Ratio, Proportion and Rates of Change

The One Thing

Ratio and proportion describe multiplicative relationships: quantities may be compared, shared, scaled, or used to determine missing values. Percentages and compound measures extend these ideas to relative change and rates, provided that original values, formulas, and units are handled consistently.

Definitions and Results

• Ratio: A comparison of two or more quantities using the same units, written using a colon, such as $2:3$. Before working with a ratio, convert all quantities to the same units.
• Simplifying a ratio: Divide every part of a ratio by the same common factor, preferably the highest common factor, to produce an equivalent ratio in its simplest form.
• Equivalent ratios: Ratios representing the same comparison, such as $2:3$ and $8:12$. Multiplying or dividing every part by the same factor preserves the comparison.
• Sharing in a ratio: Divide a total into parts according to the numbers in the ratio. To share a total $T$ in the ratio $a:b$, the total number of parts is $a+b$, and each part is $T÷(a+b)$. For $a:b:c$, each part is $T÷(a+b+c)$.
• Scale factor: The multiplier used to enlarge or reduce a quantity, shape, or measurement. Scaling all quantities by the same factor preserves their ratio.
• Percentage: A number expressed out of 100; $25\%$ means 25 out of 100.
• Percentage change: The change in a value expressed as a percentage of the original value:

$$\text{percentage change}=\frac{\text{change}}{\text{original value}}\times 100\%.$$

• Multiplier: A number used to calculate a percentage change. An increase by $p\%$ uses

$$1+\frac{p}{100},$$ while a decrease by $p\%$ uses $$1-\frac{p}{100}.$$

• Percentage increase:

$$\text{original value}\times \left(1+\frac{\text{percentage increase}}{100}\right).$$

• Percentage decrease:

$$\text{original value}\times \left(1-\frac{\text{percentage decrease}}{100}\right).$$ Repeated percentage changes must be applied successively using multipliers; equal percentage increases and decreases do not usually cancel.

• Direct proportion: A relationship in which one quantity is a constant multiple of another:

$$y=kx,k=\frac{y}{x}.$$ Thus $y/x$ remains constant. If $x$ is multiplied by a factor, $y$ is multiplied by the same factor. A graph of direct proportion passes through the origin and has a constant gradient when $y$ is plotted against $x$.

• Inverse proportion: A relationship in which one quantity increases as another decreases:

$$y=\frac{k}{x},xy=k.$$ Here $k$, the constant of proportionality, is fixed. If $x$ is multiplied by a factor, $y$ is divided by that factor. An inverse-proportion graph is curved and shows that $xy$ remains constant.

• Constant of proportionality: The fixed value $k$ linking two proportional quantities. For direct proportion, $k=y÷x$; for inverse proportion, $k=xy$.
• Compound measure: A measurement formed by combining two or more different units, such as kilometres per hour or kilograms per cubic metre.
• Speed: Distance travelled per unit of time:

$$\text{speed}=\frac{\text{distance}}{\text{time}},\text{distance}=\text{speed}\times \text{time},\text{time}=\frac{\text{distance}}{\text{speed}}.$$

• Density: Mass per unit volume:

$$\text{density}=\frac{\text{mass}}{\text{volume}},\text{mass}=\text{density}\times \text{volume},\text{volume}=\frac{\text{mass}}{\text{density}}.$$

• Pressure: Force per unit area:

$$\text{pressure}=\frac{\text{force}}{\text{area}},\text{force}=\text{pressure}\times \text{area},\text{area}=\frac{\text{force}}{\text{pressure}}.$$

• Unit conversion: Changing a measurement into an equivalent unit. Important conversions are:

$$1\text{ kilometre}=1000\text{ metres},$$ $$1\text{ hour}=60\text{ minutes},1\text{ minute}=60\text{ seconds},$$ $$1\text{ litre}=1000\text{ cubic centimetres}.$$

Worked Methods

Simplifying a ratio

• Confirm that all quantities use the same units.
• Find the highest common factor of every part.
• Divide every part by that factor.
• Write the resulting ratio in its simplest form.

Equivalent ratios may also be formed by multiplying or dividing every part by the same factor. For example, $2:3$ and $8:12$ are equivalent because both parts of $2:3$ have been multiplied by $4$.

Sharing a total in a ratio

• Add the ratio parts.
• Divide the total by this sum to find the value of one part.
• Multiply the value of one part by each ratio number.
• Check that the resulting quantities add to the original total.

For a ratio $a:b$, the value of one part is $$\frac{T}{a+b}.$$ For a ratio $a:b:c$, it is $$\frac{T}{a+b+c}.$$

Solving a proportion

• Convert quantities to consistent units.
• Find the value of one unit, if using a unitary method.
• Scale this value to obtain the required quantity.

Alternatively, form equivalent ratios by multiplying or dividing all corresponding quantities by the same factor. A sensible estimate can be used to check whether the result is reasonable.

Calculating percentage change

• Find the change by subtracting the original value from the new value, using the magnitude of the change where appropriate.
• Divide the change by the original value.
• Multiply by $100\%$:

$$\text{percentage change}=\frac{\text{change}}{\text{original value}}\times 100\%.$$

The denominator must be the original value.

Applying a percentage increase or decrease

• Convert the percentage to a multiplier.
• Multiply the original value by that multiplier.
• Interpret the result as the increased or decreased value.

For an increase: $$\text{new value}=\text{original value}\times \left(1+\frac{\text{percentage increase}}{100}\right).$$

For a decrease: $$\text{new value}=\text{original value}\times \left(1-\frac{\text{percentage decrease}}{100}\right).$$

For repeated changes, apply each multiplier successively rather than adding or subtracting the percentages. Equal percentage increases and decreases do not usually cancel.

Solving direct proportion

• Express the relationship as

$$y=kx.$$

• Calculate the constant of proportionality:

$$k=\frac{y}{x}.$$

• Substitute the known value of $x$ or $y$.
• Check that $y/x$ remains constant.

If $x$ is multiplied by a factor, $y$ must be multiplied by the same factor. A graph of the relationship passes through the origin and has constant gradient $k$.

Solving inverse proportion

• Express the relationship as

$$y=\frac{k}{x}$$ or $$xy=k.$$

• Calculate the constant of proportionality using

$$k=xy.$$

• Substitute the known value of $x$ or $y$.
• Check that the product $xy$ remains constant.

If $x$ is multiplied by a factor, $y$ must be divided by that factor. The graph is curved rather than a straight line through the origin.

Calculating speed, distance, or time

• Ensure that distance and time use consistent units.
• Select the appropriate formula:

$$\text{speed}=\frac{\text{distance}}{\text{time}},$$ $$\text{distance}=\text{speed}\times \text{time},$$ $$\text{time}=\frac{\text{distance}}{\text{speed}}.$$

• Convert units before substituting values.
• Substitute and calculate.
• State the answer with the correct compound unit.

For example, distance in kilometres and time in hours produce speed in kilometres per hour.

Calculating density, mass, or volume

• Identify the known quantities.
• Use the relevant rearrangement:

$$\text{density}=\frac{\text{mass}}{\text{volume}},$$ $$\text{mass}=\text{density}\times \text{volume},$$ $$\text{volume}=\frac{\text{mass}}{\text{density}}.$$

• Convert units where necessary.
• Substitute the values and check the resulting units.

Calculating pressure, force, or area

• Identify the required quantity.
• Use the relevant rearrangement:

$$\text{pressure}=\frac{\text{force}}{\text{area}},$$ $$\text{force}=\text{pressure}\times \text{area},$$ $$\text{area}=\frac{\text{force}}{\text{pressure}}.$$

• Ensure that the units are compatible.
• Substitute, calculate, and check the units.

Where It Goes Wrong

• Using a ratio before converting all quantities to the same units.
• Simplifying only one part of a ratio instead of dividing every part by the highest common factor.
• Sharing a total by the ratio numbers directly without first finding the total number of parts $a+b$ or $a+b+c$.
• Calculating percentage change using the new value rather than the original value.
• Treating repeated percentage changes as if the percentages can simply be added or cancelled; successive changes require successive multipliers.
• Confusing direct proportion, where $y/x$ is constant, with inverse proportion, where $xy$ is constant.
• Substituting into speed, density, or pressure formulas without rearranging correctly or converting units first.
• Forgetting that speed calculations require consistent units, such as kilometres with hours or metres with seconds.

What Gets Asked

• Simplify a ratio and identify or generate equivalent ratios, including examples such as $2:3$ and $8:12$.
• Share a total in a ratio $a:b$ or $a:b:c$.
• Solve a proportion using a unit value or equivalent ratios.
• Calculate a percentage change, increase, or decrease.
• Apply repeated percentage changes using multipliers.
• Identify whether a relationship is direct or inverse proportion.
• Find the constant of proportionality $k$, use $y=kx$, or use $y=k/x$ and $xy=k$.
• Interpret or sketch graphs of direct proportion and inverse proportion.
• Calculate speed, distance, or time, including unit conversions between kilometres and metres, hours and minutes, and minutes and seconds.
• Calculate density, mass, or volume, including conversions involving litres and cubic centimetres.
• Calculate pressure, force, or area.
• Check a result using estimation, units, the constant ratio $y/x$, or the constant product $xy$.