Proportion

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Proportion


Ratio


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Summary

Proportion

In a nutshell

Proportion is a way of describing a relationship between two quantities. In particular, two quantities in direct proportion will always increase with each other. But, if two quantities are in inverse proportion to each other, one will only increase if the other decreases.



Direct proportion

Definition

If two quantities yy​ and xx​ are in direct proportion with one another, then yy is directly proportional to xx. ​Other ways of writing this are:

  • yxy\propto x
  • y=kxy=kx​, where kk​ is called the constant of proportionality.

This also means that if yxy\propto x, then the graph of xx​ and yy​ will be a straight line through the origin.

Maths; Proportion; KS3 Year 7; Proportion


The key property of direct proportion

The key property of direct proportion is that two quantities in direct proportion will always increase and decrease by the same factor. So, if one is doubled, then the other is doubled. And if one is divided by 44​, the other is also divided by 44​.


Direct proportion questions

To solve direct proportion questions, it is easiest to approach the simpler ones just by using the key property. However, harder proportion questions may require a more algebraic approach.


Example 1

If it costs £30£30​ an hour to employ 44​ people to perform a job, how much would it cost to employ 1010​ people?


This question is using direct proportion because employing more people requires more money - the two quantities are increasing with each other. Hence, the number of employees is directly proportional to the cost. To find the cost for 10 employees, use the key property of direct proportion.

It is given that 44​ people = £30.£30.


Divide both sides by 44​ to find the cost of 11​ employee:

11​ person = £30÷4=£7.50£30\div4=£7.50


Multiply both sides by 1010​ to get the cost of employing 1010​ people:

1010​ people = £7.50×10£7.50 \times 10

1010​ people = £75£75

It will cost £75\underline{£75} to employ 1010​ people.



Inverse proportion

Definition

If two quantities yy and xx​ are in inverse proportion with one another, then yy is inversely proportional to xx. Other ways of writing this are:

  • y1xy\propto \frac{1}{x}
  • y=kxy=\frac{k}{x}, where kk is the constant of proportionality.

If yy and xx​ are inversely proportional to each other, then their graph is going to look like this:

Maths; Proportion; KS3 Year 7; Proportion


This is called a reciprocal graph, it is the graph of y=kxy=\frac{k}{x}​.


The key property of inverse proportion

The key property of two quantities in inverse proportion is that if one quantity increases by a factor, then the other quantity decreases by the same factor. So, if one quantity is doubled, the other is halved. If one quantity is multiplied by 3, the other is divided by 3.


Inverse proportion questions

To know whether or not you need to use inverse proportion, think about whether or not it makes sense for the two quantities to be inversely proportional. Will doubling one quantity halve the other? If so, then it's likely that you have to use inverse proportion.


Example 2

It takes 44​ builders 1515​ weeks to build a house.


i) How long would it take for 1212​ builders to build the same house?

ii) Find a formula for the number of builders, bb, in terms of the time taken in weeks, tt.


i)

The more builders there are, the less time it would take to build the house. Hence, it can be assumed that the number of builders is inversely proportional to the time taken to build the house. So, use the key property to find the time taken for 1212​ builders.

44​ builders = 1515​ weeks


Divide the number of builders by 44​ to find our the time for 11​ builder. Due to the inverse relation, you must multiply the time by 44​:

11​ builder = 15×4=6015\times4=60 weeks


Multiply the number of builders by 1212​ to find the time for 1212​ builders. Again, this means divide the time by 1212​:

1212​ builders = 60÷12=560\div12=5 weeks


It will take 1212​ builders 5 weeks\underline{5\ \text{weeks}}​ to build the same house.


ii)

The values of bb and tt are inversely proportional to one another.  This gives a formula that involves the two quantities:

b=ktb=\frac{k}{t} ()(\star)​​


To find the value of kk​, substitute a known value of bb and tt. When b=4,t=15b = 4, t = 15. Hence:

4=k15k=4×15=60\begin{aligned}4&=\frac{k}{15}\\k&=4\times15=60\end{aligned}


Put this value of kk​ into the original equation (\star) to give a full formula for bb:

b=60t\underline{b=\frac{60}{t}}


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FAQs - Frequently Asked Questions

How do you find the constant of proportionality?

What is the key property for two quantities in inverse proportion?

What is the key property for two quantities in direct proportion?

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