Rate and Unit Rate | Moosa Academy

A ratio compares two quantities. When those quantities carry different units — kilometres and hours, riyals and months — the comparison is called a rate. Reduce the bottom number to 1 and it becomes a unit rate, the form that is easiest to judge.

Concept What makes a rate

Divide one quantity by another whose unit is different, and the result is a rate.

\text{rate} = \frac{\text{first quantity}}{\text{second quantity}}

Litres per day, kilometres per hour, riyals per month — in each case the two units differ. That difference is what separates a rate from an ordinary ratio.

Example Drinking water

A person drinks 3 litres of water a day. Why is this a rate?

First quantity: 3 litres, a unit of volume.

Second quantity: 1 day, a unit of time.

The units differ, so the comparison is a rate.

⟹ 3 litres per day

Concept The unit rate

A unit rate is the same rate rewritten so the denominator is 1. Dividing 500 by 4 gives 125, so the car covers 125 km in each single hour.

Example Speed of a car

A car travels 500 kilometres in 4 hours. What is its speed?

Write the rate:

\dfrac{500 \text{ km}}{4 \text{ h}}

Divide to make the denominator 1:

500 \div 4 = 125

Check:

125 \times 4 = 500

⟹ 125 km per hour

Example Spending and production

Someone spends 8000 riyals every 5 months. A factory makes 1200 pieces in 6 days. Find each unit rate.

Spending:

8000 \div 5 = 1600

⟹ 1600 riyals per month

Production:

1200 \div 6 = 200

⟹ 200 pieces per day

Both follow one method: divide the first quantity by the second, then attach the units in the same order.

Concept Why unit rates are useful

Unit rates make comparison immediate. Faced with 8000 riyals over 5 months and 3600 riyals over 2 months, the totals tell you little. Reduce both to one month — 1600 against 1800 — and the answer is obvious at a glance.

Note Mistakes to avoid

Dividing the quantities the wrong way round.

Dropping the units from the answer.

Calling a comparison of two same-unit quantities a rate.

Comparing two rates without first reducing both to unit rates.

Forgetting to check by multiplying back.

Summary

A rate compares two quantities with different units.
A unit rate is a rate whose denominator has been reduced to 1.
Find it by dividing the first quantity by the second.
500 km in 4 h gives 125 km per hour.
Unit rates make two different offers directly comparable.