Find the size first
Before working out 49 × 21 exactly, find out roughly how big the answer is. Round each number to the nearest ten: 49 rounds up to 50, and 21 rounds down to 20.
50 × 20 is easy: 50 × 2 = 100, and 100 × 10 = 1000. So 49 × 21 is about 1000. The sign means "is approximately equal to", so this is written . An answer found this way is called an estimate.
50 × 20 = 1000, halfway along a line from 0 to 2000.
The exact answer is close by
The exact answer is 49 × 21 = 1029, which is close to 1000. Each number moved only a little when it was rounded, 49 up by 1 and 21 down by 1, so the answer moved only a little too.
An estimate does not give the exact answer. It gives the size of the answer: about a thousand, and not about a hundred or about ten thousand.
The exact answer, 49 × 21 = 1029, is just past the estimate of 1000.
Catch a wrong answer
A friend says that 49 × 21 = 249. Estimate it: . 249 is nowhere near 1000, so it cannot be right, and the working has a mistake in it somewhere.
Estimate before you calculate, then compare. An estimate catches an answer that is far too big or far too small, such as one with a digit missing or one with a digit too many.
249 is far below the estimate of 1000.
One up and one down
For 38 × 52, round 38 up to 40 and 52 down to 50: 40 × 50 = 2000. The exact answer is 1976, only 24 away.
A multiplication is the area of a rectangle. Rounding 38 up adds a strip to the rectangle, and rounding 52 down cuts a strip off. The added strip and the cut strip are nearly the same size, so the estimate lands close. If both numbers are rounded up, both strips are added, and the estimate is further off.
one factor rounded up and one down: the added strip and the missing strip nearly cancel, 35 × 55 − 1976 = −51, −2.6%
Round both factors to the nearest ten and get the error under 2%
The dashed rectangle is 38 by 52, and its area is the exact answer, 1976. Drag the width and the height to round them to the nearest ten, and compare the gold rectangle with the dashed one.
Worked example: A Calculator Answer Tested with an Estimate
Question Ravi works out 49 × 62 on a calculator. The calculator shows 338. (a) Estimate 49 × 62 by rounding both numbers to the nearest ten. (b) The answer on the calculator is wrong. What is 49 × 62?
1.Round each number to the nearest ten. 49 rounds up to 50 and 62 rounds down to 60.
49 rounds up to 50 and 62 rounds down to 60. 2.(a) 5 × 6 = 30, so 50 × 60 = 3000. The answer should be about 3000.
(a) The estimate is 50 × 60 = 3000. 3.338 is nowhere near 3000. It is about ten times too small, so it cannot be right.
338 is about ten times too small, so it cannot be right. 4.Find 50 × 62 first. 100 × 62 = 6200, and half of 6200 is 3100.
50 × 62 is half of 6200, which is 3100. 5.(b) 49 lots of 62 is one 62 less than 50 lots: 3100 − 62 = 3038. This is close to the estimate of 3000.
(b) 49 × 62 = 3100 − 62 = 3038, close to the estimate.
Answer: (a) 3000; (b) 3038
Common mistakes
- Accepting 338 because 8 is the correct last digit of 49 × 62. The last digit does not show the size of a number. The estimate of 3000 shows that a digit is missing.
- Taking off 49 instead of 62 and writing 3100 − 49 = 3051. 50 lots of 62 is one lot of 62 too many, so 62 is taken off.
More multiplying and dividing in your head problems, worked step by step →