\), between \(880\) and \(890\), and is closer to \(890\).

Answer

a) A: below \(520\); above \(530\); nearer \(520\) b) B: below \(600\); above \(610\); nearer \(610\) c) C: below \(880\); above \(890\); nearer \(890\)
5158463
Study the number line. Marker A lies exactly on a tick. a) What number does A represent? b) Which two consecutive multiples of \(10\) surround A? c) Is A closer to one of them, or exactly halfway? d) Why is a convention needed here, and what does A round to?
Figure for problem 515846

Hints

- Use the endpoint labels and 1-unit tick spacing to read A. - Compare the number of tick intervals from A to each endpoint. - If the distances are equal, recall the convention for a halfway value.

Solution

1. The tick spacing is \(1\), and A is at the middle tick between \(470\) and \(480\), so A represents \(475\). 2. The neighboring multiples of \(10\) are \(470\) and \(480\). 3. A is \(5\) away from each endpoint, so it is exactly halfway. 4. Distance alone does not choose an endpoint, so a convention is needed. A halfway value rounds to the greater multiple of \(10\), so \(475\) rounds to \(480\).

Answer

a) \(475\) b) \(470\) and \(480\) c) Exactly halfway d) The distances are equal, so a convention is needed; A rounds to \(480\).
5164373
For each expression, estimate the difference by rounding every number to the nearest \(10\) and subtracting from left to right. Then find the exact difference and state whether the estimate is greater than or less than the exact value. a) \(782-211-349\) b) \(943-418-222\) c) \(567-153-104\)

Hints

- Round each number to a nearby ten before estimating. - Find the exact value separately after estimating. - Compare the estimate with the exact value and consider how the rounding changes affected the result.

Solution

1. a) The estimate is \(780-210-350=220\). The exact difference is \(782-211-349=222\), so the estimate is less than the exact value. 2. b) The estimate is \(940-420-220=300\). The exact difference is \(943-418-222=303\), so the estimate is less than the exact value. 3. c) The estimate is \(570-150-100=320\). The exact difference is \(567-153-104=310\), so the estimate is greater than the exact value.

Answer

a) Estimate: \(220\); exact: \(222\); estimate is less. b) Estimate: \(300\); exact: \(303\); estimate is less. c) Estimate: \(320\); exact: \(310\); estimate is greater.
5503773
Noah says, “\(342\) rounds to \(350\) to the nearest \(10\) because the tens digit is \(4\), so I round up.” What is Noah's error? Give the correct rounded number.

Hints

- Identify the two consecutive multiples of \(10\) around the number. - Ask which place tells you where the number lies between those two tens. - Check whether the proposed rounded number is actually the nearer ten.

Solution

1. To round to the nearest \(10\), use the ones digit, not the tens digit, to decide whether to round up or down. 2. The ones digit of \(342\) is \(2\), so the number is closer to \(340\) than to \(350\). 3. Therefore, \(342\) rounds to \(340\).

Answer

Noah looked at the wrong digit. For rounding to the nearest \(10\), the ones digit controls the decision. Since the ones digit is \(2\), \(342\) rounds to \(340\).
5158473
Use the number line. The labeled endpoints are consecutive multiples of \(10\). List every whole-number tick strictly between the endpoints that rounds to the left endpoint when rounded to the nearest \(10\).
Figure for problem 515847

Hints

- Read the endpoint and midpoint labels from the number line. - Consider the whole-number ticks on each side of the midpoint. - Remember what happens to the tick exactly halfway between the two tens.

Solution

1. The endpoints are \(230\) and \(240\), with midpoint \(235\). 2. The whole-number ticks strictly between the endpoints are \(231\) through \(239\). 3. The values \(231\), \(232\), \(233\), and \(234\) lie to the left of the midpoint and are closer to \(230\). 4. The midpoint \(235\) rounds to \(240\), so it is not included.

Answer

\(231\), \(232\), \(233\), and \(234\)
5501853
A food drive announcement says, “We collected about \(480\) cans,” with the count rounded to the nearest \(10\). a) What is the least possible exact number of cans? b) What is the greatest possible exact number of cans?

Hints

- Think about the halfway points on both sides of \(480\). - Include a boundary value only if it rounds to \(480\). - Check the next whole number after your proposed greatest value.

Solution

1. \(475\) is halfway between \(470\) and \(480\), so it rounds to \(480\). 2. Values through \(484\) are closer to \(480\) than to \(490\). 3. \(485\) is halfway between \(480\) and \(490\) and rounds to \(490\). 4. Therefore, the least possible count is \(475\) and the greatest is \(484\).

Answer

a) \(475\) b) \(484\)

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