Lesson Guide
Explanation
To estimate without a calculator, round every number to one significant figure, then do the easy calculation. Three hundred and eighty-seven rounds to four hundred, and fifty-two rounds to fifty. Four hundred times fifty is twenty thousand, which is two times ten to the four. The exact answer is twenty thousand one hundred and twenty-four, so the estimate is very close. Rounding to one significant figure keeps the size of the number: three hundred and eighty-seven becomes four hundred, not four.
Your Turn
Estimate 6 120 × 29. Check: 6000 × 30 = 180 000 = 1.8 × 105.
Watch Out
387 to 1 s.f. is 400, not 4. A rounded number must keep its place value.
Explanation
When the numbers are already in scientific notation, round just the front numbers to one significant figure and keep the powers of ten exactly as they are. Four point eight rounds to five, and two point one rounds to two. Five times two is ten. Ten to the six times ten cubed is ten to the nine. Ten times ten to the nine is ten to the ten. The exact answer is one point nought nought eight times ten to the ten, so the estimate has the right size.
Your Turn
Estimate (3.2 × 104) × (2.9 × 105). Check: 3 × 3 × 109 = 9 × 109.
Watch Out
Round the fronts, never the powers. 106 stays 106.
Explanation
For division, round the front numbers, divide them, and subtract the powers. Six point three rounds to six and two point nine rounds to three. Six divided by three is two. For the powers, minus four take away three is minus seven. So the estimate is two times ten to the minus seven. Be careful when the power you subtract is negative: ten to the five divided by ten to the minus two is ten to the seven, because five take away minus two is seven.
Your Turn
Estimate (8.1 × 105) ÷ (1.9 × 10−2). Check: 8 ÷ 2 = 4 and 5 − (−2) = 7, so 4 × 107.
Watch Out
105 ÷ 10−2 is 107, not 103. Subtracting a negative index adds.
Explanation
Estimates are most useful for real quantities. A stadium crowd of eighty-two thousand five hundred people each pay forty-eight dollars. Round the crowd to eight times ten to the four, and the price to five times ten to the one. Eight times five is forty, and the powers add to ten to the five, so that is forty times ten to the five, or four times ten to the six: about four million dollars. The exact takings are three million nine hundred and sixty thousand dollars. One number was rounded down and the other up, so the errors partly cancelled.
Your Turn
Estimate the takings for 4 820 tickets at $18.50. Check: 5000 × 20 = 100 000 = 1 × 105 dollars.
Watch Out
Write $48 as 5 × 101, not 5. Leaving out the power of ten loses a zero from the answer.
Explanation
Always check a calculator answer against an estimate. Take nought point nought nought nought four nine times eight thousand one hundred. Round to five times ten to the minus four, and eight times ten cubed. Five times eight is forty, and minus four plus three is minus one, so the estimate is forty times ten to the minus one, which is four. The calculator shows three point nine six nine, close to four, so it is believable. If it had shown thirty-nine point six nine, the estimate would tell you something went wrong. Quiz and exercises now.
Your Turn
Estimate 0.032 × 0.0019. Check: 0.03 × 0.002 = 0.000 06 = 6 × 10−5.
Watch Out
An estimate checks the size, not the digits. If the estimate and the answer differ by a power of ten, work it out again.
📌 Key Points — this lesson in a nutshell
- Round every number to 1 significant figure first, then calculate. Rounding keeps the size: 387 becomes 400, not 4.
- In scientific notation, round only the front numbers; keep the powers of ten exact.
- Multiply: fronts multiply, powers add. Divide: fronts divide, powers subtract — and subtracting a negative power adds.
- An estimate checks the power of ten. If a calculator answer has a different power from your estimate, something went wrong.
Step 1: Introduction
Step 1 of 5
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