Ratios
Ratios are used to numerically compare two or more values of the same type, that is, that have the same units.
The ratio between two values \(a\) and \(b\) is written as \(a:b\).
For example, a class of 24 students consists of \(14\) girls and \(10\) boys. The ratio of girls to boys is therefore \(14:10\).
The order this is written is important. If \(a\) is the number of girls and \(b\) is the number of boys, then the ratio \(a:b\) must be written \(14:10\). If it were written as \(10:14\), then it would mean \(10\) girls and \(14\) boys, which is not accurate.
Use this page to revise the following concepts within ratios:
Simplest form
Ratios in their simplest form use the lowest possible integer values to represent the ratio.
Ratios can be expressed in their simplest form by dividing or multiplying each value by a common factor to find the smallest possible integer values. Again, one must ensure that both quantities are expressed in the same unit.
For example, the class of \(24\) students consisting of \(14\) girls and \(10\) boys has a ratio of \(14:10\).
Both of these values are divisible by \(2\), so the simplest form of this ratio is \(7:5\).
See below for more examples.
Worked Example
A recipe calls for \(1.5\text{ kg}\) or flour and \(360\text{ g}\) of cocoa powder. Express the ratio of flour to cocoa powder in its simplest form.
Convert both values to the same unit
\[1.5\text{ kg}=1500\text{ g}\]
Write the ratio
\[1500:360\]
Divide by common factors until the simplest form is obtained.
\[{1500}:{360} = {1500 \div 60}:{360 \div 60} = {25}:{6}\]
Hence the ratio of flour to cocoa powder is \(25:6\).
Worked Example
Write the ratio \(1.52:2.5\) in its simplest form.
Multiply both values by \(100\) to obtain whole numbers
\[152:250\]
Divide by common factors until the simplest form is obtained.
\[152:250 = 152 \div 2 : 250 \div 2 = 76:125\]
Hence the ratio in simplest form is \(76:125\)
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Dividing quantities in given ratios
Sometimes it is necessary to take a quantity and split it into parts using ratios. Perhaps a pastry chef is making cakes and knows their total quantity of sugar, and that they require 4 times as much sugar for the cake as they do for the icing. If they divide they total amount of sugar into this ratio they can find out how much sugar to allocate to the cake and icing, and how many cakes they can make from their total sugar.
Generally, to divide a quantity into a ratio:
- Determine how many total units are in the ratio by adding the parts of the ratio
- Divide the total quantity by the total number of units.
- Multiply the resulting value to obtain the number of units per each part of the ratio
Worked Example
A \(2\text{ L}\) container of cordial consists of a \(1:7\) ratio of concentrate to water. Determine the amount of concentrate and the amount of water in the cordial.
Determine how many units are in the ratio
\[1+7=8 \text{ units}\]
Divide the quantity by the total units
\[2\text{ L} \div 8 =2000\text{ mL} \div8=250\text{ mL}\]
Multiply this value by the number of units in each part of the ratio to obtain the amount of each part
\[1\times250=250\text{ mL}\]
\[7\times250=1750\text{ mL}\]
So, the cordial is comprised of \(250\text{ mL}\) of concentrate and \(1750\text{ mL}\) of water.
Worked Example
A prize pool of \(\$500\) is to be split between the places 1st, 2nd and 3rd with a ratio of \(13:5:2\).
Determine how much money each place wins.
First, determine how many units are in the ratio
\[13+5+2=20 \text{ units}\]
Divide the quantity by the total units
\[\$500\div20=\$25\]
Multiply this value by the number of units in each part of the ratio to obtain the amount of each part
- \(13\times\$25=\$325\) for first place
- \(5\times\$25=\$125\) for second place
- \(2\times\$25=\$50\) for third place
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Unitary method
Ratios can be used to calculate unit values, the value of one unit. For example, this is often this is used for listing the prices of items. Supermarkets will list the price of an item and also the price per kilogram of that item so that easy comparisons can be made.
If the total value of item \(X\) is known, the unitary method helps determine the value of one unit. That is, if the value of \(X\) items is a certain value, then the value per units of \(X\) can be used to determine the value of \(Y\) items.
Worked Example
A car takes \(1\text{ h } 30\text{ min}\) to travel \(80\text{ km}\). How long would it take to travel \(500\text{ km}\).
Use the unitary method to determine how long it would take to travel \(1\text{ km}\)
\[90\text{ min}\div80\text{ km}=1.125\text{ min/kilometre}\]
Multiply this up to determine the time to travel 500km.
\[1.125\times500=562.5 \text{ minutes}\]
Or \(9\text{ h } 22.5\text{ min}\).
Worked Example
A particular brand of beef stock can be bought in containers of \(400\text{ mL}\) for \(\$1.80\) or \(1\text{ L}\) for \(\$4.00\).
Which of these containers is the best value?
Use the unitary method to determine the value of 1mL in each case.
\[\$1.80\div400\text{ mL}=0.45\text{ c per mL}\]
\[\$4.00\div1000\text{ mL}=0.4\text{ c per mL}\]
Hence, it is better value to buy the \(1\text{ L}\) container.