Introduction

Welcome to Collins Further Maths Practice Book. This book will help you to progress with ease through more challenging maths work, with detailed examples and plenty of practice in the key areas needed to succeed in maths at a higher level. This book is suitable to support the AQA Level 2 Certificate in Further Mathematics.

Key Points And Worked Examples

Remember the important points of each topic with key words and succinct topic explanations. Gain a greater understanding of each area by studying the detailed worked examples before each exercise.

Graded Exercises

Consolidate your knowledge of topics and know exactly what level you are working at, with hundreds of graded practice questions. Stretch and challenge yourself with questions targetting grades 5 to 9.

Hints And Tips

Find valuable hints and tips boxes throughout the book that give you handy methods to try when answering questions.

Exam Preparation

Ensure you are ready for your exams by tackling the exam-style questions at the end of each chapter.

Answers

All answers to the practice exercises and exam-style questions in this book can be found at the back of the book.

1.1 Number: Recall and extension

THIS SECTION WILL SHOW YOU HOW TO ...

✓ understand and use the correct hierarchy of operations (BIDMAS / BODMAS)

✓ understand and use decimals, fractions and percentages

✓ understand and use ratio and proportion

✓ understand and use numbers in index form and standard form

✓ understand rounding and give answers to an appropriate degree of accuracy

KEY WORDS

✓ ratio

✓ proportion

✓ standard form

Most of the calculations in this chapter will require the combination of at least two basic number skills. For example, questions involving fractions will also test order of operations, a conversion of units may have several steps and a ratio problem may be combined with a percentage problem.

Questions may be answered with or without a calculator. If you use a non-calculator method, it is good practice to use a calculator to check your answers.

HINTS

Remember from BIDMAS / BODMAS that multiplication is carried out before addition.

It is sensible to insert brackets before carrying out the calculation.

EXAMPLE 3

For every 3 boys in a class, there are 5 girls. There are 6 more girls than boys. How many students are there in the class altogether?

The ratio of boys to girls is 3 : 5, giving 8 ‘parts’.

So the difference is 5 – 3 = 2 parts, which is equivalent to 6 students.

1 part is equivalent to 3 students.

There are 9 boys and 15 girls in the class, giving 24 students altogether.

EXAMPLE 4 x : y = 3 : 5 a) Write y in terms of x. b) Show that x is 30% of 2y. a) x y = 3 5 ⇒ 5x = 3y, y = 5x 3 b) x = 3 5 y ⇒ x = 3 10 of 2y and 3 10 = 30%

EXAMPLE 5

Calculate 43 82 ÷ 210

Give your answer as a mixed number.

Remember from BIDMAS / BODMAS that division is carried out before subtraction.

1. Work out each of these. Give your answers as mixed numbers.

Use a calculator to check your answers.

2. Work out each of these. Give your answers as mixed numbers.

Use a calculator to check your answers.

3. The ratio of A : B = 5 : 4 and the ratio of B : C = 3 : 7. Work out the ratio of A : C. Give your answer in its simplest form.

4. z is 10% more than y. y is 20% more than x. By what percentage is z more than x?

5. At a show, two-fifths of the people are men and the rest are women. 30% of the men are over 60 years old. 40% of the women are over 60 years old.

What proportion of all the people at the show are over 60?

6. Work out the number of minutes in January. Give your answer in standard form.

7. In a school, there are equal numbers of boys and girls. At the end of the school year, 20% of the boys leave and 25% of the girls leave. There are now 450 girls in the school.

Work out the number of boys now in the school.

8. How many times is 2.56 × 1027 bigger than 5 × 105? Give your answer in standard form.

9. A fuel tank is three- quar ters full . Then 10% of the fuel is taken out. There are now 54 litres in the tank. How much will the tank hold when full?

10. The price of a loaf is increased by 14%. This week a shop sells 12% fewer loaves than last week.

Does the shop take more or less money from selling loaves this week? You must show your working.

11. These three numbers are written in standard form.

2.3 × 107 8.1 × 105 5.7 × 106 a) Work out the smallest answer when two of these numbers are multiplied together. b) Work out the largest answer when two of these numbers are multiplied together. Give your answers in standard form.

12. At the start of the year, a train is half full. Each week the number of people catching this train increases by 1%.

How many weeks will it be before the train is full?

13. a : b = 1 : 4 a) Write b in terms of a. a) Work out the ratio 3a + b : a + 5b, giving your answer in its simplest form.

14. Given that x% of 2y = 25, work out the value of y% of 3x.