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

1.Simplify √45.

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

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

5.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.

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

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

8.Expand and simplify √3(2 + √12).

9.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.

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

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

12.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.

13.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.

14.3/5 of a 5/6 litre bottle of juice is poured out. Work out the exact volume poured out, in litres.

15.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.