Rate of Change

A rate tells you how one quantity changes compared with another. You meet rates every day, often without noticing:

• A car travels at $60$ km per hour — that is distance compared with time.
• Apples cost \3$ per kilogram — that is cost compared with weight.
• You type $40$ words per minute — that is words compared with time.

The word per is the giveaway — it means "for each". A rate of $60$ km per hour means that for each hour, $60$ km is covered.

A rate always has compound units — two units joined together, like km/h or $/kg. The slash also means "per".

Rates are powerful because they let you predict. If you know a steady rate, you can work out the total for any amount of time, weight or quantity.

Speed is the most familiar rate of all. It tells you how much distance is covered for each unit of time: $\text{speed} = \frac{\text{distance}}{\text{time}}.$

If a cyclist rides $30\text{ km}$ in $2$ hours, their speed is $30 \div 2 = 15$ km/h.

The three quantities — speed, distance and time — are linked, so you can find any one from the other two. A formula triangle is a handy way to remember how:

Cover the letter you want with a finger:

• Cover D: you see $S \times T$, so $\text{distance} = \text{speed} \times \text{time}$.
• Cover S: you see $D$ over $T$, so $\text{speed} = \text{distance} \div \text{time}$.
• Cover T: you see $D$ over $S$, so $\text{time} = \text{distance} \div \text{speed}$.

One important habit: keep the units consistent. If the speed is in km/h, the time must be in hours and the distance in km. Mixing minutes with hours is the most common mistake — convert first.

A distance-time graph turns a journey into a picture. Time always runs along the horizontal axis and distance up the vertical axis.

Reading the graph tells you the whole story of a journey:

Here is how to read every part:

• A sloping line going up — the object is moving away. The steeper it is, the faster.
• A flat (horizontal) line — the distance is not changing, so the object has stopped.
• A sloping line coming back down — the object is returning towards the start.

The graph above shows someone moving away from home, stopping for a while (the flat part), then moving away again, faster this time (the steeper part).

A graph like this carries far more information than a list of numbers — you can see the whole journey at a glance.

The real power of a graph is in its steepness. The steepness of a line — its gradient — tells you the rate of change.

On a distance-time graph, the steepness is the speed. You find it by comparing how far the line rises with how far it goes across: $\text{speed} = \frac{\text{change in distance}}{\text{change in time}}.$

For example, if a line rises $40\text{ km}$ over $2$ hours, the speed is $40 \div 2 = 20$ km/h.

The same idea works for any graph that compares two quantities. The steepness always tells you the rate:

• On a graph of cost against weight, the steepness is the price per kilogram.
• On a graph of water in a tank against time, the steepness is the filling rate.

So whenever you see a straight-line graph, ask "how steep is it?" — that steepness is the rate, the heart of this whole topic.

Speed gets the most attention, but the rate idea is everywhere — anytime one quantity changes against another.

Some everyday rates:

• Cost rate — price per kilogram, or cost per ticket. If $5$ kg of rice costs \20$, the rate is$$4$ per kg.
• Flow rate — litres of water per minute filling a bath or pool.
• Pay rate — money earned per hour of work.
• Population growth — people added per year in a town.

Every one of these is found the same way: divide the total change by the amount it is compared against. Cost rate is total cost ÷ total weight; pay rate is total pay ÷ hours worked.

A handy check: if you double the time (or weight, or hours), a steady rate means the total doubles too. Rates and the ratio-and-proportion topic are close cousins — both are about quantities changing together in step.

The rate-of-change ideas you have built this year carry straight into harder maths:

• Gradients of straight lines — the steepness of any graph line is exactly the rate of change.
• Linear functions and graphs — equations like $y = mx + c$ use $m$ as the rate.
• Compound measures — speed, density and pressure are all rates you will use again.
• In everyday life — speed limits, prices, wages, mobile-data usage and fuel economy are all rates.

If a later graphs topic feels tricky, the rate idea — how much one thing changes for each unit of another — is almost always underneath it. Keep this guide close.