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Tutor: Alice

Direct and inverse proportion

​​​​In a nutshell

Proportion is a way of describing relationships between two sets of quantities. It can be described as being direct or indirect.

Direct proportion

​​Definition

If two quantities, $x$ and $y$ are directly proportional, as one increases the other does too. Both increase by the same factor - this means that as one value doubles, so does the other. If one value halves, the other does too.

Proportion can be written in two ways - both mean exactly the same thing:

• Using the following symbol: $\propto$ which means 'is proportional to'.
• As an equation where k is some constant known as the constant of proportionality.

When $x$ is directly proportional to $y$, this creates a straight line on a graph. See below for some graphed examples.

 Written using $\propto$ symbol​ Written as an equation Graph representation ​'$y$​ is directly proportional to $x$​'​​​​ ​$y \propto x$​​​y \propto xy \propto x​ ​​$y=kx$​ ​'$y$​​ is directly proportional to the square of $x$​​'​​​ ​​$y\propto x^2$​ ​​$y=kx^2$​​​ ​​'$y$​​ is directly proportional to the square root of $x$​​'​​​ ​​$y \propto \sqrt{x}$​ ​​$y=k\sqrt{x}$​

Note: Sometimes, the word 'directly' will be omitted and a question might just say 'is proportional to'. This means the same as 'is directly proportional to'.

Example 1

Grace buys $1000g$​ of flour which is enough to bake $40$ cakes. Flour costs $30p$ per $250g$. How much would Grace need to spend on flour to make $55$ cakes?

Work out how much flour is in one cake.

$1000\div 40 =25g$

Work out how much flour is in $55$ cakes.

$55\times25=1375g$

Work out how many lots of $250g$ are in $1375g$.

$1375\div250=5.5$​​

Multiply $5.5$​ by the cost of $250g$.

$30p \times 5.5 = \underline{£1.65}$

​​

Inverse proportion

​​Definition

If two quantities, $x$ and $y$ are inversely proportional, as one increases the other decreases. As one increases by a factor, the other decreases by the same factor  - this means that as one value doubles, the other halves. If one value is multiplied by four, the other is divided by four (or multiplied by $\frac{1}{4}$​ ).

Just like with direct proportion, inverse proportion can be written in two ways - however this time $x$ is replaced by $\frac{1}{x}$. Again see the below graphed examples.

 ​ ​Written using $\propto$​ symbol​ ​Written as an equation​ ​Graph representation​ ​​'$y$​​ is inversely proportional to $x$​​'​​​​ ​​​$y \propto \frac{1}{x}$​​ ​​​$y=\frac{k}{x}$​ ​​​'$y$​​​ is inversely proportional to the cube of $x$​​​'​​​ ​​​$y \propto \frac{1}{x^3}$​ ​​​$y=\frac{k}{x^3}$​

Example 2

It takes six decorators twelve weeks to decorate a mansion. How long would it take for $18$ decorators to decorate the same mansion?  Find a formula for the number of decorators, d, in terms of the number of weeks, t.

More decorators = less time so we are using inverse proportion to answer this question.

$6$ decorators $= 12$ weeks

$18$ decorators $= x$ weeks

Work out what six has been multiplied by.

$18\div6=3$

Divide the weeks by the same number.

$12\div3=x$

$x=\underline4$ weeks

To find a formula use the equation for inverse proportionality, using the letters from the question.

$d=\frac{k}{t}$​​

Substitute in the numbers you know and find the constant, k.

\begin{aligned}&&6&=\frac{k}{12}\\\times 12&&&&\times12\\&&72&=k\end{aligned}

Substitute k back into the original formula.

$\underline{d=\frac{72}{t}}$​​​