Rates and Direct Proportion

What you will learn

• Find a rate, such as a price per item or a speed.
• Solve direct proportion problems with the unitary method.
• Compare unit prices to find the best buy.

Before you start: you should know ratios and be able to divide decimals. See Ratios and Sharing in a Ratio and Multiplying and Dividing Decimals.

The idea

A rate compares two different kinds of quantity: pounds per kilogram, miles per hour, words per minute. Two quantities are in direct proportion when doubling one doubles the other. Pens at a fixed price are like that: twice as many pens cost twice as much.

The unitary method

1. Find the amount for one unit by dividing.
2. Multiply to get the amount you need.

If $4$ pens cost £$6$, one pen costs $6 \div 4 = £1.50$, and $10$ pens cost $10 \times 1.50 = £15$.

Speed, distance and time

Speed is a rate: distance per unit of time. At a steady speed, distance is in direct proportion to time.

$\text{distance} = \text{speed} \times \text{time}$

Best buys

To compare packs, find the price for one unit (one item, or $100$ g). The smaller unit price is the better buy, if you can use it all.

Worked examples

Example 1: Unitary method

Three loaves cost £$4.50$. What do $5$ loaves cost?

1. One loaf: $4.50 \div 3 = 1.50$.
2. Five loaves: $5 \times 1.50 = 7.50$.

Answer: £$7.50$

Example 2: Speed

A cyclist rides $120$ km in $1.5$ hours. What is the average speed?

1. Speed is distance over time: $120 \div 1.5$.
2. Multiply both by $2$: $240 \div 3 = 80$.

Answer: $80$ km per hour

Example 3: Best buy

Juice: $2$ litres for £$3.20$, or $3$ litres for £$4.50$. Which is cheaper per litre?

1. First: $3.20 \div 2 = 1.60$ per litre.
2. Second: $4.50 \div 3 = 1.50$ per litre.

Answer: The $3$ litre bottle.

Watch out: not everything is in direct proportion. A taxi with a fixed starting fee does not double in price when the distance doubles.

Why it matters

Fuel economy, wages per hour, recipes scaled for more people, currency and map scales all use rates and proportion. It is also the start of linear graphs, where $y = kx$ is a straight line through the origin.