IB Maths AI HLNumber ToolkitPaper 1 & 2ε formula~6 min read
Percentage Error
Percentage error measures how far an estimated or rounded value sits from the true value, written as a percentage of that true value. One formula does all the work — and because it uses an absolute value, the answer is always positive.
📘 What you need to know
Percentage error compares an approximate value with the exact value, as a percentage.
The formula is ε = |(vA − vE) ÷ vE| × 100% — it is in the formula booklet.
vE is the exact value; vA is the approximate value. The exact value always goes on the bottom.
The | | means absolute value — percentage error is always positive.
It does not matter whether the estimate is too big or too small — only the size of the gap counts.
The further the estimate is from the truth, the larger the percentage error.
What is percentage error?
An estimate or a rounded value is rarely the true value. Percentage error quantifies the gap between them: it is the size of the difference, expressed as a percentage of the exact value.
It answers “how wrong is this estimate, relative to the real number?” A gap of 10 matters far more against an exact value of 50 than against 5000 — the percentage captures that.
The error is the absolute difference between the values; dividing it by the exact value and multiplying by 100% turns it into a percentage error.
The percentage error formula
Percentage error ε uses a single formula. The exact value vE always sits on the denominator, and the absolute value bars guarantee a positive result.
Percentage errorε = |vA − vEvE| × 100%
vE = exact value · vA = approximate value — given in the formula booklet
Reading the result
Because of the absolute value, an over-estimate and an under-estimate of the same size give the same percentage error. A larger gap always means a larger percentage error, so the formula is a fair way to compare two competing estimates — the smaller percentage error is the better estimate.
Exact value first: work out vE in full — as a surd, fraction or unrounded decimal — before substituting. Rounding it early adds error of its own.
🧠Recipe — percentage error in five steps
Identify the values: the exact value vE and the approximate value vA.
Find the differencevA − vE.
Divide by the exact valuevE — it always goes on the bottom.
Take the absolute value and multiply by 100%.
Round the answer (3 s.f. unless told otherwise) — it must be positive.
Worked examples
WE 1
A basic percentage error
A jar contains exactly 250 sweets. A student estimates that there are 240. Find the percentage error in the student’s estimate.
vE = 250 (exact), vA = 240 (estimate)ε = |(240 − 250) ÷ 250| × 100%= |−10 ÷ 250| × 100%ε = 4%the absolute value turns the −10 into a positive error.
WE 2
Percentage error from rounding
A length is measured as 7.86 m. It is rounded to 7.9 m. Find the percentage error caused by the rounding, correct to 3 significant figures.
vE = 7.86 (exact), vA = 7.9 (rounded)ε = |(7.9 − 7.86) ÷ 7.86| × 100%= |0.04 ÷ 7.86| × 100% = 0.50890…%ε ≈ 0.509% (3 s.f.)rounding is just another source of approximation — the same formula applies.
WE 3
An over-estimate
A runner’s exact time for a race is 12.5 s. A stopwatch records the time as 12.8 s. Find the percentage error in the stopwatch reading.
vE = 12.5 (exact), vA = 12.8 (reading)ε = |(12.8 − 12.5) ÷ 12.5| × 100%= |0.3 ÷ 12.5| × 100%ε = 2.4%the estimate is too big, but the error is still positive — only the gap matters.
WE 4
Percentage error of an approximation
The value of π is often approximated by 3.14. Find the percentage error in using 3.14 for π, correct to 3 significant figures.
vE = π = 3.14159… (exact), vA = 3.14ε = |(3.14 − π) ÷ π| × 100%= |−0.0015926… ÷ π| × 100% = 0.050695…%ε ≈ 0.0507% (3 s.f.)use the full value of π from your GDC — never a rounded one — for vE.
WE 5
Comparing two estimates
The exact length of a bridge is 60 m. Anya estimates 57 m and Ben estimates 64 m. Use percentage error to decide whose estimate is better.
Anya: vA = 57, vE = 60ε = |−3 ÷ 60| × 100% = 5%Ben: vA = 64, vE = 60ε = |4 ÷ 60| × 100% = 6.67% (3 s.f.)Anya’s estimate is better — 5% < 6.67%the smaller percentage error means the closer estimate.
WE 6
Full question: exact value then error
Let Q = a sin(2b), where a = 6 and b = 30°. (a) Calculate the exact value of Q. (b) Find the percentage error when 5 is used as an estimate for Q, correct to 3 s.f.
(a) substitute a = 6 and b = 30°Q = 6 sin(60°) = 6 × √32Q = 3√3 (exact value)(b) vE = 3√3 = 5.19615…, vA = 5ε = |(5 − 3√3) ÷ 3√3| × 100% = 3.7749…%(a) Q = 3√3 · (b) ε ≈ 3.77% (3 s.f.)keep the exact surd 3√3 as vE — substitute it straight into the formula.
💡 Top tips
The exact value vE always goes on the denominator — never the estimate.
The answer is always positive; if you get a negative number, you missed the absolute value.
Find the exact value first — surd, fraction or full decimal — then substitute.
Don’t forget the × 100% — a percentage error is a percentage, not a decimal.
The formula is in the booklet, so there is no need to memorise it — but know which value is which.
âš Common mistakes
Dividing by the approximate value instead of the exact value vE.
Leaving the answer negative — the absolute value makes percentage error positive.
Forgetting × 100% — giving 0.04 instead of 4%.
Rounding the exact value too early, which introduces extra error before you start.
Swapping vA and vE — the exact value is the true one, the approximate value is the estimate.
Next up: Accuracy & Estimation — exact values, sensible estimation for rough checks, and using logic to pick the right answer. Percentage error feeds straight in: it is how you measure whether an estimate was good enough.
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