# Topic 4: Simplifying ratios

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```Topic 4: Simplifying ratios
When writing ratios, it is expected that they will be written in their simplest form containing whole
numbers. Remember ratios can be written as fractions and simplifying fractions ideas are used to
simplify ratios.
Examples:
Simplify 12:20
To simplify, the Highest Common Factor of 12 and 20 is required. Remember the HCF is the
highest number that goes into 12 and 20, the HCF is 4. If the common factor used is not the HCF,
then simplifying may take two or more stages.
12÷4 : 20 ÷4
= 3:5
Simplify 35:75
In this question the HCF is 5.
35 ÷5 : 75 ÷5
= 7 : 15
Simplify 300mm:1.2m
In this question the units are different, so the first step is make the units the same. Generally, it is
better to express both in the smaller unit. (1.2m = 1.2 x 1000 =1200 mm)
300 : 1200
= 300 ÷300 : 1200 ÷300
= 1: 4
Simplify 0.5:1.25
Because both numbers are expressed as decimals, multiplying both sides by 10, 100, 1000, etc is
required to make both whole numbers. The number with the most decimal place is 1.25, that is 2
decimal places, multiplying by 100 is required to remove decimals.
0.5 : 1.25
= 0.5×100 : 1.25×100
= 50 : 125
Now the HCF of 50 and 125 is 25, the ratio can be simplified.
50 : 125
= 50 ÷25 : 125 ÷25
= 2:5
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Simplify 2
1 3
:
2 5
With numbers expressed as fractions, it is important that mixed numbers are changed to improper
fractions. Both fractions should then be expressed with a common denominator. Each fraction can
then be multiplied by that common denominator and then inspected to see if more simplifying can
take place.
1 3
2 :
Change any mixed numbers to improper fractions
2 5
5×5 3×2
= ×5 : ×2 Express both over a common denominator of 10
2
5
25 6
=
:
Multiply both sides by the common denominator 10
10 10
= 25 : 6
This cannot simplify any further
When doing comparisons, writing ratios as unit ratios (either ... : 1 or 1: …) gives a way of deciding
best or worst case situations.
Examples:
At Aisville High School, there are 23 teaching staff for 385 students. At a neighbouring school,
Beeston High School, there are 44 teaching staff for 672 students. Use unit ratios to determine the
school with the best teacher student ratio.
Aisville High School
Beeston High School
23 : 385 divide both by 23
to obtain a ratio 1: ...
÷23
= 23 : 385 ÷23
= 1: 16.74 (to 2 decimal places)
44 : 672 divide both by 44
to obtain a ratio 1 : ...
÷44
= 44 : 672÷44
= 1: 15.27 (to 2 d.p.)
The best teacher student ratio is at Beeston High School because there are fewer students for
each teacher.
It is accepted that a doctor patient ratio of 1 : 1200 is acceptable for providing adequate patient
care. In an outback district, there are 6 doctors for a population of 10430 people. Is there an
acceptable number of doctors in this district?
Acceptable ratio
District ratio
1:1200
6 : 10430 divide both by 6
to obtain a ratio 1 : ...
÷6
= 6 : 10430 ÷6
= 1: 1738 (to nearest whole number)
No, there is an unacceptable number of doctors in this district. The doctor patient ratio for the
outback district exceeds the acceptable level.
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[last edited on 16 July 2015] CRICOS Provider: 01241G
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Numeracy
Video ‘Simplifying Ratios’
Activity
1.
2.
Simplify the following ratios
(a)
15 : 35
(b)
49 : 84
(c)
0.8 : 2
(d)
0.125 : 0.5
(e)
2 1
:
5 4
(f)
2 1
2 :1
3 6
(g)
15 mm : 5 cm
(h)
1.5g : 250mg
The Greeks believed that rectangles that had a length to width ratio of 1.618 (approx)
: 1 were the most pleasing to the eye. This ratio is known as the Golden Ratio.
(a) A computer screen has a length of 37cm and a width of 23cm, how close is it to
having the Golden Ratio?
(b) An envelope measures 21cm by 14cm, how close is it to having the Golden
Ratio? To make it closer to the Golden Ratio, would the length or width need to
be increased?