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

1.Expand and simplify √3(2 + √12).

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.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?

4.A square tile has an area of 72 cm². Work out the exact perimeter of the tile, giving your answer in the form k√2 cm.

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

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

7.Simplify √45.

8.A shape is made by joining a rectangle measuring 10 cm by 6 cm to a semicircle of diameter 6 cm along one of the rectangle's shorter sides. Work out the exact area of the shape, in terms of π.

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

10.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.

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

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

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

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

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