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Rates, Ratio, Proportions and Percentages

1.0 Numbers · 1.5 Rates, Ratio, Proportions and Percentages

Syllabus tag: Kenya CBC | Grade 8 Mathematics | Strand 1.0 Numbers | Sub-Strand 1.5 Rates, Ratio, Proportions and Percentages (14 lessons)

Lesson objectives

By the end of this sub-strand, you should be able to:

  • Identify and work out rates in real-life situations
  • Compare ratios and divide quantities in given ratios
  • Work out percentage increase and decrease of quantities
  • Identify and work out direct and indirect proportions

Rates, Ratio, Proportions and Percentages

These four ideas all compare quantities. They appear together because the same thinking runs through all of them.

a) Rates

A rate compares two quantities of different kinds, such as distance and time.

Working out a rate Working out a rate rate = quantity / time one thing per another speed = 300 km / 4 h a bus journey speed = 75 km/h kilometres in one hour

The word "per" signals a rate. Kilometres per hour, shillings per kilogram, litres per minute.

Always state the units. A rate of 75 means nothing; 75 km/h means something.

b) Ratio

A ratio compares quantities of the same kind. It has no units.

Sharing 4 500 in the ratio 2 : 3 Sharing 4 500 in the ratio 2 : 3 2 parts 1 800 3 parts 2 700 KES 4 500

Write a ratio in its simplest form. 10 : 15 becomes 2 : 3 by dividing both by 5.

Order matters. 2 : 3 is not the same as 3 : 2.

c) Dividing in a ratio

Dividing in a ratio Dividing in a ratio total parts = 2 + 3 add the ratio numbers total parts = 5 five equal shares one part = 4500 / 5 divide the amount one part = 900 each share is worth this the shares = 1800 and 2700 2 × 900 and 3 × 900

Add the ratio numbers to find how many equal parts there are. Divide the amount by that total to find one part. Then multiply out each share.

Check by adding the shares back. They must give the original amount.

d) Percentages

A percentage is a fraction out of 100.

Percentage increase Percentage increase increase = 4600 − 4000 new minus old increase = 600 the change percentage = 600 / 4000 × 100 compare with the ORIGINAL percentage = 15% the answer

For an increase or decrease, always compare the change with the original amount, not the new one. Dividing by the wrong figure is the usual mistake.

For a decrease the method is identical, using old minus new.

e) Direct proportion

Two quantities are in direct proportion when one rises as the other rises, in the same ratio.

If 3 books cost KES 150, then 6 books cost KES 300. Double the books, double the cost.

Find the cost of one first, then multiply. That single step handles almost every direct proportion question.

f) Indirect proportion

Two quantities are in indirect proportion when one rises as the other falls. Their product stays the same.

If 4 workers take 6 days, the work is 24 worker-days. With 3 workers it takes 8 days, because 3 × 8 is also 24.

More workers means fewer days. More speed means less time. Ask yourself which way the answer should move before calculating, then check it moved that way.

g) Where this is used

A driver works out fuel consumption. A builder mixes cement and sand in a ratio. A trader raises prices by a percentage. A farmer estimates how long a job takes with more hands.

Words to know

  • Rate -- a comparison between two different kinds of quantity, such as distance per time.
  • Ratio -- a comparison between two or more quantities of the same kind.
  • Direct proportion -- a relationship where two quantities increase or decrease together at the same rate.
  • Indirect (inverse) proportion -- a relationship where one quantity increases as the other decreases, keeping their product constant.

:::checkpoint Check yourself

  1. A car travels 240 km in 3 hours. What is its speed?
  2. Divide KES 3 500 in the ratio 3 : 4.
  3. A price rises from KES 800 to KES 920. Find the percentage increase.
  4. If 5 workers take 12 days, how long would 6 workers take? :::

Bridge to practice

Try the exercises below -- on rates, simplifying and dividing ratios, percentage change, and direct and indirect proportion.

Check yourselfPractise Rates, Ratio, Proportions and Percentages — 10 questions →