Real life rarely hands us a neat 10% or 25%. We might face 15%, 17%, or 35%. The strategy is to break the target percent into benchmarks we already know and add the pieces. Two outcomes are possible:
- If our chosen benchmarks add up to exactly the target percent, the result is exact.
- If we round the target to a nearby easy benchmark, the result is an estimate.
Example 1 — Find 15% of $80 exactly. Split 15% into 10% + 5%. Since those add to exactly 15%, the result 8 + 4 = $12 is exact.
Example 2 — Find 17% of $80 exactly. Think of 17% as 10% + 5% + 1% + 1%:
- 10% of 80 = 8
- 5% of 80 = 4
- Two lots of 1% of 80 = 0.80 + 0.80 = 1.60
- 17% of 80 = 8 + 4 + 1.60 = $13.60
Because 10% + 5% + 1% + 1% = 17% on the nose, the answer $13.60 is exact, not an approximation. If we only need a quick number to say out loud, we can round it to about $14.
Example 3 — Estimate 17% of $80 by rounding the percent. When a ballpark is enough, we can round 17% to the nearest easy benchmark — say 15% — and compute 10% + 5% = $12. This is genuinely an estimate, because 15% != 17%, but it is fast and close.
Example 4 — Find 35% of $200 exactly. Split 35% into 25% + 10%, giving 50 + 20 = $70. Exact again, since 25% + 10% = 35%.
The takeaway: decomposing into benchmarks that sum to the target gives an exact answer; rounding the target itself gives an estimate. Both are useful — we just need to know which one we are doing.