3.3 Ratios, Rates, Proportion & Unit Conversion

Ratios, rates, proportion and unit conversion

Ratios compare quantities

A ratio of 3:5 means that for every three equal parts of the first quantity there are five equal parts of the second. It does not mean the first quantity is three fifths of the total. The total contains eight parts, so the first quantity is 3/8 of the combined amount.

To share 320 in the ratio 3:5:

1. total parts = 3 + 5 = 8;
2. one part = 320 ÷ 8 = 40;
3. shares = 3 × 40 = 120 and 5 × 40 = 200.

Check that the shares add to 320 and preserve 120:200 = 3:5.

Before simplifying a ratio, use the same units. The ratio 2 metres to 50 centimetres is 200 cm:50 cm = 4:1, not 2:50.

Rates compare unlike units

A rate might be kilometres per hour, litres per minute, items per person, or dollars per unit. Write the units as part of the number.

For motion:

• distance = speed × time;
• speed = distance ÷ time;
• time = distance ÷ speed.

If a vehicle travels at 72 km/h for 25 minutes, convert time to hours: 25/60 hour. Distance = 72 × 25/60 = 30 km.

If one part of a journey uses a different speed, calculate each segment separately. The average speed is total distance divided by total time, not usually the simple mean of the speeds.

Direct proportion

Two quantities are directly proportional when multiplying one by a factor multiplies the other by the same factor. The ratio y/x remains constant.

If five identical containers hold 60 litres, one holds 12 litres and eight hold 96 litres. A unit-rate method is often clearer than cross-multiplication:

60 litres ÷ 5 containers = 12 litres per container.

Then multiply by the requested count.

Cross-multiplication is useful when the relationship is confirmed:

5/60 = 8/x, so 5x = 480 and x = 96.

Inverse proportion

Under ideal assumptions, some quantities move in opposite directions while their product remains constant. If six identical workers complete a fixed task in ten hours at a constant individual rate, twelve workers would take five hours:

workers × time = 6 × 10 = 60 worker-hours.

This model applies only if workers are interchangeable, work can be divided, and no coordination losses occur. A word problem may explicitly grant those assumptions. In real work they may fail, but an aptitude model uses the facts supplied.

Similarly, for a fixed distance, speed × time is constant. Increasing speed reduces time. Do not use inverse proportion when the task itself grows or when a fixed setup time is included unless the problem accounts for it.

Combined rates

Add rates when sources work simultaneously in the same direction. If one pump transfers 120 L/min and another transfers 80 L/min, the combined ideal rate is 200 L/min. A 1,000-litre volume then takes 1,000 ÷ 200 = 5 minutes.

For fill-and-drain problems, assign signs. A fill rate of 50 L/min and a drain rate of 20 L/min produce a net increase of 30 L/min.

For people or machines completing a task, convert each time to a fraction of the job per unit time. A machine completing a job in four hours works at 1/4 job per hour. Another taking six hours works at 1/6 job per hour. Together they work at 1/4 + 1/6 = 5/12 job per hour, so the ideal combined time is 12/5 = 2.4 hours.

Dimensional analysis

Write conversion factors so unwanted units cancel:

90 km/h × 1,000 m/1 km × 1 h/3,600 s = 25 m/s.

Kilometres cancel and hours cancel, leaving metres per second. This method prevents multiplying when division is required.

Useful metric relationships include:

• 1 kilometre = 1,000 metres;
• 1 metre = 100 centimetres;
• 1 litre = 1,000 millilitres;
• 1 cubic metre = 1,000 litres;
• 1 hour = 60 minutes = 3,600 seconds.

For area and volume, conversion factors are squared or cubed. Because 1 m = 100 cm, 1 m² = 10,000 cm² and 1 m³ = 1,000,000 cm³. A common error is applying only the linear factor.

Scale and maps

A scale of 1:500 means one unit on the drawing represents 500 of the same units in reality. A 6 cm line represents 3,000 cm = 30 m. Keep units consistent before interpreting the result.

Final checks

Label the result. Ask whether the direction is sensible: more identical containers should hold more, more workers should reduce time in an inverse model, and a conversion from kilometres to metres should increase the numeric value. Substitute the answer into the original ratio or rate to verify it.