R13, R14 - Proportion equations and graphs | AQA GCSE Mathematics

R13 connects direct and inverse proportion equations with fixed ratios, products and reciprocals. R14 connects these relationships to graphs and interprets a straight-line gradient as a rate of change. You will use both the shape and the scale of a graph to explain what it represents.

What a direct proportion equation says

A hire charge is $C=3t+5$ , where $t$ is time in hours and $C$ is cost in pounds. The charge rises by $\pounds 3$ every hour. Does that make cost directly proportional to time? A constant rate of change and a constant ratio are different tests.

You already know

Direct proportion keeps a ratio fixed; inverse proportion keeps a product fixed. In $y=mx+c$ , $m$ is the gradient and $c$ is the vertical intercept. Substitute a value to use a given equation.

For $y=4x$ , each unit of $x$ gives four units of $y$ . The points $(2,8)$ and $(5,20)$ both have $y\div x=4$ . The symbol $\propto$ means “is proportional to”: $y\propto x$ says that $y$ is directly proportional to $x$ .

Interpreting direct proportion

y\propto x\quad\Longleftrightarrow\quad y=kx

Here $k$ is a fixed constant of proportionality. For $x\ne0$ , $\frac yx=k$ . There is no added non-zero constant; the graph passes through the origin.

Not on the formulae sheet: learn it.

The coefficient can be a fraction or a decimal. It need not be a whole number or greater than one.

Worked example 1: reading a constant ratio

Question. Positive quantities satisfy $y=2.5x$ . Write down the constant of proportionality and work out $y$ when $x=4$ . Non-calculator.

Plan. Read the multiplier of $x$ , then substitute $4$ .

\begin{aligned} k&=2.5 &&\text{coefficient of }x\\ y&=2.5\times4 &&\text{substitute }x=4\\ &=10 &&\text{evaluate} \end{aligned}

Why is $2.5$ the constant? For any positive $x$ , dividing $2.5x$ by $x$ gives $2.5$ .

Answer. $k=2.5$ , and $y=10$ .

Check. $10\div4=2.5$ .

Add only a constant $3$ to the rule. The graph stays straight, but its ratio property changes.

Worked example 2: a straight line that is not proportion

Question. Positive quantities satisfy $y=2.5x+3$ . Is $y$ directly proportional to $x$ ? Show working to support your answer. Non-calculator.

Plan. Compare $y\div x$ for two values of $x$ .

\begin{aligned} x=2&:\ y=2.5(2)+3=8 &&\text{first pair}\\ \frac yx&=\frac82=4 &&\text{first ratio}\\ x=4&:\ y=2.5(4)+3=13 &&\text{second pair}\\ \frac yx&=\frac{13}{4}=3.25 &&\text{different ratio} \end{aligned}

Why are two unequal ratios enough? Direct proportion would require the same ratio for every allowed pair. These two pairs disprove it.

Answer. No. The ratios $4$ and $3.25$ are different. Equivalently, the graph has vertical intercept $3$ , so it does not pass through the origin.

Check. Doubling $x$ from $2$ to $4$ changes $y$ from $8$ to $13$ , not $16$ .

The opening charge has the same issue: its extra $\pounds 5$ is a fixed starting charge. The $\pounds 3$ hourly increase alone does not make the whole charge proportional to time.

Watch out

Wrong: “The gradient is constant, so the quantities are directly proportional.” Right: a straight line also needs to pass through the origin to represent direct proportion.

Test a new rule by checking its intercept or its ratios.

Inverse proportion as proportion to a reciprocal

The reciprocal of $2$ is $\frac12$ , and the reciprocal of $4$ is $\frac14$ . As a positive number doubles, its reciprocal halves. That is exactly the multiplier behaviour of inverse proportion.

For $y=12\div x$ , the equation can also be written $y=12\times\frac1x$ . It is a direct-proportion rule between $y$ and the new quantity $\frac1x$ . For example, $x=2,4,8$ gives reciprocals $\frac12,\frac14,\frac18$ and outputs $6,3,1.5$ .

Inverse proportion and a reciprocal

y\propto\frac1x\quad\Longleftrightarrow\quad y=\frac{k}{x}\quad\Longleftrightarrow\quad xy=k

Here $k$ is fixed and $x\ne0$ . Thus “ $y$ is inversely proportional to $x$ ” means “ $y$ is directly proportional to $\frac1x$ ”. For positive physical quantities, use the positive values allowed by the context.

Not on the formulae sheet: learn it.

This gives a neat change of viewpoint: a curved graph of $y$ against $x$ becomes a straight line through the origin when you plot $y$ against $\frac1x$ . You have changed the horizontal quantity, not the data.

Worked example 3: using the fixed product

Question. Positive quantities satisfy $y=18/x$ . Work out $y$ when $x=6$ , and state the value of the fixed product $xy$ . Non-calculator.

Plan. Substitute into the supplied inverse rule, then multiply back.

\begin{aligned} y&=18\div6=3 &&\text{substitute the input}\\ xy&=6\times3=18 &&\text{check the product} \end{aligned}

Why is the product $18$ ? Multiplying $y=18/x$ by non-zero $x$ cancels the division by $x$ .

Answer. $y=3$ ; the fixed product is $18$ .

Check. $y\div(1/x)=3\div\frac16=18$ , showing direct proportion to the reciprocal.

Change only the unknown: the same rule now gives $y$ , and you need $x$ .

Worked example 4: finding the denominator

Question. Positive quantities satisfy $y=18/x$ . Work out $x$ when $y=4.5$ . Non-calculator.

Plan. Use the fixed product $xy=18$ , then divide by $y$ .

\begin{aligned} 4.5x&=18 &&\text{multiply the equation by }x\\ x&=18\div4.5 &&\text{divide by }4.5\\ &=4 &&\text{evaluate} \end{aligned}

Why multiply by $x$ first? It removes the unknown from the denominator, leaving a familiar linear equation.

Answer. $x=4$ .

Check. $18\div4=4.5$ , as required.

The denominator is not a number you may set to zero. The inverse rule gives no value at $x=0$ .

Watch out

Wrong: $y\propto1/x$ means $y=1/x$ . Right: the multiplier may be any fixed constant; here it is $18$ , so $y=18/x$ . The proportionality symbol does not mean equality.

Now find the denominator in another given rule.

Gradient as a rate of change

A straight-line gradient says how much the vertical quantity changes for each one-unit increase in the horizontal quantity.

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Number

Algebra

Ratio, proportion and rates of change

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Probability

Statistics

R13, R14 - Proportion equations and graphs

What a direct proportion equation says

Inverse proportion as proportion to a reciprocal

Gradient as a rate of change

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