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1 Number
Foundation Student
Book reference
What is covered in this section
1.1 Calculations
• Estimate answers to calculations.
• Write an inequality to represent an error interval.
• Use a calculator for complex calculations.
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.
1.3 Fractions
• Add, subtract, multiply and divide mixed numbers.
1.4 Indices, powers and roots
• Simplify expressions involving indices, roots and surds.
• Understand and use zero and negative indices.
1.5 Standard form
• Write very small and very large numbers in standard form.
• Calculate with numbers in standard form.
1.6 Mixed exercise
• Consolidate your learning with more practice.
Unit 1
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Unit 1
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Objectives
Units 1, 18
Unit 18
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1.1 Calculations
Units 4, 18
• Estimate answers to calculations.
• Write an inequality to represent an error interval.
• Use a calculator for complex calculations.
Key point 1
M
A 10% error interval means that a measurement m could be
up to 10% larger or smaller than the one given.
<m<
210%
45 kg
110%
50 kg
55 kg
SA
Key point 2
To estimate the answer to a calculation, you can round every number to 1 significant figure (s.f.).
c 9.1
d 7.58
2 Estimate the answer to these calculations by first rounding each value to 1 s.f.
___________
1.7 + 12.2
a ________
​ 
 
  
​
   
​
b √​  84.37 + 18.6 
5.9
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 50
c Copy and complete this inequality to show the possible number of matches, m, in a box.
<m<
Warm up
1 Round these numbers to 1 s.f.
a 456
b 24
1
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.
Q4 A common error is to use
, instead of < signs.
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.
Q6 A common error is to write
‘Yes’ or ‘No’ and to not show
calculations to explain why.
O
FS
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.
__________
3.​2​ 2​ + 9.17
2.​17​​ 2​ + 57.2
_________
a ​ 
​ 
 
= 3.98 (3 s.f.)
   
​  
 ​ 
= 4.99​(3 s.f.)b
​ __________
6.19 − 2.3
3.​9​ 2​
5436 − ​82​​ 2​
9.​8​ 3​ × 3.7
 
c __________
​ 
​ = 71.5​(3 s.f.)
 ​  
= 149​(3 s.f.)d
​ ________2  
5.36 + 12.97
1.8 × 5.​2​  ​
PR
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√ 
9 Use a calculator to work out the correct answer to the
calculations in Q8 that were wrong. Round to 3 s.f.
Q9 A common error is to round
to 3 d.p. instead of 3 s.f.
10 Use a calculator to work these out. Round your answers to the number of s.f. given.
__________
4.​1​ 2​ + 2.​15​​ 3​
__________
​ 
   
​  
 (2
3.​8​ 2​
24 − 3.​29​​  ​
s.f.)b
​√ ​ 
 ​ 
 ​ (3 s.f.)
√ 
11.7 + 3.2
−3)​​  ​ + 2 × 4 × 3 ​
3 + ​√ ​(  
​5​  ​ + 13.​2​  ​
__________________
​ 
 
   
​  (3 s.f.)d
 
​  (4 s.f.)
​   
​√ ________
19.2
6
a ​
_________
3
 
2
3
 
______________
2
Q10 hint Estimate the
answers first, in order to
check your calculations.
PL
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c
_________
2
3 _________
 
 
 
1.2 Factors and multiples
Objectives
M
• Write a number as a product of its prime factors.
• Use prime factor decomposition to find the HCF and LCM of two numbers.
SA
Key point 3
Warm up
All numbers can be written as a product of prime factors. This is called prime factor decomposition.
1 Write as a product of powers
2×5×3×5×2×2
2 Find
a the HCF of 12 and 32
b 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.
2
Q1 hint 2 × 3 × 5 84
2
105
5
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 factors
of 84
Prime factors
of 72
Q5 A common error is to give the LCM
answer for the HCF or the HCF for the LCM.
2
Q7 hint O
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6 Find the HCF and LCM of
28 and 70c
132 and 60
a 105 and 84b
7 Two numbers have HCF 8 and LCM 120.
What are the two numbers?
8
8 Problem-solving  Find two numbers with LCM 70 and HCF 5.
PR
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9 A = 22 × 5 × 7 B = 3 × 52 × 11
2
C = 2 × 13
D = 2 × 32 × 72
a Find the HCF of
i A and D ii A and C
iii B and D
b Which two numbers have LCM
i 1820
ii 42 900
iii8820
c Which two numbers have no common factors?
×8×
Q9 hint Draw Venn diagrams.
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?
PL
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= 120
Q10 A common
error is to not
realise that this is
an LCM question.
1.3 Fractions
M
Objective
• Add, subtract, multiply and divide mixed numbers.
SA
Key point 4
To add, subtract, multiply or divide mixed numbers, write them as improper fractions first.
2 Cancel then multiply
a ​ _56 ​ × ​ _37 ​
6
c __
​ 12
  ​ × ​ _23 ​
b
d
​ _47 ​ × ​ _38 ​
3
_
​ 79 ​ × ​ __
14  ​
3 Work out
a ​2​ _15 ​ + 1​ _25 ​b
​3​ _17 ​ + 2​ _35 ​
3
_1
c ​5_​ 14 ​ + 1​ _49 ​d
​4​ __
10  ​ + 1​ 3 ​
d _​ 23 ​ ÷ ​ _38 ​
3   ​​​​ × 5 __
​​1​ ​ ⟋
5 × 3 ______
Q2 hint _____
​ 
 ​  
= ​   ​
 ​  = ​  2
6 × 7 ​​ ​ ​ ⟋
6   ​​​​ × 7
Warm up
1 Work out
a ​ _56 ​ + ​ _35 ​
b ​ _49 ​ − ​ _16 ​
c _​ 34 ​ × ​ _57 ​
Give your answers as mixed numbers where appropriate.
Q3 hint Write your answers as mixed numbers.
3
4 Work out
7
a 2
​ _​ 35 ​ + 1​ __
3
​ _​ 58 ​ + 4​ _34 ​
10  ​b
11
_5
c ​5_​ 67 ​ + 2​ _23 ​d
​1​ __
12 ​ + 3​ 9 ​
Q4 A common error when adding or
subtracting fractions is to add/subtract
the numerators and denominators.
5 Work out
8
_2
a ​5​ _49 ​ − 1​ _15 ​b
​3​ _34 ​ − 1​ _58 ​c
​6​ _67 ​ − 3​ _13 ​d
​4​ __
11  ​ − 3​ 5 ​
6 Work out
5
_2
a ​4_​ 14 ​ − 1​ _35 ​b
​3​ _17 ​ − 2​ _14 ​c
​5​ _23 ​ − 2​ _79 ​d
​4​ __
12  ​ − 2​ 3 ​
7 Work out
a ​3_​ 15 ​ × ​ _23 ​b
​ _25 ​ × 2​ _14 ​
c _​ 57 ​ × 3​ _12 ​d
​ _34 ​ × 2​ _25 ​
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8 Work out
a ​2​ _14 ​ ÷ ​ _15 ​b
1
​ ​ _12 ​ ÷ ​ _47 ​
c ​3_​ 23 ​ ÷ ​ _56 ​d
1
​ ​ _78 ​ ÷ ​ _58 ​
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.
PR
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9 Work out
a ​2​ _12 ​ × 1​ _16 ​b
​1​ _23 ​ × 1​ _18 ​c
​1​ _17 ​ × 3​ _34 ​d
​3​ _15 ​ × 2​ _12 ​
10 Work out
a ​2​ _34 ​ ÷ 2​ _13 ​b
​1​ _45 ​ ÷ 1​ _34 ​c
2​​ _14 ​÷ 3​_​ 35 ​d
2
​ _​ 23 ​ ÷ 1​ _79 ​
PL
E
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.
1 5 kg
8
2 11 kg
12
3 9 kg
10
M
12 Problem-solving  Ailsa has ​8​ _35 ​metres of wire.
a How many 1
​ _​ 12 ​metre lengths can she cut from it?
b How much is left over?
1.4 Indices, powers and roots
SA
Objectives
• 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.
4