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

1.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?

2.Simplify √45.

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

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

5.Expand and simplify (2 + √3)², giving your answer in the form a + b√3.

6.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.

7.The two shorter sides of a right-angled triangle are √12 cm and √24 cm. Work out the exact length of the hypotenuse.

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

9.3/5 of a 5/6 litre bottle of juice is poured out. Work out the exact volume poured out, in litres.

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

11.Expand and simplify √3(2 + √12).

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

13.Work out 1 1/2 ÷ 3/4 exactly, giving your answer in its simplest form.

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

15.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?