An exact answer keeps a fraction, a surd or π in the result. An approximate answer is a rounded decimal. The skill is deciding which one a question wants, and never rounding until the last step.
This lesson belongs to number sense and exact arithmetic and underpins area, volume, trigonometry and standard form later in the course.
When is an answer exact, and when is it approximate?
An answer is exact when it is written in a form that loses no information: 13/6, 36π or √18. It becomes approximate the moment you round it, for example 2.17 or 113.1.
Exact form is required when the question says “exact”, “in terms of π” or “leave your answer as a fraction”. Otherwise, the paper’s instructions decide the accuracy, so read them before you start.
How to decide and check, step by step
- Read the wording. Look for “exact”, “in terms of π”, or a required accuracy such as “to 1 decimal place”.
- Estimate first. Round each number to one significant figure and work the estimate in your head.
- Calculate with full values. Keep every digit or keep the fraction or π symbol until the last step.
- Round once, at the end, to the accuracy asked.
- Compare with your estimate. The two should be in the same neighbourhood.
Worked example
A circular table top has radius 6 cm on a model. Find its area, first as an exact answer and then to 1 decimal place.
Step 1, formula: area = π × r² = π × 6² = π × 36.
Step 2, exact answer: 36π cm².
Step 3, estimate: using π ≈ 3, 36 × 3 = 108, so the area should be a little over 108.
Step 4, approximate answer: 36 × π = 113.0973… on a calculator, which is 113.1 cm² to 1 decimal place.
Check: 113.1 is a little over 108, which matches the estimate. Using 3.14 instead of the π key gives 36 × 3.14 = 113.04, close but not identical, which is why the calculator’s π is preferred.
The mistake to watch for
A common slip is to round a value early and then round again at the end.
Mistaken answer: 2/3 of 30 is 0.67 × 30 = 20.1
The student replaced 2/3 with 0.67 at the start, so the tiny error carried through.
Exactly, 2/3 × 30 = 20. Rounding 2/3 to two decimal places made the answer wrong in the third significant figure. The fix is to keep 2/3 as a fraction, or keep every digit on the calculator, and round only the final answer.
Check yourself
Try these, then open each answer.
1. Estimate 7.9 × 3.02, then find the exact product.
Show answer
Estimate: 8 × 3 = 24. Exact: 7.9 × 3 = 23.7 and 7.9 × 0.02 = 0.158, so 23.7 + 0.158 = 23.858. It is close to the estimate of 24.
2. A circle has radius 5 cm. Give its area exactly, then to 3 significant figures.
Show answer
Area = π × 5² = 25π cm². On a calculator, 25π = 78.5398…, which is 78.5 cm² to 3 significant figures.
3. Is 0.33 equal to 1/3? Give a reason.
Show answer
No. 1/3 = 100/300 and 0.33 = 99/300, so they differ by 1/300. Writing 0.33 for 1/3 is an approximation, so use ≈.
Where this leads next
Next, check operation order in a multi-step calculation, where careful keeping of exact values matters across several steps. Then try the number sense practice set. The non-calculator working trainer shows why an exact step differs from a rounded one.
If you often know the method but lose marks on the final form of an answer, a teacher in online one-to-one Mathematics tuition can go through your past papers with you and mark where the decision was made.