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Compound Proportions and Rates of Work

Numbers · Compound Proportions and Rates of Work

Syllabus tag: Kenya CBC | Grade 9 Mathematics | Strand 1.0 Numbers | Sub-Strand 1.4 Compound Proportions and Rates of Work (12 Lessons)

Lesson objectives

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

  • Distinguish between direct and inverse proportion in real situations.
  • Work out compound proportions involving more than two quantities.
  • Calculate rates of work for individuals and for groups working together.
  • Apply proportional reasoning to problems involving labour, time, cost and output.

Compound Proportions and Rates of Work

Two farmers share a harvest. Fifteen workers dig a trench that six workers started. A tap fills a tank while another drains it. All three are the same kind of problem.

Once you can see who gets what, and how fast work gets done, these questions become routine.

a) Dividing into proportional parts

A ratio tells you how to cut something up. Read 3 : 5 as three parts for one person and five parts for the other.

Sharing in the ratio 3 : 5 Sharing in the ratio 3 : 5 3 parts KES 9 000 5 parts KES 15 000 8 parts in all = KES 24 000

The important step is counting the total parts first. Three plus five is eight, so the money is being cut into eight equal pieces, not two.

Sharing a profit Sharing a profit total parts = 3 + 5 add the ratio numbers total parts = 8 the whole is 8 parts one part = 24 000 / 8 divide the money by 8 one part = KES 3 000 the value of one part first share = 3 × 3 000 multiply by her parts first share = KES 9 000 her share

Check your answer by adding the shares back together. 9 000 plus 15 000 is 24 000, so nothing has gone missing.

b) Comparing ratios

Ratios stay the same when you multiply or divide both sides by the same number. So 3 : 5 is the same sharing as 6 : 10 and as 30 : 50.

Always simplify before you work. Smaller numbers mean fewer mistakes.

To compare two ratios, write both as fractions. 3 : 5 becomes 3/5 and 4 : 7 becomes 4/7. Then compare the fractions in the usual way.

c) Compound proportion

Sometimes two things change together. More workers means fewer days. That is a compound proportion.

The safest method is to find the total amount of work first.

Compound proportion Compound proportion 6 workers = 10 days what we are told 15 workers = ? days what we want total work = 6 × 10 workers multiplied by days total work = 60 worker-days the job is fixed days = 60 / 15 share the job out days = 4 days more workers, fewer days

The unit here is the worker-day, meaning one worker labouring for one day. The job needs 60 of them however you arrange it.

Ask yourself whether the answer should be bigger or smaller before you divide. More workers must give fewer days. If your answer goes the wrong way, you have divided upside down.

d) Rates of work

A rate of work is the fraction of a job finished in one unit of time.

If Amina digs a shamba in 6 days, then in one day she digs one sixth of it. That single idea solves every question in this section.

Rates of work Rates of work Amina per day = 1 / 6 she takes 6 days Brian per day = 1 / 4 he takes 4 days together per day = 1/6 + 1/4 add the two rates together per day = 5 / 12 common denominator 12 days needed = 12 / 5 turn the fraction over days needed = 2.4 days the answer

Two rules are worth memorising. To combine workers, add their rates. To turn a rate back into a time, turn the fraction upside down.

If something works against you, such as a leaking tank, subtract its rate instead of adding it.

e) Where this is used

Business partners split profits by their capital contributions. A contractor quoting for a road works out worker-days before pricing the job. A farmer sharing a harvest with a labourer uses proportional parts to settle it fairly.

Words to know

  • Direct proportion -- a relationship in which two quantities increase or decrease together, keeping a constant ratio.
  • Inverse proportion -- a relationship in which one quantity increases as the other decreases, keeping a constant product.
  • Compound proportion -- a situation in which a quantity depends on two or more others at the same time.
  • Unitary method -- finding the value for a single unit first, then scaling to the required quantity.
  • Rate of work -- the fraction of a task completed in one unit of time.

:::checkpoint Check yourself

  1. Share KES 36 000 between two people in the ratio 4 : 5.
  2. Simplify the ratio 18 : 24.
  3. If 8 workers take 15 days to build a wall, how long would 10 workers take?
  4. A tap fills a tank in 3 hours. Another fills it in 6 hours. How long do both take working together? :::

Bridge to practice

The exercises start with straightforward direct and inverse proportion, then build to compound situations with two changing quantities, and finish with combined rates of work including a phased problem. For every question, say aloud whether the answer should be larger or smaller than the figure you were given, and use that as your check before selecting an option.

Check yourselfPractise Compound Proportions and Rates of Work — 10 questions →Next in MathematicsTime, Distance and Speed →