1. Fraction Operations
A Formal Summary
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2. Every fraction is a division Every division is a fraction
Fact
Every division is a fraction
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3. To convert any fraction into decimal form, carry out the division:
To convert any decimal into a fraction,
write it as a fraction out of 10, 100, 100 etc:
Note that recurring decimals need some algebraic manipulation before converting
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4. The rule for Addition & Subtraction of fractions is:
For example:
When the denominators of the fractions are different we must find a common denominator before adding or subtracting them.
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5. The rule for Addition & Subtraction of fractions is:
For example:
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6. Multiplication of fractions follows the rule:
For example:
What is the physical meaning of fraction multiplication?
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7. Find of Find of This is in analogy of multiplication of decimals:
0.7 x 0.8
finds 0.7 of 0.8
or 0.8 of 0.7
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8. Division of fractions follows the rule:
Reciprocals
Two fractions such that one is obtained by turning the other “upside down”
Reciprocals are said to be inverses of each other with respect to the operation of multiplication
The reciprocal of is because:
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9. Division of fractions follows the rule:
Reciprocals
Two fractions such that one is obtained by turning the other “upside down”
Reciprocals are said to be inverses of each other with respect to the operation of multiplication
The reciprocal of is because:
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10. Division of fractions follows the rule:
Reciprocals
Two fractions such that one is obtained by turning the other “upside down”
Reciprocals are said to be inverses of each other with respect to the operation of multiplication
The reciprocal of is 3 because:
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11. Division of fractions follows the rule:
Reciprocals
Two fractions such that one is obtained by turning the other “upside down”
Reciprocals are said to be inverses of each other with respect to the operation of multiplication
The reciprocal of 7 is because:
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12. Division of fractions follows the rule:
Reciprocals
Two fractions such that one is obtained by turning the other “upside down”
Reciprocals are said to be inverses of each other with respect to the operation of multiplication
The reciprocal of 2.33 is because:
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13. Division of fractions follows the rule:
Examples of fraction division:
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15. Three adverts appear on a page of a newspaper.
The 1st advert covers 1/4 of the page.
The 2nd avert covers 1/8 of the page.
The 3rd advert covers 3/16 of the page.
What fraction of the page is not covered by adverts?
x 4
x 2 1 4 1 8 3 16 4 16 2 16 3 16 9 16 + + = + + =
x 4
x 2 9 16
If is covered by adverts
then is not covered by adverts. 7 16
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17. How many quarters in 2 ? How many quarters in 3 ?
How many tenths in 2 ?
How many tenths in 1 ?
How many twelfths in 1 ? 3 4 2 3 4 = 11 4 1 2 3 10 3 5 1 3
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18. How many quarters in 2 ? How many quarters in 3 ?
How many tenths in 2 ?
How many tenths in 1 ?
How many twelfths in 1 ? 3 4 2 3 4 = 11 4 1 2 3 1 2 = 7 2 = 14 4 3 10 3 5 1 3
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19. How many quarters in 2 ? How many quarters in 3 ?
How many tenths in 2 ?
How many tenths in 1 ?
How many twelfths in 1 ? 3 4 2 3 4 = 11 4 1 2 3 1 2 = 7 2 = 14 4 3 10 2 3 10 = 23 10 3 5 1 3
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20. How many quarters in 2 ? How many quarters in 3 ?
How many tenths in 2 ?
How many tenths in 1 ?
How many twelfths in 1 ? 3 4 2 3 4 = 11 4 1 2 3 1 2 = 7 2 = 14 4 3 10 2 3 10 = 23 10 3 5 1 3 5 = 8 5 = 16 10 1 3
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21. How many quarters in 2 ? How many quarters in 3 ?
How many tenths in 2 ?
How many tenths in 1 ?
How many twelfths in 1 ? 3 4 2 3 4 = 11 4 1 2 3 1 2 = 7 2 = 14 4 3 10 2 3 10 = 23 10 3 5 1 3 5 = 8 5 = 16 10 1 3 1 1 3 = 4 3 = 16 12
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23. Method 1 6 + 3 + 6 + 3 = 19 + + = 19 + = 19 + = 19 + 1 = 20 cm
A rectangle is 6⅝ cm long by 3½ cm wide.
1. Calculate it perimeter.
2. Calculate its area.
Method 1
to find the perimeter:
3½ cm 5 8 1 2 5 8 1 2 6 + 3 + 6 + 3 = 5 8 5 8 19 + + =
6⅝ cm
÷ 2 10 8 19 + =
÷ 2 5 4 19 + = 1 4 19 + 1 = 1 4 20 cm
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24. Method 2 6 + 3 + 6 + 3 = + + + = + = + = = 20 = 20 cm
A rectangle is 6⅝ cm long by 3½ cm wide.
1. Calculate it perimeter.
2. Calculate its area.
Method 2
to find the perimeter:
3½ cm 5 8 1 2 5 8 1 2 6 + 3 + 6 + 3 = 53 8 7 2 53 8 7 2 + + + =
6⅝ cm
106 8 14 2
x 4 + =
x 4
106 8 56 8 + =
162 8 2 8 = 20 = 1 4 20 cm
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25. 6 x 3 = x = = 23 cm2 A rectangle is 6⅝ cm long by 3½ cm wide.
1. Calculate it perimeter.
2. Calculate its area.
to find the area: 5 8 1 2 6 x 3 =
3½ cm 53 8 7 2 x = 2 3
6⅝ cm
371 16 = 1 6 3 7 1 5 3 3 16 23
cm2
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27. Roulla used half of her exercise book in the autumn term.
So far this term she has used a further one sixth of it.
1. What fraction of her exercise book has she used so far?
2. How many pages does her exercise book have if she
has 30 pages left?
x 3
÷ 2 1 2 1 6 3 6 1 6 4 6 2 3 + = + = =
x 3
÷ 2
If she has used of her exercise book
she must have of it left.
If of her exercise book is 30 pages
then the entire exercise book must have 90 pages 2 3 1 3 1 3
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29. Johnny spent his monthly allowance as follows:
of it on lunch and food snacks
of it on two music CDs
of it on a new book
1. What fraction of his monthly allowance has he got left?
2. If he is left with £6 what is his monthly allowance? 2 5 1 3 1 6
x 6
x 10
x 5
÷ 3 2 5 1 3 1 6 12 30 10 30 5 30 27 30 9 10 + + = + + = =
x 6
x 10
x 5
÷ 3
Johnny has spent of his monthly allowance
so he must have of his monthly allowance left.
If of his monthly allowance is £6
then his monthly allowance must be £60 9 10 1 10 1 10
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31. Edmonton is 3⅝ miles away from Southgate.
Barnet is 2¾ miles away from Southgate.
How much further from Southgate is Edmonton than Barnet?
x 2 5 8 3 4 29 8 11 4 29 8 22 8 7 8 3 – 2 = – = – =
x 2
Edmonton is ⅞ of a mile further from Southgate than Barnet is.
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33. Adding the two figures:
Ethan works day and night shifts in alternating weeks.
He works:
5 days for 7¾ hours per day during his “day-shift” week.
5 nights for 6½ hours per night in his “night-shift” week.
Calculate how many hours he works a fortnight.
Week 1: 3 4 31 4
155 4 3 4 5 x 7 = 5 x = = 38
Week 2: 1 2 13 2 65 2 1 2 5 x 6 = 5 x = = 32
Adding the two figures: 3 4 1 2 1 4 1 4 38 + 32 = 70 + 1 = 71
Ethan works 71¼ hours every fortnight
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35. 1 2 3 4
Calculate 4 ÷
[You may shade the shapes below to help you with your answer]
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36. 1 2 3 4
Calculate 4 ÷
[You may shade the shapes below to help you with your answer] 1 2 3 4 9 2 3 4 9 2 4 3 36 6 6 4 ÷ = ÷ = x = =
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38. 7 ÷ = ÷ = = = Bethany uses ⅝ metres of ribbon to wrap up a gift box.
How many identical gift boxes can she wrap using a 7½ metre roll of ribbon? 1 2 5 8 15 2 5 8 15 2 8 5
120 10 12 7 ÷ = ÷ = x = =
Bethany can wrap up 12 such boxes
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40. There is of the original amount left
Tony drank of a full carton of orange juice.
His sister Alice drank of what was left. 1 3 3 4
What fraction of the original amount of juice remains in the carton?
Who drunk the most juice? [you must show full workings]
If Tony drank of a full carton there is of a full carton left.
Therefore Alice drank of of a full carton
The operation that finds “something” of “something” is:
multiplication 1 3 2 3 3 4 2 3 3 4 2 3 6 12 1 2 x = =
Alice drunk half the carton
[who drank the most?]
x 2
x 3 1 3 1 2 2 6 3 6 5 6
There is of the original amount left 1 6 + = + =
x 2
x 3
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41. Let us solve the problem pictorially
Tony drank of a full carton of orange juice.
His sister Alice drank of what was left. 1 3 3 4
What fraction of the original amount of juice remains in the carton?
Who drunk the most juice? [you must show full workings]
Let us solve the problem pictorially
Tony drunk of a full carton
Alice drunk of what was left
We are left with of a full carton
Tony drunk of a full carton
Alice drunk of a full carton
Which is the same as 1 3 3 4 1 6 1 3 3 6 1 2
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43. The school day in Northgate School starts at 08:45 and finishes at 15:30.
What fraction of a 24 hour day does the school day take up? [Give your answer in its simplest form.]
08:45
09:00
15:00 to
09:00
15:00
15:30
15 minutes
6 hours
30 minutes
6 hours, 45 minutes
6¾ hours
6.75 hours
x 2
x 2
÷ 3
6.75 24
13.5 48 27 96 9 32 = = =
x 2
x 2
÷ 3
x 100
÷ 25
÷ 3
6.75 24
675
2400 27 96 9 32 = = =
x 100
÷ 25
÷ 3
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45. To make a litre of a certain fruit punch, fruit juices are mixed in the following proportions:
type of juice
amount
x 5
x 4
apple
of a litre 5 12 5 12 4 15 25 60 16 60 41 60 + = + =
x 5
x 4
cranberry
of a litre 4 15 1 41 60
kiwi
of a litre 60 41 60 19 60 – = 19 60 – =
total: 1 litre
What fraction of a litre corresponds to kiwi fruit juice?
What is the ratio of these juices as apple : cranberry : kiwi ?
What fraction of a litre from each fruit juice is contained in 2½ litres of this fruit punch?
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46. To make a litre of a certain fruit punch, fruit juices are mixed in the following proportions:
type of juice
amount
x 5
x 4 5 12 4 15 19 60 : :
apple
of a litre 5 12
x 5
x 4
cranberry
of a litre 4 15 25 60 16 60 19 60 : :
kiwi
of a litre 19 60
x 60
x 60
x 60
total: 1 litre
25 : 16 : 19
What fraction of a litre corresponds to kiwi fruit juice?
What is the ratio of these juices as apple : cranberry : kiwi ?
What fraction of a litre from each fruit juice is contained in 2½ litres of this fruit punch?
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47. What fraction of a litre corresponds to kiwi fruit juice?
To make a litre of a certain fruit punch, fruit juices are mixed in the following proportions:
type of juice
amount
apple
of a litre 5 12 5 12
of a litre x 5 2 25 24 =
cranberry
of a litre 4 15 4 15 5 2 x 20 30 2 3
of a litre = =
kiwi
of a litre
of a litre 19 60 19 60 x 5 2 95
120 19 24 = =
total: 1 litre
of a litre x 5 2 = 5 2
What fraction of a litre corresponds to kiwi fruit juice?
What is the ratio of these juices as apple : cranberry : kiwi ?
What fraction of a litre from each fruit juice is contained in 2½ litres of this fruit punch?
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48. Exam Question
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49. Calculate the area of the following trapezium:5
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50. © T Madas