Learn | Grasp Maths

Lesson overview

Number — form and approximation

Different contexts require different forms of answer. Exact answers (surds, fractions, expressions) preserve mathematical precision and are preferred in theoretical work and algebra. Approximate answers (decimals, rounded values) are appropriate for real-world measurements, contexts with given data uncertainty, and practical problems. Key decisions: leave as a surd if the problem involves geometry or algebra; use a fraction for exact rational values; round sensibly for measurements. Always consider the context and what the question asks for.

Exact answer — leave as surd

Find the length of the hypotenuse of a right-angled triangle with legs 3 cm and 5 cm. Give an exact answer.

The question asks for an exact answer. Leave it as $\sqrt{34}$ rather than 5.83... cm.

Approximate answer — round sensibly

A ladder reaches 4 m up a wall when its foot is 3 m from the wall. How long is the ladder (to 2 d.p.)?

For practical measurement, round sensibly. Here $\sqrt{25}$ = 5 exactly, so to 2 d.p. is 5.00 m.

Exact fraction — do not convert to decimal

Solve $3x = 7$ . Give your answer as a fraction.

If asked for an exact answer, give $\frac{7}{3}$ . Only convert to decimal if specifically asked or if it's a practical context.

Deciding based on context

A recipe requires π metres of fabric. Should you give the answer exactly or approximately?

Exact formπ m

Use in theoretical or algebraic problems.

Approximate form3.14 m or 3.142 m

Use when buying actual fabric — you need a practical measurement.

Context decides: math theory → exact; practical purchasing → approximate to sensible precision.

Worked example — choosing the right form

A sector of a circle has radius 10 cm and angle 60°. The perimeter of the sector is the arc length plus two radii. Calculate the exact perimeter, then give a suitable approximation for a real-world context.

Identify what's asked: 'Calculate exact perimeter' — give the exact form first.
Use the arc length formula: angle/360 × 2πr = $\frac{60}{360}$ × 2π(10) = (10π)/3 cm.
Add the two radii: perimeter = (10π)/3 + 20 cm.
Exact form: (10π + 60)/3 cm or (10π)/3 + 20 cm. Both are equally valid.
For approximation: (10π + 60)/3 ≈ 30.47 cm (to 2 d.p.), suitable for measurement.
Always show the exact form first, then approximate only if the context demands it.

Try it

Use the examples carefully, then choose the answer.

Question 1

Solve 2x = 5. Which form is the 'exact answer'?

💡 Exact means no rounding. Keep it as a fraction.

Question 2

A circle has area 50 ${m}^{2}$ . What is the radius (exact answer)?

💡 Exact form uses surds and π; do not approximate.

Question 3

You are buying fabric for a project and the calculation gives π metres. What should you measure out?

💡 Real-world purchasing needs a practical approximation, not an exact surd.

Question 4

$\sqrt{18}$ = 3 $\sqrt{2}$ . If a geometry problem asks for an exact answer, which form is correct?

💡 Exact means simplified surd form, not a decimal approximation.

Question 5

A survey finds the average rainfall is (27 + $\sqrt{3}$ )/4 mm. For a weather report, what form should you use?

💡 News reports use practical approximations. Show 2–3 significant figures.

Common mistakes

Watch for these when working through the lesson.

Converting exact answers (surds, fractions) to decimals when the question asks for 'exact'.
Leaving answers like $\sqrt{18}$ unsimplified when they should be 3 $\sqrt{2}$ .
Forgetting to include π in exact answers for circle problems — writing 50 instead of 50π or (50/π).
Over-rounding in theoretical contexts (saying 3.1 instead of keeping π in an algebra problem).

Choosing exact values or estimates appropriately

Lesson overview

Worked example — choosing the right form

Try it

Common mistakes

Related topics

Independent practice will plug in here