Exact calculation: fractions, surds and π worksheet — GCSE Higher

1.3/5 of a 5/6 litre bottle of juice is poured out. Work out the exact volume poured out, in litres.

2.Rationalise the denominator of 6/√3, giving your answer in its simplest form.

3.Simplify √8 + √18, giving your answer in the form k√2.

4.A quarter-circle has a radius of 6 cm. Work out the exact perimeter of the quarter-circle, giving your answer in terms of π.

5.A circle has a radius of 3 cm. Which of these is the exact area of the circle?

6.Work out 3/4 − 5/12 exactly, giving your answer in its simplest form.

7.A semicircle has a diameter of 8 cm. Work out the exact area of the semicircle, in terms of π.

8.Simplify √45.

9.A circle has a circumference of 8π cm. Work out the exact area of the circle in terms of π.

10.A cake recipe needs 3/4 of a kilogram of sugar. Aisha wants to make half the recipe. Work out how much sugar she needs, giving your answer as a fraction of a kilogram in its simplest form.

11.A student works out the exact area of a circle with radius 4 cm by squaring the radius but forgetting to multiply by π. Work out the correct exact area of the circle, in terms of π.

12.Rationalise the denominator of 10/(4 − √6), giving your answer in its simplest form.

13.In a school, 3/5 of the students study French. Of these students, 2/3 also study Spanish. What fraction of all the students study both French and Spanish?

14.Work out √3 × √12, giving your answer as an integer.

15.Chloe wants to work out 3 1/4 − 1 2/3. She converts both mixed numbers to twelfths, then subtracts the whole numbers and the fraction parts separately, without checking whether she needs to exchange first. Work out the correct value of 3 1/4 − 1 2/3, giving your answer as a mixed number in its simplest form.