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Number · Rounding & bounds

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Estimation

Why estimating means rounding everything to 1 significant figure BEFORE calculating, how to divide by decimals cleanly, and how to reason about whether an estimate is an over- or underestimate.

Number · Rounding & bounds

Estimation

Why estimating means rounding everything to 1 significant figure BEFORE calculating, how to divide by decimals cleanly, and how to reason about whether an estimate is an over- or underestimate.

Why it works

An estimate trades precision for speed: round every number to 1 significant figure first, then calculate with the easy numbers. The rounding happens before the arithmetic — that's the whole point, and the whole mark scheme. Calculating exactly and then rounding is a different activity (and on a non-calculator paper, a slow way to lose the method marks).

48.6×21.29.7≈50×2010=100010=100.\frac{48.6 \times 21.2}{9.7} \approx \frac{50 \times 20}{10} = \frac{1000}{10} = 100.

Why 1 s.f. and not "the nearest ten"? One significant figure keeps one digit of information at the right size — every number becomes (digit) × (power of ten), and the tens cancel and combine by the index laws. Rounding 0.19 "to the nearest ten" would give 0 and destroy the sum; to 1 s.f. it gives 0.20.2, which is exactly usable.

Dividing by a decimal: scale both. 3000.2\frac{300}{0.2} — multiply top and bottom by 10: 30002=1500\frac{3000}{2} = 1500. Dividing by a number less than 1 must give something bigger; if your estimate shrank, the decimal ate a step.

Roots: trap the number between squares. 83.7×1.2≈80×1\sqrt{83.7 \times 1.2} \approx \sqrt{80 \times 1}… better, spot that 83.7×1.2≈10083.7 \times 1.2 \approx 100, and 100=10\sqrt{100} = 10. For a bare root, bracket it: 85\sqrt{85} lies between 81=9\sqrt{81} = 9 and 100=10\sqrt{100} = 10, nearer 9.

Over or under? Track the directions. Each rounding pushes the answer a known way: making the top of a fraction bigger, or the bottom smaller, pushes the result up; the reverses push it down. Estimating 29741.3\frac{297}{41.3} as 30040=7.5\frac{300}{40} = 7.5: the top went up AND the bottom went down — both pushes are upward, so 7.5 is an overestimate, and you can say why in one sentence. When the pushes conflict, the honest answer is "can't tell without more care" — exam questions are built so they don't conflict.