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

1.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.

2.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.

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

4.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?

5.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.

6.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?

7.Expand and simplify √3(2 + √12).

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

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

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

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

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

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

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

15.Simplify √45.