Estimate first
Round hard, compute fast, and get an answer that's close enough to act on. Estimating is also the check that catches a mistyped calculator, which is the one error nobody notices.
THE IDEA
Keep the first digit or two, bin the rest
Round each number to the one or two digits that carry its size, then work with those. 312 becomes 300, 19 becomes 20. Multiply those and you get 6,000, while the true answer is 5,928. You were never more than 2% out, and it took no effort at all.
A price of 487,000 is 'about half a million'. A distance of 31 km is 'about 30'. That is enough to decide almost anything.
Will 300,000 cover this basket?
47,000 → 50
128,000 → 130
89,000 → 90
32,000 → 30
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300 (thousand)
real total: 296,000about 300,000 — it's going to be tightWork in thousands and the numbers shrink to something you can hold. The estimate was 4,000 high, which is exactly the kind of error that doesn't matter and exactly the kind of answer you needed.
Check yourself
Without calculating exactly: 612 × 48 is closest to which of these?
- 3,000
- 300,000
- 30,000
- 24,000
Show the answer
30,000
Yes. 600 × 50 = 30,000, and the true answer is 29,376. Close enough to spot a wrong result instantly.
SIZE FIRST
Get the number of zeros right before anything else
Most real mistakes are not 'the answer was 5,928 and I said 5,918'. They are 'the answer was 5,928 and I said 592'. So estimate the size first: multiply the leading digits, then count the zeros in both numbers and put them all on the end.
612 × 48: leading digits 6 × 5 = 30. 612 has two zeros' worth behind the 6, 48 has one behind the 5. 30 with three zeros is 30,000.
Check yourself
Estimating 8 × 39 as 8 × 40 = 320 tells you the true answer must be a bit under 320.
Show the answer
True
True, and knowing the direction is half the value. You rounded 39 up, so the estimate is high; the real answer is 320 − 8 = 312. An estimate you can label 'a bit high' or 'a bit low' is worth far more than a bare number.
Is this answer plausible? Three questions
- Someone tells you 27 × 34 = 6,918
You don't need to redo it. You need to decide whether it could be true.
- How big should it be?
Round both to 30: 30 × 30 = 900. So the answer belongs in the hundreds, and 6,918 is nearly eight times too big. Something lost a decimal place.
- Does the last digit fit?
7 × 4 = 28, so the answer must end in 8. It does, which is why the mistake wasn't in the digits; it was in the size.
- Say what you think it is
Around 900, ending in 8. The real answer is 918. You got within 2% without doing the multiplication.
Check yourself
Match each calculation to the size of its answer
Show the answer
- 31 × 29 → about 900
- 198 × 51 → about 10,000
- 487 + 512 → about 1,000
- 6,000 ÷ 19 → about 300
Estimate, or work it out exactly?
An estimate is enough
When the decision is yes or no: can I afford this basket, is this deal cheaper, will this fit in the car, is that news figure believable? You need the size, not the digits.
You need the exact number
When someone is owed money or a form is being filled in: change from a note, splitting a bill so nobody is short, a tax return. Here an estimate tells you whether the exact answer looks right.
Check yourself
Estimate, or exact?
- Will my card cover this basket?
- Splitting a bill so nobody is short
- Is this phone plan cheaper than that one?
- Counting the change from a note
- Could a stadium really hold that many?
- Paying back what you owe a friend
Show the answer
An estimate will do: Will my card cover this basket?, Is this phone plan cheaper than that one?, Could a stadium really hold that many?
Has to be exact: Splitting a bill so nobody is short, Counting the change from a note, Paying back what you owe a friend
Make it a habit
- Estimate before you calculate, not after. An estimate made afterwards quietly agrees with whatever you already got.
- Round to one or two digits, and know which way you rounded.
- Get the zeros right first. Size errors are the ones that cost real money.
- Work in thousands or millions so the numbers fit in your head, and put the units back at the end.
Check yourself
The hard one. A 48-metre roll of fabric costs 1,250,000 in total. Roughly what is one metre?
- about 2,600
- about 26,000
- about 260,000
- about 60,000
Show the answer
about 26,000
Right. Round 48 to 50: 1,250,000 ÷ 50 is 2,500,000 ÷ 100 = 25,000. Since 48 is a little less than 50, the true figure is a little more: about 26,000.
Lesson recap
- Round each number to one or two digits, calculate with those, and you'll usually be within a few percent.
- Get the size right first: multiply the leading digits and count the zeros.
- Round one number up and one down where you can, and always know which way you leaned.
- Before accepting any answer, ask how big it should be and what its last digit ought to be.