Compound Proportions and Rates of Work
Numbers · Compound Proportions and Rates of Work
Syllabus tag: KCSE | Mathematics | Form 3 | Topic 12 Compound Proportions and Rates of Work
Lesson objectives
By the end of this topic, you should be able to:
- Solve problems involving compound proportions using the unitary and ratio methods.
- Apply ratios and proportions to real life situations.
- Solve problems involving rates of work.
- Handle mixtures and proportional parts.
Compound Proportions and Rates of Work
Compound proportion involves three or more quantities varying together. Rates of work applies the same thinking to jobs done at different speeds.
a) Recognising the direction
Before calculating, decide for each quantity whether it is direct or inverse.
More workers means fewer days: inverse.
More work means more days: direct.
More hours per day means fewer days: inverse.
Getting a direction wrong flips the whole answer, so decide first and write it down.
b) The unitary method
Reduce to a single unit, then scale up.
For workers and days, the product worker-days is the total work and stays constant.
So 6 workers for 8 days is 48 worker-days. The same job with 4 workers takes 48 ÷ 4 = 12 days.
c) The ratio method
Multiply the original value by a ratio for each changing quantity.
For a direct relation, put the larger new value on top.
For an inverse relation, put it on the bottom.
Say 6 workers take 8 days for one wall. Then 4 workers building 3 walls take 8 × (6/4) × (3/1) = 36 days.
The workers ratio is inverted because fewer workers means more days. The walls ratio is not, because more walls means more days.
d) Checking the answer
Ask whether each change should raise or lower the answer, then confirm it did.
Fewer workers should raise the days. More walls should raise them further. An answer of 36 against the original 8 is consistent with both.
That check catches nearly every error in this topic.
e) Rates of work
If a job takes t hours, the rate is 1/t of the job per hour.
Working together, add the rates.
The time taken is the reciprocal of the combined rate.
Do not average the times. Two workers taking 6 and 12 hours do not take 9 hours together; they take 4.
f) Work against each other
If one process fills and another empties, subtract the rates.
A tap fills in 6 hours and a leak empties in 12. That gives 1/6 − 1/12 = 1/12, so the tank fills in 12 hours.
If the emptying rate is larger, the result is negative and the tank never fills. Say so rather than reporting a negative time.
g) Mixtures and proportional parts
To divide a quantity in a ratio, add the parts, divide, then multiply out.
For mixtures, work with the amount of each component, not the percentages.
Mix 20 litres at 30% with 30 litres at 50%. That gives 6 + 15 = 21 litres of the component in 50 litres, which is 42%.
Percentages cannot simply be averaged unless the quantities are equal. Here averaging 30 and 50 would give 40%, which is wrong.
h) Where this is used
Job scheduling and labour costing. Filling and draining tanks. Blending fuels, feeds and fertilisers. Any planning with several varying factors.
Words to know
- Compound proportion -- proportion involving three or more quantities changing together.
- Unitary method -- solving by first finding the value for a single unit.
- Rate of work -- the fraction of a job completed per unit time.
- Person-days -- a unit measuring total work or supply as people multiplied by days.
- Mixture -- a combination of quantities of differing unit values.
:::checkpoint Check yourself
- If 5 workers take 12 days, how long would 3 workers take?
- A alone takes 4 hours and B alone takes 6 hours. How long together?
- A tap fills in 5 hours and a leak empties in 10 hours. How long to fill?
- Why can percentages not simply be averaged when mixing unequal quantities? :::
Bridge to practice
The exercises begin with compound proportion by both methods, move through rates of work and staged work, and finish with mixtures and supply problems. For each changing quantity, write down whether it acts directly or inversely before forming any fraction.