IB Maths AI SLTopic 1 — Number & AlgebraPaper 1 & 2In formula booklet~5 min read
Percentage Error
Percentage error tells you how far off an approximate value is from the exact one, as a percentage. One formula, always the same recipe — just plug in the numbers.
📘 What you need to know
The formula (in the booklet): ε = |vA − vE| / |vE| × 100%
vA = approximate value (the rounded / estimated / measured one)
vE = exact value (the true one — divide by THIS, not by vA)
The | | bars make everything positive — % error is never negative.
Don’t forget × 100% — without it you’d have a decimal, not a percentage.
Default: 3 s.f. on the final answer unless told otherwise.
The exact conversion of $250 at today’s rate is €228.50. A traveller estimated €230. Find the percentage error.
v_A = 230 (estimate), v_E = 228.50 (exact)ε = |230 − 228.50| / |228.50| × 100% = 1.50 / 228.50 × 100% = 0.6565…%ε ≈ 0.657% (3 s.f.)“exact” is whatever the question treats as true — here, the actual converted amount.
WE 4
Rounded edge → volume error
A cube has edge 7.5 cm. Someone rounds the edge to 8 cm and calculates the volume. Find the percentage error in this volume.
Volumesv_E = 7.5³ = 421.875 cm³ (exact)v_A = 8³ = 512 cm³ (using rounded edge)Apply formulaε = |512 − 421.875| / 421.875 × 100% = 90.125 / 421.875 × 100% = 21.36…%ε ≈ 21.4% (3 s.f.)a 6.7% rounding error in the edge → 21% error in the volume. Errors compound when you cube.
WE 5
Time estimate
A driver estimated a journey would take 2 hours. The actual time was 2 hours 7 minutes. Find the percentage error in the estimate.
Convert actual time to a single unitv_E = 2 h 7 min = 127/60 h = 2.11667… hv_A = 2 hApply formulaε = |2 − 2.11667| / 2.11667 × 100% = 0.11667 / 2.11667 × 100% = 5.5118…%ε ≈ 5.51% (3 s.f.)always convert both values to the SAME unit before subtracting.
WE 6
Budget vs actual
A party was budgeted at $180. The actual cost came to $195.50. Find the percentage error of the budget.
v_A = 180 (budget), v_E = 195.50 (actual)ε = |180 − 195.50| / |195.50| × 100% = 15.50 / 195.50 × 100% = 7.928…%ε ≈ 7.93% (3 s.f.)even though the budget was UNDER the actual cost, ε is still positive — the | | bars take care of that.
💡 Top tips
Divide by vE, not vA — exact value goes on the bottom.
Keep π and √ symbolic in working; switch to decimal only at the last step.
Always × 100% — easy to forget when racing through a paper.
Sanity-check the size: if vA and vE are close, ε should be small (≪ 10%). A huge ε means you’ve used the wrong value on the bottom.
âš Common mistakes
Dividing by vA: 0.2 / 48 instead of 0.2 / 47.8 → wrong answer by a small but examiner-detected amount.
Giving a negative answer: % error must be positive — the | | bars are not optional.
Forgetting × 100%: 0.004184 is the ratio, not the percentage.
Mixed units: e.g. minutes vs hours, cm vs m. Convert first, then plug in.
Up next: Accuracy & Estimation. Learn to spot whether an answer is roughly the right size before you commit to a precise calculation — invaluable for catching mistakes under exam pressure.
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