Transcript Dividing fractions

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Adding and subtracting fractions
When fractions have the same denominator it is quite easy
to add them together.
For example,
3
5
+
1
5
=
3+1
5
=
4
5
We can show this calculation in a diagram:
+
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=
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Adding and subtracting fractions
Similarly, when fractions have the same denominator it is
quite easy to subtract them.
÷4
7
8
For example,
–
3
8
7–3
8
=
=
4
8
=
1
2
÷4
Fractions should always be cancelled to their lowest terms.
We can show this calculation in a diagram:
–
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=
=
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Adding and subtracting fractions
Adding fractions with the same denominator is quite easy.
1
7
4
+
+
9
9
9
1+7+4
=
=
9
12
9
Improper fractions should be written as mixed numbers.
÷3
12
9
=
9
9
3
+
9
=
1
3
9
=
1
1
3
÷3
+
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+
=
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Fractions with common denominators
Fractions are said to have a common denominator if they
have the same denominator.
11 4
5
,
and
For example,
have a common denominator
12 12
12
of 12.
Adding these together gives us a total of:
11
4
5
11 + 4 + 5
20
=
+
+
=
=
12
12
12
12
12
1
8
=
12
=
1
6
1
2
3
Subtracting these gives us a total of:
11
4
5
11 – 4 – 5
–
–
=
12
12
12
12
6 of 40
=
2
12
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Fractions with common denominators
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Fractions with different denominators
Fractions with different denominators are more difficult to
add and subtract.
If fractions have different denominators we need to do
something to make the denominators the same.
5
2
How would you calculate
–
?
6
9
–
5
6
8 of 40
×3
15
=
18
×3
=
2
9
×2
4
=
18
×2
15 – 4
11
=
18
18
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Using diagrams
When fractions that are being added have different
denominators, we need to find a common denominator.
3
3
How would you calculate
+
?
5
4
Find a common denominator and make equivalent fractions.
+
12
20
+
=
15
20
=
=
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12 + 15
20
20
7
+
20
20
=
=
27
20
1
7
20
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Using diagrams
When fractions that are being added have different
denominators, we need to find a common denominator.
How would you calculate
1
1
7
–
?
4
10
Find a common denominator and make equivalent fractions.
–
25
20
10 of 40
–
=
14
20
=
25 – 14
11
=
20
20
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Using a common denominator
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Equivalent fractions
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Adding and subtracting fractions
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Fraction cards
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Multiplying integers by fractions
Multiplying an integer by a fraction can be done by following a
specific set of instructions.
Multiply by the numerator and divide by the denominator.
It doesn’t matter which order you perform the tasks in, so long
as you multiply and divide the correct part of the fraction.
4
54 ×
9
= 54 ÷ 9 × 4
or
4
54 ×
9
= 54 × 4 ÷ 9
=6×4
= 216 ÷ 9
= 24
= 24
4
This is equivalent to
of 54.
9
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Multiplying integers by fractions
In some cases, the question gives an idea about the most
logical order to perform the multiplication and division.
5
What is 12 ×
?
7
5
12 ×
= 12 × 5 ÷ 7
7
= 60 ÷ 7
60
=
7
=
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8
It is difficult to divide 12
by 7, so in this example,
it is better to perform
the multiplication first.
4
7
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Using cancellation to simplify calculations
To make calculations simpler, it can sometimes be a good
idea to cancel down before performing the multiplication.
7
What is 16 ×
?
12
7
We can write 16 ×
as:
12
4
16
7
28
×
=
3
1
123
=
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9
1
3
If the number we are
multiplying by and the
denominator of the fraction
share a common factor,
cancel the common factor
before multiplying.
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Using cancellation to simplify calculations
Cancellation is particularly useful in multiplications that
involve big numbers that are difficult to work out mentally.
8
What is
× 40?
25
8
We can write
× 40 as:
25
8
8
40
64
The number we are
×
=
25 5
1
5
multiplying by and the
denominator of the fraction
4
share a common factor,
=
5
allowing us to cancel the
common factor.
12
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Multiplying integers by fractions
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Multiplying a fraction by a fraction
When multiplying two fractions together, multiply the
numerators together and multiply the denominators together.
3
2
What is
×
?
8
5
÷4
3
4
×
8
5
=
12
40
=
3
10
÷4
Alternatively, cancel the numerator of
the fraction we are multiplying by and
the denominator of the first fraction.
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3
4 1
3
=
×
8
5
10
2
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Multiplying a fraction by a fraction
Dealing with mixed numbers
requires an extra step to make
the calculation slightly easier.
What is
5
5
12
×
?
6
25
Start by writing the calculation with
any mixed numbers as improper fractions.
To make the calculation easier, cancel any numerators
with any denominators.
Convert the improper fraction back to a mixed number.
7
2
35
14
12
×
=
=
6 1 25 5 5
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2
4
5
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Multiplying fractions
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Dividing an integer by a fraction
Dividing an integer by a fraction with a value less than 1 will
always result in a larger number.
1
What is 4 ÷
?
3
How many thirds are there in 4?
Here are 4 rectangles:
If they are divided into thirds, you can see that there are 12
thirds in the four rectangles.
1
4÷
= 12
3
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Dividing an integer by a fraction
Calculations where an integer is divided by a fraction can be
thought of as ‘how many of the fraction are in the integer?’
2
What is 4 ÷
?
5
How many two fifths are there in 4?
Here are 4 rectangles:
Divide them into fifths.
Count the number of two fifths.
2
4÷
= 10
5
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Dividing an integer by a fraction
It’s not always possible to represent fractions using diagrams.
We need another way to perform the calculations.
3
What is 6 ÷
?
4
How many three quarters are there in six?
1
6÷
= 6 × 4 = 24
4
There are 4 quarters in each whole.
3
= 24 ÷ 3
=8
4
We can check this by multiplying.
6÷
3
8×
=8÷4×3 =6
4
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Dividing integers by fractions
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Dividing a fraction by a fraction
Dividing a fraction by a fraction works in much a similar way.
1
1
What is
÷
?
8
2
How many eighths are in one half?
This diagram shows half
of a rectangle.
If the shape is now
divided into eighths, you
can see how many eights
there are in one half.
1
1
÷
=4
8
2
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Dividing a fraction by a fraction
With more complex fraction divisions, we can write an
equivalent calculation to help perform the division.
4
2
What is
÷
?
5
3
4
2
How many
are in
?
5
3
An equivalent calculation involves multiplying by the
denominator and dividing by the numerator.
4
5
2
2
can be written as:
÷
×
5
4
3
3
÷2
5
2
×
=
4
3
10
12
=
5
6
We have swapped
the numerator and
denominator and
multiplied by this.
Remember to simplify.
÷2
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Dividing a fraction by a fraction
With mixed numbers, convert the mixed number to an
improper fraction before continuing with the calculation.
What is
3
3
6
3
÷
?
7
5
3
18
=
5
5
18
6
÷
=
5
7
21
=
5
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=
4
3
18
7
×
5
61
1
5
Cancelling the
fractions at this stage
makes the next stages
of the calculation
slightly easier.
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Dividing fractions
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Multiplying and dividing by fractions
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How much was spent?
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Bills to pay
Mary takes home £1,200 a month in her
job working at a bank.
Of the money she takes home, she spends:
4
paying the rent on her flat
7
2
of the remainder paying for food.
5
How much does she pay in rent?
How much does she spend on food?
What fraction of her wages is she left with after paying
her rent and spending money on food?
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Fenced in
John is painting the fence in his back garden.
5
His 10 litre tin of paint is
full.
6
How much paint is in his tin?
8
Each fence panel requires
9
of a litre of paint.
How many fence panels can he paint?
John has 18 panels to paint in total, so he buys another
8 litre tin of paint. Does he have enough paint to complete
all the panels in the fence? Show your working.
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Healthy bones
Joe likes to drink a lot of milk. Every day during
the week, he drinks a set amount.
2
In the morning, he has
of a pint.
3
In the afternoon, he has another 2
3
of a pint.
At the weekend, he drinks 3 of a
4
pint at breakfast.
How much milk does he drink each week?
A 1 pint carton costs £0.43, a 2 pint carton costs £0.76
and a 4 pint carton costs £1.12. Investigate the cheapest
way for Joe to get as much milk as he needs each week.
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Fill in the gaps
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Mixed up information
Some students have some information about their year group.
What fraction of the students were boys?
What fraction had neither a blazer nor a jumper?
What fraction had Science as their favourite subject?
What fraction voted for Jamie as student rep?
What is the smallest possible number of students in the
year group?
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