1.7.2 Working with Ratios
Sharing in a ratio:
To share an amount in a ratio
- Add the ratios together to get the total number of parts
- Divide the total by the number ofparts
- To calculate eachperson’s share, multiply by the ratio
- Check that these add up to the original amount
Worked example:
Divide \( $ \ 24 \) in the ratio \( 7 \): \( 5 \)
Add the ratios together to get the number of parts
\( 7+5=12 \)
Divide the total by number of parts 🠦 \( \frac{24}{12}=2 \)
Multiply \( 7 \) by \( 2 \) 🠦 \( 7 \times 2=14 \)
Multiply \( 5 \) by \( 2 \) 🠦 \( 5 \times 2=10 \)
\( $ \ 14 \),\( $ \ 10 \)
Difference in a ratio:
Sometimes, you could be told the difference between two shares rather than the total amount to be shared.
To solve such questions
- Calculate difference in theparts of the ratio
- Difference in theparts of the ratio equals difference in the amount
- Calculate onepart of the ratio by dividing both sides by the difference in the ratio
- Multiply the result by the share to get the amount
- Calculate sum of theparts of the ratio
- Multiply sum of the ratio by onepart of the amount to get total amount
Worked example:
\( Ali \) and \( Mo \) share a sum of money in the ratio
\( Ali \) : \( Mo \ =7 \): \( 5 \)
\( Ali \) receives \( $ \ 600 \) more than \( Mo \).
Calculate how much each receives?
Calculate difference in the ratio?
\( 9-7=2 \)
This means that 🠦 \( 2 \) parts=\( $ \ 600 \)
Dividing both sides by \( 2 \) 🠦 \( 1 \) part=\( $ \ 300 \)
Calculating sum of the ratio 🠦 \( 9+7=16 \)
Total amount will be 🠦\( 16 \times $ \ 300 \)=\( $ \ 4800 \)
Ali receives 🠦 \( 9 \times $ \ 300 \)=\( $ \ 2700 \)
\( Mo \) receives 🠦 \( 7 \times $ \ 300 \)=\( $ \ 2100 \)
\( Ali \) and \( Mo \) receives\( $ \ 2700 \) and \( $ \ 2100 \) respectively
Finding the value of the otherpart:
In some cases, you could be told the value of one side of the ratio rather than the total amount to be shared.
To find the value of the otherpart of the ratio
Calculate the value of onepart of the ratio by using the given data
Multiply the result by the share to get the amount
Calculate sum of theparts of the ratio
Multiply sum of the ratio by onepart of the amount to get total amount
Worked example:
Kristian and Stephanie share some money in the ratio \( 3\): \( 2 \).
Kristian receives \( $ \ 72 \).
Work out how much Stephanie receives.
The ratio of shares of Kristian to Stephanie is
\( 3\): \( 2 \)
Kristian receives \( $ \ 72 \), So
\( 3 \) parts\( =72 \)
Divide both sides by \( 3 \) to get onepart
\( 1 \) part\( = 24 \)
Stephanie’s share is \( 2 \), So
\( 2 \times 24=48 \)
Stephanie receives\( $ \ 48 \)
Three-part ratio:
You may be given two separate ratios that are linked together in some way so that you can form a three-part ratio.
To do this
- Figure out whichpart of the ratios can be used as a link
- Multiply/divide both ratios so they both compare to the same number
- Link them together
- Simplify
Worked example:
Some visitors to a campsite are in the ratio
\( men \) : \( women\ =5 \):\( 4 \) and \( women \) : \( children \ = 3 \):\( 7 \)
- Calculate the ratio \( men \) : \( women \) : \( children \) in its simplest form.
Given ratios are
\( men \) : \( women \ = 5 \):\( 4 \)
\( women \) : \( children\ = 3 \):\( 7 \)
Women features in both ratios, so we will use it as a link
Multiplying \( men \) : \( women \ = 5 \):\( 4 \) by \( 3 \) and \( women \) : \( children \ = 3 \):\( 7 \) by \( 4 \)
So both ratios are comparing to \( 12 \) women
\( men \) : \( women\ = 15 \):\( 12 \)
\( women \) : \( children\ = 12 \):\( 28 \)
Now we can link them together
\( men \) : \( women \) : \( children\ = 15 \):\( 12 \):\( 28 \)
There are \( 224 \) Children at the campsite.
- Calculate the total number of men and women
The ratio is
\( men \) : \( women \) : \( children\ = 15 \):\( 12 \):\( 28 \)
Number of children is \( 224 \) so
\( 28 \) parts\( = 224 \)
Divide by \( 28 \) to get one part
\( 1 part = 8 \)
Number of men 🠦 \( 8 \times 15 = 120 \)
Number of women 🠦
\( 8 \times 12 = 96 \)
Total number of men and women
\( 120+96 = 216 \)
Total number of men and women is \( 216 \).