Which ten is nearer?
A box holds 47 pencils, and someone asks roughly how many there are. 47 lies between two tens, 40 and 50. It is 7 away from 40 and only 3 away from 50, so 50 is the nearer ten. To the nearest ten, 47 rounds up to 50.
Halfway between 40 and 50 is 45. Every number above 45 is nearer to 50 and rounds up. Every number below 45 is nearer to 40 and rounds down, so 42 rounds down to 40.
47 is above the halfway mark of 45, so it is nearer to 50 and rounds up to 50.
Exactly halfway
45 is 5 away from 40 and 5 away from 50, so neither ten is nearer. For this tie there is a rule that everyone follows: a number exactly halfway rounds up. So 45 rounds to 50, and 65 rounds to 70.
48 > 45, above the halfway mark: it is 2 from 50 but 8 from 40, so 48 rounds up to 50
Find the smallest number that rounds up
Slide the number between 40 and 50. Below the halfway mark it rounds down to 40; from 45 onward it rounds up to 50.
One digit decides
The ones digit says which side of the halfway mark a number is on, so it decides on its own. A ones digit of 5, 6, 7, 8 or 9 rounds up, and 1, 2, 3 or 4 rounds down. A number whose ones digit is 0 is already a multiple of ten and stays as it is.
Bigger numbers round the same way; only the two tens change. 362 lies between 360 and 370, and its ones digit is 2, so it rounds down to 360. Take care when the tens digit is 9. 596 lies between 590 and 600, because the ten after 590 is 600. Its ones digit is 6, so it rounds up to 600.
Rounded numbers are quick to add in your head, so rounding each number to the nearest ten gives a fast estimate of a total.
Worked example: A Shopping Total Estimated by Rounding Each Price
Question Mrs Tan buys a bag for $47, a shirt for $33 and a pair of shoes for $68. She rounds each price to the nearest ten dollars to estimate the total. (a) What is her estimate? (b) How many dollars more than the exact total is her estimate?
1.Round each price to the nearest ten dollars. $47 rounds up to $50, $33 rounds down to $30 and $68 rounds up to $70.
$47 rounds up to $50, $33 rounds down to $30 and $68 rounds up to $70. 2.(a) Her estimate is 50 + 30 + 70 = $150.
(a) The estimate is 50 + 30 + 70 = $150. 3.Add the real prices to find the exact total: 47 + 33 + 68 = $148.
The exact total is 47 + 33 + 68 = $148. 4.(b) Her estimate is 150 − 148 = $2 more than the exact total. Check: rounding added $3 and $2 and took away $3, and 3 + 2 − 3 = 2.
(b) The estimate is 150 − 148 = $2 more than the exact total.
Answer: (a) $150; (b) $2
Common mistakes
- Rounding every price up. $33 is nearer to $30 than to $40, so it rounds down.
- Saying the estimate and the exact total are the same. Two prices were rounded up by $5 altogether and one was rounded down by only $3, so the estimate is a little more.
Worked example: One Number Rounded to the Nearest Ten and to the Nearest Hundred
Question A library has 3849 books. (a) Round 3849 to the nearest ten. (b) Round 3849 to the nearest hundred.
1.For the nearest ten, look at the ones digit. 3849 is between 3840 and 3850. The ones digit is 9, so round up.
Between 3840 and 3850, each mark is 1. The ones digit of 3849 is 9. 2.(a) To the nearest ten, 3849 is 3850.
(a) 3849 is nearer to 3850, so it rounds up to 3850. 3.For the nearest hundred, go back to 3849. It is between 3800 and 3900, and the halfway number is 3850.
Start again from 3849. It is between 3800 and 3900, and the halfway number is 3850. 4.The tens digit of 3849 is 4, so 3849 is less than 3850. Round down.
3849 is just below the halfway number 3850, so it rounds down. 5.(b) To the nearest hundred, 3849 is 3800. Check: 3849 is 49 from 3800 and 51 from 3900, so 3800 is nearer.
(b) To the nearest hundred, 3849 is 3800.
Answer: (a) 3850; (b) 3800
Common mistakes
- Rounding the rounded number: taking 3850 from part (a) and rounding it to 3900. Always start again from 3849, which is less than the halfway number 3850.
- Looking at the ones digit 9 when rounding to the nearest hundred. For the nearest hundred the tens digit decides, and the tens digit is 4.