Chapter 1
Significant figures, percentages and fractions
What you should know

You should know how to:
★★ round to a given number of significant figures
★★ use percentages to calculate compound interest, appreciation and
depreciation
★★ use reverse percentages to calculate an original quantity
★★ add, subtract, multiply and divide common fractions including
mixed numbers
(and work with combinations of these operations).
Rounding
Answers to calculations are often rounded to an appropriate degree of
accuracy. For example, answers to money calculations are often rounded
to 2 decimal places (to the nearest penny). For example, 17∙5% of £29∙99 =
£5∙24825 = £5∙25 to 2 decimal places. A number rounded to n decimal
places, has n digits to the right of the decimal point.
Another kind of rounding uses significant figures. All figures in a
­number are significant except:
●● zeros at the end of a whole number
●● zeros at the start of a number with a decimal point.
Example
f
1 70 518 has 5 significant figures.
70 518 = 70 520 correct to 4 sig. figs.
70 518 = 70 500 correct to 3 sig. figs.
70 518 = 71 000 correct to 2 s.f.
70 518 = 70 000 correct to 1 s.f.
2 0∙003907 has 4 significant figures
0∙003907 = 0∙00391 correct to 3 sig. figs.
0∙003907 = 0∙0039 correct to 2 s.f.
0∙003907 = 0∙004 correct to 1 s.f.
Zeros at the end of a whole
number are not significant.
Zeros at the start of a number
with a decimal point are not
significant.
Compound interest
Interest is money paid to you by a bank or building society for saving with
them (or charged to you for borrowing from them).
The original amount you save (or borrow) is called the principal.
Compound interest is interest paid on both the principal and any interest
already added.
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Chapter 1
Example
f
Morag invests £8000 in a savings account for 3 years at an interest rate of 4% per annum.
Calculate the amount in Morag’s account at the end of the 3 years.
Solution
Each year the amount is 104% (100% + 4%) of the previous year’s amount.
104
After 1 year:
amount = 1∙04 × 8000 = £8320
After 2 years:
amount = 1∙04 × 8320 = £8652∙80
(1∙04)2 × 8000 = £8652∙80
or
After 3 years:
Since 104% = 100 = 1∙04
amount = (1∙04)3 × 8000 = £8998∙91 (to the nearest penny)
Appreciation
When the value of something increases we say its value has appreciated.
Example
f
A house valued at £180 000 at the end of 2011 appreciated in value by 5% in 2012 and by 4∙5% in
2013. Calculate its value at the end of 2013.
Solution
After 1 year the house was worth 105% (100% + 5%) of its value at the end of 2011.
After 2 years the house was worth 104∙5% (100% + 4∙5%) of its value at the end of 2012.
End of 2012: value = 1∙05 × 180 000 = £189 000
Since 105% = 105 = 1∙05
End of 2013: value = 1∙045 × 189 000 = £197 505
Since 104∙5% =
100
104.5
= 1∙045
100
Depreciation
When the value of something decreases we say its value has depreciated.
Example
f
A car bought for £15 000 depreciated by 20% during its first year and by a further 15% during its
second year. Calculate its value after two years.
Solution
After 1 year the car was worth 80% (100% – 20%) of its original value.
After 2 years the car was worth 85% (100% – 15%) of its value after 1 year.
80
8
After 1 year: value = 0∙8 × 15 000 = £12 000
Since 80% =
= = 0∙8
100 10
After 2 years: value = 0∙85 × 12 000 = £10 200
85
Since 85% = 100 = 0∙85
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Significant figures, percentages and fractions
Reversing a percentage change
f
Example
1 The price of a family room for two nights at the Bay Hotel is £210 including Value Added Tax
(VAT) at 20%.
Calculate the price excluding VAT.
Solution
Price including VAT = 120% (100% + 20%) of price excluding VAT.
This means
£210 = 1∙2 × price excluding VAT.
So price excluding VAT = £210 ÷ 1∙2
= £175.
2 There are 720 pupils in Westside High School this year. This is 4% less than last year.
How many pupils were in Westside High School last year?
Solution
Number of pupils this year = 96% (100% – 4%) of number of pupils last year.
This means
720 = 0∙96 × number of pupils last year.
So number of pupils last year = 720 ÷ 0∙96
= 750.
Fractions
You will be expected to carry out calculations like these in the
non-calculator paper.
Example
f
1
2
3
1 Calculate + 7 .
Solution
1
2
3
+ 7 = 147 + 146
=
Make the denominators the same.
13
14
Add the numerators.
Do not add the denominators.
4
5
2 Calculate –
7
15.
Solution
4
5
7
– 157 = 12
15 – 15
15
Make the denominators the same.
=3 15
Subtract the numerators.
=
Divide both numerator and denominator by 5.
1
3
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Chapter 1
3 Calculate 1 49 + 2 56 .
Solution
15
1 49 + 2 56 = 1 188 + 2 18
Make the denominators the same.
Add the whole numbers and add the numerators.
23
= 3 18
= 4 185
Convert
5
23
to 1 18 .
18
4 Calculate 6 41 – 4 23 .
Solution
6 41 – 4 23 = 6 123 – 4 128
15
= 5 12
– 4 128
=
1 127
Make the denominators the same.
Convert 6 3 to 5 + 1 3 = 5 15
12
12
12
Subtract the whole numbers and subtract the numerators.
15
5 Calculate 49 × 16
.
Solution
14
9
15
16
4
×
15 5
1
×
4
39
1
5
3 × 4
=
=
= 125
Divide both the 4 and the 16 by 4.
Divide both the 9 and the 15 by 3.
Multiply numerators together and multiply denominators together.
6 Calculate 2 58 × 3 31 .
Solution
2 58 × 3 31 =
7 21
8
× 103
7
×
48
= 74 × 51
= 354
= 84
=
1
10 5
1
Divide both the 21 and the 3 by 3.
Divide both the 8 and the 10 by 2.
Multiply numerators together and multiply denominators together.
3
7 Calculate
Change the mixed numbers to vulgar fractions.
Change the vulgar fraction to a mixed number.
7
12
÷ 58 .
×
82
5
Solution
7
12
÷ 58 =
7
12
3
Change the × to a ÷ and turn the second fraction upside down.
= 73 × 25
Divide both the 12 and the 8 by 4.
= 14
15
Multiply numerators together and multiply denominators together.
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Significant figures, percentages and fractions
8 Calculate 2 23 ÷ 1 51.
Solution
2 23 ÷ 1 51 = 83 ÷ 56
48
Change the mixed numbers to vulgar fractions.
5
63
=
×
= 43 × 53
Divide both the 8 and the 6 by 2.
= 209
Multiply numerators together and multiply denominators together.
= 2 29
Change the vulgar fraction to a mixed number.
3
Change the × to a ÷ and turn the second fraction upside down.
9 Calculate 2 51 + 107 of 1 76 .
Solution
7
10
of 1 76 = 107 × 1 76
17
Calculate 7 of 1 6 first. Remember BODMAS.
10
7
= 10 × 137
Convert the mixed number to a vulgar fraction.
= 101 × 131
Divide both the 7s by 7.
13
= 10
Multiply numerators together and multiply denominators together.
= 1 103
Convert the vulgar fraction to a mixed number.
1
Now add the answer to 2 1.
5
2 51
+
1 103
=
2 102
+
1 103
51
= 3 10
= 3 21
2
Make the denominators the same.
Add the whole numbers and add the numerators.
Divide both numerator and denominator by 5.
10 Calculate 54 of (1 89 – 21 ).
Solution
1 89 – 21 = 1 16
– 189
18
Calculate 1 8 – 1 first. Remember BODMAS.
9
2
Make the denominators the same.
4
5
of
= 1 187
1 187 = 54 ×
24
Subtract the numerators.
1 187
Now calculate 4 of the answer.
5
25
× 18
Convert the mixed number to a vulgar fraction.
×
Divide both the 4 and the 18 by 2.
=
=
= 21 × 59
Divide both the 5 and the 25 by 5.
= 109
Multiply the numerators together and multiply the denominators together.
= 1 91
Convert the vulgar fraction to a mixed number.
5
2
15
9
25 5
9
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Chapter 1
For practice
s
(The answers to the following questions are given in Appendix 1.)
1 Round
a) 52 390 to 1 significant figure
b) 0∙0666 to 2 s.f.
c) 208 517 to 3 s.f.
2 Ruth invested £2400 in a savings account for 3 years.
The interest rate was 4% per annum in the first year, 3∙5% per annum in the second year and
3∙75% per annum in the third year.
Calculate how much Ruth had in the account after the 3 years.
3 A cottage bought for £80 000 three years ago appreciated in value by 5% per annum.
How much is the cottage worth now?
4 Jim buys a motor bike for £3000. Its value depreciates by 25% over the first year and by a further
20% during the second year.
How much is Jim’s motor bike worth after the two years?
5 After receiving a 6% pay rise, Julie is now paid £18 550 per year.
How much was Julie paid per year before the pay rise?
6 A store reduces all its prices by 35% in a sale. The sale price of a laptop is £299.
Calculate the price of the laptop before the sale.
7 Calculate 23 + 74 .
3
8 Calculate 2 4 + 3 51.
9 Calculate 56 – 83 .
10 Calculate 6 21 – 1 109 .
11 Calculate 127 × 89 .
12 Calculate 1 78 × 2 29 .
3
13 Calculate 4 ÷ 56 .
14 Calculate 1 53 ÷ 4 23 .
3
15 Calculate 3 4 – 23 of 1 58 .
3
16 Calculate 73 of (2 61 + 4 ).
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