EQN-06-08

Estimation & Approximation in Comparisons

Round smartly to settle a comparison in seconds — and know when rounding is too risky.

~5 min read•Difficulty 3/5

Round to decide, not to compute

You only need the order, so round both quantities and compare. If the gap survives the rounding, you are done.

A : 48 \times 21 B : 1050

A \approx 50 \times 20 = 1000... too close — round more carefully :

48 \times 21 = 1008 < 1050 \to B is greater

Round both the SAME direction with care

Rounding one quantity up and the other down can exaggerate a gap that isn't real. When the values are close, compute exactly instead of trusting the estimate.

Benchmark fractions and percents

Compare to ½, ¼, 10%, etc. "19 out of 40" is just under ½; "11 out of 20" is just over $\frac{1}{2} \to$ the second is larger, no division needed.

Know when NOT to estimate

If the two quantities are within a few percent of each other, estimation can mislead — switch to an exact or structural comparison.

Worked examples

Example 1 (difficulty 3/5)

Quantity

A : 0.48 \times 600.

Quantity B: 290. Which is greater?

◎ Predict the answer first:

Example 2 (difficulty 2/5)

Quantity A: 17/35. Quantity B: 1/2. Which is greater?

◎ Predict the answer first:

Tips

Round only enough to expose a clear gap; if it stays close, compute exactly.
Benchmark fractions against ½ by comparing the numerator to half the denominator.
Beware rounding the two quantities in opposite directions — it can fake a difference.

Common mistakes

Trusting a rough estimate when the quantities are nearly equal

Why: Estimation feels decisive.

Fix: Within a few percent, switch to exact arithmetic or a structural argument.

Rounding one value up and the other down, then declaring a winner

Why: It seems harmless.

Fix: Bias the rounding consistently, or it can reverse a true close result.

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