Last reviewed: 10 November 2026
Updated November 2026. These Grade 9 compound proportions and rates of work notes show how to share a quantity in a given ratio, solve problems where two or more quantities change at once, and work out how long people or machines take when working together. The key idea for rates of work is that you add work done per unit time: if one tap fills a tank in 6 hours and another in 3 hours, together they fill 1/6 + 1/3 = 1/2 of the tank each hour, so the job takes 2 hours.
The examples and questions are original practice for Kenyan junior school learners, not an official paper. For the other Grade 9 maths topics, see our Grade 9 maths revision guide and the Grade 9 notes hub. Basic ratio work is in our ratio, proportion and rates notes.
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What the Grade 9 Curriculum Covers
Compound proportions and rates of work is a sub-strand under Numbers in the Kenya Institute of Curriculum Development’s Grade 9 Mathematics design. Learners divide quantities into proportional parts, relate proportional parts, work out compound proportions using the ratio method, and calculate rates of work. The design is listed on the KICD Grade Nine curriculum designs page.
Proportional Parts
To share an amount in a ratio, add the parts, find the value of one part, then multiply.
Worked example
Three farmers share KSh 36,000 profit from a joint maize sale in the ratio 2 : 3 : 4 (a made-up example).
- Total parts: 2 + 3 + 4 = 9.
- One part: 36,000 ÷ 9 = KSh 4,000.
- Shares: 2 × 4,000 = KSh 8,000; 3 × 4,000 = KSh 12,000; 4 × 4,000 = KSh 16,000.
- Check: 8,000 + 12,000 + 16,000 = 36,000.
Relating ratios
If A : B = 2 : 3 and B : C = 4 : 5, make B the same in both. Multiply the first by 4 and the second by 3: A : B = 8 : 12 and B : C = 12 : 15. So A : B : C = 8 : 12 : 15.
Direct and Inverse Proportion
A proportion is a statement that two ratios are equal, as described in OpenStax Prealgebra’s section on proportions. Before solving, ask how each quantity affects the answer:
- Direct: more of one means more of the other. More bags of cement cost more money.
- Inverse: more of one means less of the other. More workers finish a job in fewer days.
Compound Proportion by the Ratio Method
In a compound proportion, two or more quantities change. Deal with each one separately and multiply by a ratio that makes the answer bigger or smaller as common sense says.
Worked example
6 workers take 8 days to weed a farm. How long would 4 workers take to weed a farm 1.5 times as big, working at the same rate?
- Start with 8 days.
- Fewer workers (6 → 4) means more days: multiply by 6/4.
- Bigger farm (1 → 1.5) means more days: multiply by 1.5/1.
- Days = 8 × 6/4 × 1.5 = 18 days.
Worked example with three quantities
5 machines working 8 hours a day pack 2,000 bags of flour. How many bags would 3 machines pack working 10 hours a day? Fewer machines → fewer bags: × 3/5. More hours → more bags: × 10/8. Bags = 2,000 × 3/5 × 10/8 = 1,500.
Rates of Work
If a person can finish a job in n days, they do 1/n of the job each day. When people work together, add their daily rates; the time is 1 ÷ (total rate).
Worked example: two people
Achieng can paint a classroom in 4 days. Kiprop can paint it in 6 days. Together they do 1/4 + 1/6 = 3/12 + 2/12 = 5/12 each day. Time = 12/5 = 2.4 days.
Worked example: filling and emptying
Tap A fills a tank in 3 hours. An outlet empties it in 6 hours. With both open, the tank gains 1/3 − 1/6 = 1/6 per hour, so it fills in 6 hours. Subtract the rate of anything working against the job.
Worked example: part of the job done
A can do a job in 10 days and B in 15 days. They work together for 3 days, then A leaves. Together they do 1/10 + 1/15 = 1/6 per day, so in 3 days they finish 3/6 = 1/2. B finishes the remaining 1/2 at 1/15 per day: (1/2) ÷ (1/15) = 7.5 days.
Original Practice Questions With Worked Answers
- Share 450 kg of fertiliser in the ratio 1 : 2 : 6. 9 parts; 50 kg per part: 50 kg, 100 kg, 300 kg.
- 8 workers build a wall in 9 days. How long would 12 workers take? 9 × 8/12 = 6 days.
- Pipe P fills a tank in 5 hours and pipe Q in 20 hours. How long together? 1/5 + 1/20 = 1/4, so 4 hours.
- If A : B = 3 : 4 and B : C = 2 : 5, find A : B : C. B : C = 4 : 10, so A : B : C = 3 : 4 : 10.
- 4 pumps working 6 hours a day drain a pond in 5 days. How many days for 3 pumps working 8 hours a day? 5 × 4/3 × 6/8 = 5 days.
Cover the answers and attempt each question first. For timed practice, use the Grade 9 revision papers or the Grade 9 CBC Challenge. These notes are free; full revision access is available with the KSh 20 Day Pass or the KSh 500 Monthly Pass.
Frequently Asked Questions
What is a compound proportion?
A problem where the answer depends on two or more quantities that change at the same time, such as the number of workers and the number of hours.
How do you calculate rates of work?
Write each worker’s rate as the fraction of the job done per unit of time, add the rates for people working together, and divide 1 by the total rate to get the time.
How do you share an amount in a ratio?
Add the parts of the ratio, divide the amount by that total to get one part, then multiply by each part of the ratio.
How do I know if a proportion is direct or inverse?
Ask whether increasing one quantity increases the answer (direct) or decreases it (inverse). More workers usually means fewer days, so that is inverse.