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

1.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.

2.A pizza is cut into 12 equal slices. Ben eats 5 slices and Mia eats 3 slices. What fraction of the pizza is left, giving your answer in its simplest form?

3.Simplify √45.

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

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

7.Expand and simplify √3(2 + √12).

8.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.

9.Work out 5/6 × 2/9, giving your answer in its simplest form.

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

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

12.3/5 of a 5/6 litre bottle of juice is poured out. Work out the exact volume poured out, in litres.

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

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