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1
Topic 2
Fractions, decimals and
percentages
Contents
2
...
2
...
2
5
2
...
8
2
...
2
...
1 Converting mixed numbers to improper fractions
...
4
...
2
...
11
12
2
...
2
...
14
17
2
...
19
Learning Objectives
By the end of this topic, you should be able to:
• calculate compound interest;
• calculate appreciation;
• calculate depreciation;
• reverse percentages;
• convert mixed numbers to improper fractions;
• convert improper fractions to mixed numbers;
• add, subtract, multiply and divide fractions
...
FRACTIONS, DECIMALS AND PERCENTAGES
2
...
The
percentage must be entered as a fraction or a decimal
...
Problem:
Express 95% as a decimal fraction
...
2
...
Solution:
106% = 106 /100 = 1·06
...
Problem:
Express 4% as a decimal fraction
...
4
...
5% as a decimal fraction
...
5 /
100
= 0·175
...
Simple interest
Problem:
Calculate the interest paid by a bank if £250 is saved for a year at 3%
...
© H ERIOT-WATT U NIVERSITY
TOPIC 2
...
...
...
...
...
Compound interest
Performing compound interest calculations
£800 is invested in a savings account
...
Interest is paid
annually at 3% per annum
...
Year 0
£800
No interest at start
Year 1
£800 × 1·03 = £824
Simple interest = £24
Year 2
£800 × 1·03 × 1·03
£800 × 1·03 2 =
£848·72
Year 3
£800 × 1·03 × 1·03 ×
1·03
£800 × 1·03 3 =
£874·18
Year 4
Compound interest =
£74·18
£800 × 1·03 × 1·03 ×
1·03 × 1·03
£800 × 1·03 4 =
£900·41
Year 5
Compound interest =
£48·72
Compound interest =
£100·41
£800 × 1·03 × 1·03 ×
1·03 × 1·03 × 1·03
£800 × 1·03 5 =
£927·42
Compound interest =
£127·42
...
FRACTIONS, DECIMALS AND PERCENTAGES
Top Tip
Remember:
Calculations involving money should have 2 decimal places unless you are told
otherwise
...
Examples
1
...
3·5% interest is paid annually
...
By the end of 4 years the amount in the bank will be 640 × 1·035 × 1·035 × 1·035 ×
1·035 = £734·41
Written in shorter form this becomes 640 × 1·035 4 = £734·41
...
Problem:
£5000 is invested in a bank for 10 years
...
Calculate the
compound interest at the end of 10 years
...
By the end of 10 years the amount in the bank will be 5000 × 1·02 10 = £6094·97
The compound interest is the principal sum subtracted from the amount in the bank at
the end of the period 6094·97 - 5000 = £1094·97
...
Q6: How much money will be in the bank after 5 years when £480 is deposited and
the interest rate is 2% per annum?
Give your answer to the nearest whole pound
...
Q7: £1980 is invested in a bank for 3 years
...
Calculate the
compound interest at the end of 3 years
...
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TOPIC 2
...
Interest is paid at 1·8% per
annum
...
...
Interest is paid at 6% per annum
...
b) Calculate the compound interest paid at the end of 7 years
...
Q10: £2472 is invested in a savings account for 2 years
...
Calculate the compound interest paid at the end of 2 years
...
2
...
Things which commonly appreciate are antiques and property
...
Things which commonly depreciate are cars and mobile phones
...
Appreciation
Problem:
A house was valued at £125,000 and sold 2 years later
...
Calculate the value of the house when it was sold
...
When sold the value is 125000 × 1·04 × 1·07 = £139100
...
Depreciation
Problem:
A computer worth £1200 loses 5% of its value each year
...
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5
6
TOPIC 2
...
Written in shorter form this becomes 1200 × 0·95 5 = £928·54
...
Appreciation and depreciation
Problem:
A car is said to depreciate most during the first 3 years
...
Give your answer to the nearest pound
...
82% = 82 /100 = 0·82
After 3 years the car is worth 10995 × 0·82 × 0·86 × 0·89 = £6901
...
Appreciation and depreciation
Problem:
A collector bought an antique clock for £1500
...
It increased in value by 3% in the second year
...
What is the clock worth after 3 years?
Solution:
1500 × 0·98 × 1·03 × 0·99 = £1498·96
...
Calculate the value of the painting now
...
Q12: A car worth £15000 loses 8% of its value a year
...
...
It depreciated by 20% in the first year and 12% in the second year
...
...
FRACTIONS, DECIMALS AND PERCENTAGES
Q14: A collector bought an antique vase for £600
...
It decreased in value by 2% in the
second year
...
What is the vase worth after 3 years?
Give your answer to the nearest penny
...
Appreciation and depreciation exercise
Q15:
a) Express 86% as a decimal
...
d) Express 2·6% as a decimal
...
Q16: Appreciation
An antique valued at £3541 appreciates by 11%
...
Q17: Appreciation
A house was valued at £215000 and sold 2 years later
...
Calculate the value of the house when it was sold
...
Q18: Depreciation
A car valued at £2027 depreciates by 10%
...
Q19: Depreciation
A yacht was bought new for £86000
...
Calculate the value of the yacht now
...
...
The vase increases in value by 3% in the first year
...
It decreases in value by 3% in the third year
...
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8
TOPIC 2
...
3
Undoing percentages
There are many methods for undoing or reversing the application of a percentage
...
Method 1
Problem:
The sale price of a bicycle after it has been reduced by 20% is £135·20
...
Solution:
After the bike has been reduced by 20% it is now priced at 80% of the original price
...
2
...
Calculate the weight of an original tin
...
We can write an expression for calculating the weight of the special offer size
...
Top Tip
Try the method that makes most sense to you in the following questions
...
Calculate the capacity of the stadium before it was increased
...
Q22: The value of a car had depreciated in value by 22% to £7683
...
...
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TOPIC 2
...
Calculate the population on the island last year
...
Q24: Undoing a percentage decrease
A school role has dropped this session by 10% to 1134
...
...
a) This was 20% increase on last year
...
How much did they pay for gas
the previous year?
...
4
Improper fractions and mixed numbers
An improper fraction is a fraction in which the numerator is larger than the denominator
...
For example, 61 /4
2
...
1
Converting mixed numbers to improper fractions
Converting mixed numbers to improper fractions
We need to convert the mixed number 1 1 /3 to an improper fraction
...
35 /8 = 3 + 5 /8 = 1 + 1 + 1 + 5 /8
35 /8 = 8 /8 + 8 /8 + 8 /8 + 5 /8
35 /8 = 24 /8 + 5 /8 = 29 /8
...
Problem:
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9
10
TOPIC 2
...
Solution:
3
3
= 5+
5
4
4
5
=
=
=
1+1+1+1+1
20 3
+
4
4
23
4
=
4 4 4 4 4
+ + + +
4 4 4 4 4
=
20
4
add
...
Problem:
Convert 42 /7 to an improper fraction
...
Top Tip
To type an answer like this into the assessment system enter 30/7
...
...
...
a) 21 /2
b) 12 /3
c) 43 /5
d) 85 /6
e) 113 /7
...
FRACTIONS, DECIMALS AND PERCENTAGES
2
...
2
Converting improper fractions to mixed numbers
Converting improper fractions to mixed numbers
Here we want to convert the improper fraction 8 /3 to a mixed number
...
25 /
4
= 4 /4 + 4 /4 + 4 /4 + 4 /4 + 4 /4 + 4 /4 + 1 /4
25 /
4
= 1 + 1 + 1 + 1 + 1 + 1 + 1 /4 = 61 /4
25 divided by 4 = 6 remainder 1, where 6 is the whole number and 1 is the quarters
...
Examples
1
...
Solution:
17 divided by 5 = 3 remainder 2
This tells us the whole number part of the answer is 3
...
17 /
5
= 32 /5
...
Problem:
Convert 32 /9 to a mixed number
...
The remainder tells us the number of ninths
...
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11
12
TOPIC 2
...
Practice converting improper fractions to mixed numbers
Q29: Convert 19 /4 to a mixed number
...
Q30: Convert 43 /6 to a mixed number
...
Converting Improper fractions to mixed numbers exercise
Q31: Improper fractions to mixed numbers
Change the improper fractions to mixed numbers
...
2
...
We need to know how to add fractions together
...
+
3
4
×
5
5
Multiply for a common denominator
...
Change the improper fraction to a mixed number
...
Example
Problem:
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TOPIC 2
...
Add the fractions
...
Add the whole
numbers
...
Now practice adding fractions
...
...
Let's look at how to subtract a fraction from a fraction
...
=
5
3
3 − 5
5
5
3 × 5 −
25
9
15 − 15
=
16
15
=
=
We need a common denominator
...
Now we can subtract
...
1
= 1 15
...
FRACTIONS, DECIMALS AND PERCENTAGES
1
4
− 3
2
3
=
21
4
=
5
21
4
63
12
19
12
=
=
−
×
3
3
−
11
3
−
11
3
×
4
4
44
12
We need improper fractions and a common
denominator
...
Now we can subtract the fractions
...
7
= 1 12
...
Q33:
Calculate 41 /2 - 13 /5
Write down your working
...
Adding and subtracting fractions exercise
Q34: Adding fractions
Give your answers as mixed numbers
...
b) Calculate 22 /3 + 1 /2
Hint: Find a common denominator for the fractions then add the whole numbers
and add the fractions
...
...
a) Calculate 2 /3 - 1 /4
Hint: Make a common denominator of 3 and 4
...
c) Calculate 32 /5 - 11 /2
Hint: Make both fractions improper fractions then find a common denominator
...
2
...
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TOPIC 2
...
4
30
Simplify the answer
...
Key Point
To multiply fractions we multiply the numerators and multiply the denominators
together
...
=
2
3
Simplify
...
...
Q36:
Calculate 13 /5 × 31 /2
Write down your working
...
We can't divide fractions but we can change the divide to a multiply
...
=
4
3
×
5
3
Change the ÷ to a × and flip the 2nd
fraction over
...
2
9
Change the improper fraction to a mixed
number
...
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16
TOPIC 2
...
e
...
Example
Problem:
Calculate 51 /3 ÷32 /5
Solution:
1
3
2
5
16
3
÷
17
5
We need improper fractions
...
=
÷ 3
=
=
5
80
51
= 1
Multiply the numerators and denominators
...
...
Q37:
Calculate 41 /2 ÷ 13 /5
Write down your working
...
Multiplying and dividing fractions exercise
Q38: Multiplying fractions
Give your answers as mixed numbers
...
Calculate 2 /7 × 3 /5
Hint: Multiply numerators and denominators
...
Calculate 22 /3 × 3 /5
Hint: Make the first fraction an improper fraction then multiply numerators and
denominators
...
Calculate 11 /7 × 22 /3
Hint: Make both fractions improper fractions then multiply numerators and
denominators
...
Q39: Dividing fractions
Give your answers as mixed numbers
...
Calculate 2 /3 ÷ 1 /4
Hint: Change the divide to a multiply, flip the second fraction over then multiply
numerators and denominators
...
FRACTIONS, DECIMALS AND PERCENTAGES
2
...
3
...
...
7
Learning points
Compound interest
• The method of compound interest repeatedly applies a percentage increase or
decrease
...
a
...
• 12
...
• 23% depreciation ⇒ × 0·77
Undoing percentages
• To undo a percentage is to reverse the application of a percentage
...
– Find 1% by dividing by 110
...
• Method 2
– If an item has decreased by 7% it is worth 93%
...
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18
TOPIC 2
...
• 31 /4 = 12 /4 + 1 /4 = 13 /4
Converting improper fractions to mixed numbers
• A fraction is just the numerator divided by the denominator
•
25 /
3
means 25 ÷ 3 = 8 remainder 1 = 8 1 /3
Adding & subtracting fractions
• To add and subtract fractions you need a common denominator
...
• To subtract fractions which have mixed numbers change them to improper fractions
first
...
Multiplying & dividing fractions
• To multiply and divide fractions which have mixed numbers change them to
improper fractions first
...
• To divide fractions change the sign to a multiply and flip the second fraction over
...
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TOPIC 2
...
8
End of topic test
End of topic 2 test
Q40: Compound interest
Jane invests £7500 in a bank that pays 3·4% interest per annum
...
...
a) Calculate how much money Sarah would have in her account after 5 years
...
Q42: Appreciation and depreciation
Last year a company made a profit of 2
...
It expects to increase its profit by 12% this year and 18% next year
...
Give your answer to 2 significant figures
...
Q43: Appreciation and depreciation
A patient is given 400 mg of a drug at 7 a
...
35% of the drug is absorbed by the body each hour
...
m
...
Q44: Undoing percentages
Alexander bought a new car last year
...
How much did Alexander pay for the car last year?
...
What did James earn before the pay rise?
...
b) Change 5 2 /3 to an improper fraction
...
Q47: Adding and subtracting fractions
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20
TOPIC 2
...
b) Calculate 5 /6 + 21 /4 giving your answer as a mixed number
...
c) Calculate 4 /5 - 3 /8
Hint: Find a common denominator
...
Hint: Make both fractions improper fractions then find a common denominator
...
Q48: Adding and subtracting fractions
A rectangle has length 2 2 /3 cm and breadth 1 1 /8 cm
...
Give your answer as a mixed number
...
Add all the sides
together
...
Q49: Multiplying and dividing fractions
a) Calculate 2 /5 × 3 /8
Hint: Multiply the numerators then multiply the denominators
...
Hint: Make the first fraction an improper fraction then multiply
...
d) Calculate 91 /3 ÷ 25 /6 giving your answer as a mixed number
...
...
Calculate its area
...
Hint: Area of a rectangle = length × breadth
...
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ANSWERS: TOPIC 2
Answers to questions and activities
2 Fractions, decimals and percentages
Answers from page 2
...
46
Q2: 1
...
18
Q4: 0
...
30
Answers from page 4
...
30
Compound interest exercise (page 5)
Q8: £6593·61
Q9:
a) £1127·72
b) £377·72
Q10: £160·74
Appreciation and depreciation practice (page 6)
Q11: £ 610400
Q12: £ 8367
...
47
Appreciation and depreciation exercise (page 7)
Q15:
a) 0·86
b) 1·02
c) 0·06
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22
ANSWERS: TOPIC 2
d) 0·026
Q16: £3930·51
Q17: £311922
Q18: £1824
...
Q23: 71200 (82592 ÷ 1·16)
Q24: 1260 (1134 ÷ 0·90)
Q25:
a) £680·80
b) £740
Practice converting mixed numbers to improper fractions (page 10)
Q26: 22/3
Q27: 23/8
Converting mixed numbers to improper fractions exercise (page 10)
Q28:
b)
5/
2
5/
3
c)
23 /
5
d)
53 /
6
80 /
7
a)
e)
Practice converting improper fractions to mixed numbers (page 12)
Q29: 4+3 /4
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ANSWERS: TOPIC 2
23
Q30: 7+1 /6
Converting Improper fractions to mixed numbers exercise (page 12)
Q31:
a) 11 /2
b) 41 /5
c) 85 /6
d) 111 /7
e) 77 /12
Answers from page 13
...
4
Q33:
3
9
8
1
− 1
=
−
2
5
2
5
5
8
2
9
×
−
×
=
2
5
5
2
16
45
−
=
10
10
29
=
10
9
= 2
10
Adding and subtracting fractions exercise (page 14)
Q34:
a) 11 /12
b) 31 /6
c) 84 /15
Q35:
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ANSWERS: TOPIC 2
a)
5/
12
b) 18 /15
c) 19 /10
Answers from page 15
...
4
Q37:
3
9
8
1
÷ 1
=
÷
2
5
2
5
5
9
×
=
2
8
45
=
16
13
= 2
16
Multiplying and dividing fractions exercise (page 16)
Q38:
1
...
13 /5
3
...
22 /3
2
...
33 /10
End of topic 2 test (page 19)
Q40: £8291·30
Q41:
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ANSWERS: TOPIC 2
a) £2492·36
b) 16 years
Q42: £3300000
Q43: 169 mg
Q44: £15500
Q45: £17250
Q46:
a) 32 /5
b)
17 /
3
Q47:
a)
23 /
56
b) 31 /12
c)
d)
17 /
40
5/
3 12
Q48: 77 /12 cm
Q49:
a)
b)
c)
d)
3/
20
5/
2 14
15 /16
35 /17
Q50: 147 /12 m2
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