Download fractions as ratios

nuMber AnD AlgebrA • real numbers
4g
fractions as ratios
introduction to ratios
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Ratios are used in many aspects of everyday life. They are used to compare quantities of the
same kind.
Where might you hear the following comparisons? What are the quantities being
compared?
1. Michael Schumacher’s F1 car is twice as fast as Mick’s delivery van.
1
2. You will need 2 buckets of water for every 2 bucket of sand.
3. The fertiliser contains 3 parts of phosphorus to 2 parts of potassium.
4. The Tigers team finished the season with a win : loss ratio of 5 to 2.
5. Mix 1 teaspoon of salt with 4 teaspoons of flour.
In the last example we are considering the ratio of salt (1 teaspoon) to flour (4 teaspoons). We
write 1 : 4 and say, ‘one is to four’; we are actually mixing a total of 5 teaspoons. Ratios can
also be written in fraction form:
1:4 «
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1
4
Note: Since the ratios compare quantities of the same kind, they do not have a
name or unit of measurement. That is, we write the ratio of salt to flour as 1 : 4, not
1 teaspoon : 4 teaspoons. The order of the numbers in a ratio is important. In the example
of the ratio of salt and flour, 1 : 4 means 1 unit (for example a teaspoon) of salt to 4 units of
flour. The amount of flour is 4 times as large as the amount of salt. On the contrary, the ratio
4 : 1 means 4 units of salt to 1 unit of flour, which means the amount of salt is 4 times as
large as the amount of flour.
Chapter 4
rational numbers
115
nuMber AnD AlgebrA • real numbers
WorkeD exAMple 23
Look at the completed game of ‘noughts and crosses’ at right
and write down the ratios of:
a crosses to noughts
b noughts to unmarked spaces.
Think
WriTe
a Count the number of crosses and the number of noughts. Write the
a 4:3
2 numbers as a ratio (the number of crosses must be written first).
b Count the number of noughts and the number of unmarked spaces.
b 3:2
Write the 2 numbers as a ratio, putting them in the order required
(the number of noughts must be written first).
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Before ratios are written, the numbers must be expressed in the same units of measurement.
Once the units are the same, they can be omitted. When choosing which of the quantities to
convert, keep in mind that ratios contain only whole numbers.
WorkeD exAMple 24
Rewrite the following statement as a ratio:
7 mm to 1 cm.
Think
WriTe
1
Express both quantities in the same units. To obtain whole numbers,
convert 1 cm to mm (rather than 7 mm to cm).
7 mm to 1 cm
7 mm to 10 mm
2
Omit the units and write the 2 numbers as a ratio.
7 : 10
simplifying ratios
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When the numbers in a ratio are multiplied, or divided by the same number to obtain another
ratio, these two ratios are said to be equivalent. (This is similar to the process of obtaining
equivalent fractions.)
For instance, the ratios 2 : 3 and 4 : 6 are equivalent, as the second ratio can be obtained by
multiplying both numbers of the first ratio by 2.
Ratios 10 : 5 and 2 : 1 are also equivalent, as the second ratio is obtained by dividing both
numbers of the first ratio by 5.
Like fractions, ratios are usually written in simplest form; that is, reduced to lowest terms.
This is achieved by dividing each number in the ratio by the highest common factor (HCF).
WorkeD exAMple 25
Express the ratio 16 : 24 in simplest form.
Think
116
WriTe
1
Copy the ratio in your workbook. What is the largest number, by
which both 16 and 24 can be divided (that is, the highest common
factor of 16 and 24)? It is 8.
16 : 24
2
Divide both 16 and 24 by 8 to obtain
an equivalent ratio in simplest form.
2:3
Maths Quest 7 for the Australian Curriculum
nuMber AnD AlgebrA • real numbers
WorkeD exAMple 26
Write the ratio of 1.5 m to 45 cm in simplest form.
Think
WriTe
1
Write down the question.
1.5 m to 45 cm
2
Express both quantities in the same units by changing
1.5 m into cm.
(1 m = 100 cm)
150 cm to 45 cm
3
Omit the units and write the 2 numbers as a ratio.
150 : 45
4
Simplify the ratio by dividing both 150 and 45 by
15 — the HCF.
10 : 3
WorkeD exAMple 27
Simplify the following ratios.
2
5
a
7
: 10
b
5
6
5
:8
Think
a
b
WriTe
2
5
: 10
Write equivalent fractions using the lowest
common denominator: in this case 10.
=
4
10
3
Multiply both fractions by 10.
= 4:7
4
Check if the remaining whole numbers that form
the ratio can be simplified. In this case they
cannot.
1
Write the fractions in ratio form.
2
1
Write the fractions in ratio form.
2
a
7
7
: 10
5
6
:8
Write equivalent fractions using the lowest
common denominator. In this case 24.
=
20
24
3
Multiply both fractions by 24.
= 20 : 15
4
Check if the remaining whole numbers that form
the ratio can be simplified. In this case divide
each by the HCF of 5.
= 4:3
b
5
15
: 24
Dividing in a given ratio
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Consider the following situation:
Isabel and Rachel decided to buy a $10 lottery ticket. Isabel had only $3 so Rachel put in
the other $7. The ticket won the first prize of $500 000. How are the girls going to share the
prize? Is it fair that they would get equal shares?
In all fairness, it should be shared in the ratio 3 : 7.
Chapter 4
rational numbers
117
nuMber AnD AlgebrA • real numbers
WorkeD exAMple 28
Share the amount $500 000 in the ratio 3 : 7.
Think
WriTe
1
Calculate the total number of parts in the ratio.
Total number of parts = 3 + 7
= 10
2
The first share represents 3 parts out of a total of 10,
3
so find 10 of the total amount.
First share =
The second share represents 7 parts out of a total of 10,
7
so find 10 of the total amount.
Second share =
3
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3
10
ì $500 000
= $150 000
7
10
ì $500 000
= $350 000
We can mentally check our answer by adding the two shares together. The total should be the
same as the amount that was originally divided.
WorkeD exAMple 29
Concrete mixture for a footpath was made up of 1 part of cement, 2 parts of sand and 4 parts of
blue metal. How much sand was used to make 4.2 m2 of concrete?
Think
WriTe
1
Find the total number of parts.
Total number of parts = 1 + 2 + 4
=7
2
There are 2 parts of sand to be used in the mixture,
so find 2 of the total amount of concrete made.
Amount of sand = 7 ì 4.2 m2
= 1.2 m2
7
2
reMeMber
1.
2.
3.
4.
5.
6.
7.
8.
9.
118
Ratios compare quantities of the same kind.
The ratios themselves do not have a name or unit of measurement.
The order of the numbers in a ratio is important.
Before ratios are written, the numbers must be expressed in the same units of
measurement.
Ratios contain only whole numbers.
If each number in a ratio is multiplied, or divided by the same number, the equivalent
(or equal) ratio is formed.
It is customary to write ratios in the simplest form. This is achieved by dividing each
number in the ratio by the highest common factor (HCF).
To form a ratio, using fractions, convert the fractions so that they have a common
denominator and then write the ratio of the numerators.
To share a certain amount in a given ratio, find the total number of shares (parts) first.
The size of each share is given by the fraction this share represents out of the total
number of shares.
Maths Quest 7 for the Australian Curriculum