1

1 Number

1 Round these numbers to 1 s.f. a 456 b 24 c 9.1 d 7.58

2 Estimate the answer to these calculations by first rounding each value to 1 s.f.

a 1.7 + 12.2 ________ 5.9

b √ ___________

84.37 + 18.6

3 The number of matches in a box is 50 with an error interval of 6%. a Calculate 6% of 50.b Calculate

i 50 − 6 % of 50 ii 50 + 6 % of 50c Copy and complete this inequality to show the possible number of matches, m, in a box.

< m <

Key point 1

A 10% error interval means that a measurement m could be up to 10% larger or smaller than the one given.

< m < 45 kg 50 kg 55 kg

210% 110%

Key point 2

To estimate the answer to a calculation, you can round every number to 1 significant figure (s.f.).

Warm

up

1.1 CalculationsObjectives

• Estimate answers to calculations.• Write an inequality to represent an error interval.• Use a calculator for complex calculations.

What is covered in this sectionFoundation Student Book reference

1.1 Calculations• Estimate answers to calculations.• Write an inequality to represent an error interval.• Use a calculator for complex calculations.

Unit 1

1.2 Factors and multiples• Write a number as a product of its prime factors.• Use prime factor decomposition to find the HCF and LCM of two numbers.

Unit 1

1.3 Fractions• Add, subtract, multiply and divide mixed numbers.

Units 4, 18

1.4 Indices, powers and roots• Simplify expressions involving indices, roots and surds.• Understand and use zero and negative indices.

Units 1, 18

1.5 Standard form• Write very small and very large numbers in standard form.• Calculate with numbers in standard form.

Unit 18

1.6 Mixed exercise• Consolidate your learning with more practice.

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4 Shampoo bottles contain 240 ml shampoo with an error interval of 5%. Write an inequality to show the possible volume of shampoo, s, in a bottle.

5 A machine produces 20 m rolls of paper with an error interval of 10%. Write an inequality to show the possible lengths of paper in a roll.

6 Problem-solving A machine fills packs of crisps with 120 g of crisps, with an error interval of 3%.The crisps label says, ‘Minimum contents 116 g’. Is this true? Show how you get your answer.

7 Tom calculates that 892.17 ÷ 18.4 = 85.8 (1 d.p.). Estimate the answer to his calculation to show he is wrong.

8 Use estimation to see which of these calculations is wrong and which could be correct.

a 3. 2 2 + 9.17 _________ 6.19 − 2.3

= 4.99 (3 s.f.) b √ __________

2. 17 2 + 57.2 __________ 3. 9 2

= 3.98 (3 s.f.)

c 5436 − 82 2 __________ 5.36 + 12.97

= 149 (3 s.f.) d 9. 8 3 × 3.7 ________ 1.8 × 5. 2 2

= 71.5 (3 s.f.)

9 Use a calculator to work out the correct answer to the calculations in Q8 that were wrong. Round to 3 s.f.

10 Use a calculator to work these out. Round your answers to the number of s.f. given.

a √ __________

4. 1 2 + 2. 15 3 __________ 3. 8 2

(2 s.f.) b 3 √ _________

24 − 3. 29 2 _________ 11.7 + 3.2

(3 s.f.)

c 3 √ _________

5 2 + 13. 2 3 ________ 19.2

(3 s.f.) d 3 + √

______________

(−3) 2 + 2 × 4 × 3 __________________

6 (4 s.f.)

Q4 A common error is to use , instead of < signs.

Q6 A common error is to write ‘Yes’ or ‘No’ and to not show calculations to explain why.

Q9 A common error is to round to 3 d.p. instead of 3 s.f.

Q10 hint Estimate the answers first, in order to check your calculations.

1.2 Factors and multiplesObjectives

• Write a number as a product of its prime factors.• Use prime factor decomposition to find the HCF and LCM of two numbers.

1 Write as a product of powers2 × 5 × 3 × 5 × 2 × 2

2 Find a the HCF of 12 and 32b the LCM of 6 and 10.

3 a Copy and complete these factor trees.b Write 84 and 105 as products of their prime factors.

Q3 hint Write the products of factors using index notation, smallest factor first.

Key point 3

All numbers can be written as a product of prime factors. This is called prime factor decomposition.

War

m u

p

Q1 hint 2 × 3 × 5

84

2

105

5

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3

4 What is the prime factor decomposition of 72?

5 Copy and complete this Venn diagram to find the HCF and LCM of 84 and 72.

Prime factorsof 84

Prime factorsof 72

2

6 Find the HCF and LCM ofa 105 and 84 b 28 and 70 c 132 and 60

7 Two numbers have HCF 8 and LCM 120. What are the two numbers?

8 Problem-solving Find two numbers with LCM 70 and HCF 5.

9 A = 22 × 5 × 7 B = 3 × 52 × 11C = 22 × 13 D = 2 × 32 × 72

a Find the HCF ofi A and D ii A and C iii B and D

b Which two numbers have LCMi 1820 ii 42 900 iii 8820

c Which two numbers have no common factors?

10 Problem-solving Number 6 buses stop at the railway station every 9 minutes.Number 15 buses stop at the railway station every 12 minutes.At 10 am a number 6 bus and a number 15 bus both stop at the station.When will a number 6 bus and a number 15 bus next stop at the station together?

Q5 A common error is to give the LCM answer for the HCF or the HCF for the LCM.

Q7 hint

8

× 8 × = 120

Q9 hint Draw Venn diagrams.

Q10 A common error is to not realise that this is an LCM question.

1.3 FractionsObjective

• Add, subtract, multiply and divide mixed numbers.

1 Work out

a 5 _ 6 + 3 _ 5 b 4 _ 9 − 1 _ 6 c 3 _ 4 × 5 _ 7 d 2 _ 3 ÷ 3 _ 8

Give your answers as mixed numbers where appropriate.

2 Cancel then multiply

a 5 _ 6 × 3 _ 7 b 4 _ 7 × 3 _ 8

c 6 __ 12 × 2 _ 3 d 7 _ 9 × 3 __ 14

3 Work out

a 2 1 _ 5 + 1 2 _ 5 b 3 1 _ 7 + 2 3 _ 5

c 5 1 _ 4 + 1 4 _ 9 d 4 3 __ 10 + 1 1 _ 3

Key point 4

To add, subtract, multiply or divide mixed numbers, write them as improper fractions first.

Warm

up

Q2 hint 5 × 3 _____ 6 × 7

= 1 3 ⟋ × 5 ______ 2 6 ⟋ × 7

= __

Q3 hint Write your answers as mixed numbers.

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4 Work out

a 2 3 _ 5 + 1 7 __ 10 b 3 5 _ 8 + 4 3 _ 4

c 5 6 _ 7 + 2 2 _ 3 d 1 11 __ 12 + 3 5 _ 9

5 Work outa 5 4 _ 9 − 1 1 _ 5 b 3 3 _ 4 − 1 5 _ 8 c 6 6 _ 7 − 3 1 _ 3 d 4 8 __ 11 − 3 2 _ 5

6 Work outa 4 1 _ 4 − 1 3 _ 5 b 3 1 _ 7 − 2 1 _ 4 c 5 2 _ 3 − 2 7 _ 9 d 4 5 __ 12 − 2 2 _ 3

7 Work out

a 3 1 _ 5 × 2 _ 3 b 2 _ 5 × 2 1 _ 4

c 5 _ 7 × 3 1 _ 2 d 3 _ 4 × 2 2 _ 5

8 Work out

a 2 1 _ 4 ÷ 1 _ 5 b 1 1 _ 2 ÷ 4 _ 7

c 3 2 _ 3 ÷ 5 _ 6 d 1 7 _ 8 ÷ 5 _ 8

9 Work outa 2 1 _ 2 × 1 1 _ 6 b 1 2 _ 3 × 1 1 _ 8 c 1 1 _ 7 × 3 3 _ 4 d 3 1 _ 5 × 2 1 _ 2

10 Work outa 2 3 _ 4 ÷ 2 1 _ 3 b 1 4 _ 5 ÷ 1 3 _ 4 c 2 1 _ 4 ÷ 3 3 _ 5 d 2 2 _ 3 ÷ 1 7 _ 9

11 Problem-solving Karl has these bags of sand.He needs 8 kg of sand to make cement.Does he have enough? Show working to explain.

12 Problem-solving Ailsa has 8 3 _ 5 metres of wire. a How many 1 1 _ 2 metre lengths can she cut from it?b How much is left over?

Q4 A common error when adding or subtracting fractions is to add/subtract the numerators and denominators.

Q7 A common error is to not to change to improper fractions first.

Q8 A common error when dividing fractions is to forget to ‘flip’ the fraction you are dividing by, when you change the calculation to multiplication.

158

kg 21112

kg 3 910

kg

1.4 Indices, powers and rootsObjectives

• Simplify expressions involving indices, roots and surds.• Understand and use zero and negative indices.

Key point 5

Any number (or term) raised to the power 0 is equal to 1.Any number (or term) raised to the power –1 is the reciprocal of the number.To raise a power of a number (or term) to another power, multiply the indices.

Key point 6

3 −2 is the same as ( 3 −1 ) 2 .To find a negative power, find the reciprocal and then raise the value to the positive power.

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