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

1.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.

2.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?

3.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?

4.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.

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

6.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.

7.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.

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

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

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

11.Expand and simplify √3(2 + √12).

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

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

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

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