MISSION · Number sense
Estimating to Check an Answer
The five-second calculation that catches the big mistakes
LEVEL 1 · EXPLORE
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LEVEL 2 · LEARN
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The idea in one sentence
Before or after you calculate exactly, work out roughly what the answer should be, so a wrong answer has somewhere to be caught.
Why bother when you have a calculator
Because calculators are obedient, not careful. A misplaced decimal point, a slipped finger, a missed zero — the machine reports all of them with total confidence. The only defence is knowing what the answer should look like before you read it.
The errors estimation catches are the big ones: answers ten times too large, answers that came out smaller when they should have grown. Those are also the errors that matter most.
How to estimate
Round each number to something you can handle in your head, usually one or two significant figures, then do the easy calculation.
\[312 \times 19 \approx 300 \times 20 = 6000\]The exact answer is 5928. Close enough to know that 59,280 or 592 would have been wrong.
\[4718 \div 58 \approx 4800 \div 60 = 80\]Notice the rounding there: 4718 went up to 4800 rather than down to 4700, because 48 divides by 6 and 47 doesn’t. Estimation is allowed to be opportunistic. The goal is a number you can compute instantly, not the nearest round number.
Knowing which way you’re wrong
Rounding both numbers up makes the estimate too big; both down makes it too small. Round one each way and the errors partly cancel, which is usually the better choice.
In \(300 \times 20\) the first went down and the second up, so the estimate is close but its direction isn’t obvious. In \(300 \times 20 = 6000\) against the true 5928, the 19 → 20 move mattered more, so the estimate came out high.
The reasonableness check
This is estimation’s real job, and it’s a habit rather than a calculation. After every answer, ask one question: could that be right?
- Adding two numbers near 50 should give something near 100.
- Multiplying by a number bigger than 1 makes things bigger.
- A discount price should be less than the original.
- An average has to land somewhere between the smallest and largest value.
None of that needs arithmetic. It needs only a second of attention, and it catches a large fraction of all mistakes.
Estimating in real life
Shopping totals, whether the paint will be enough, how long a journey takes, whether a news figure is plausible. In most of these an estimate is not a rough version of the real answer — it is the answer, because nobody needs the exact one.
Common mix-ups
- Estimating after looking at the answer, which turns into agreeing with it.
- Rounding so hard the estimate is useless, like 312 × 19 ≈ 0 × 0.
- Rounding mid-calculation and then calling it exact. Keep the two separate.
- Rounding only one of the numbers, which leaves the arithmetic just as hard.
Check yourself
Estimate 312 × 19. Then judge: a calculator shows 48.7 + 51.2 = 998.7. Reasonable?
300 × 20 = 6000, so the true answer of 5928 is fine. For the second, two numbers close to 50 must add to close to 100 — 998.7 is ten times too big, probably a mistyped decimal point.
LEVEL 3 · REFLECT
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