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JUNIOR SEC โ†’ GRADE 9 โ†’ Mathematics โ†’ COMPOUND PROPORTIONS AND RATES OF WORK

COMPOUND PROPORTIONS AND RATES OF WORK Notes, Quizzes & Revision

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Mathematics โ€” Numbers
Subtopic: Compound Proportions and Rates of Work
(Age ~14, Kenyan context)

Learning goals:

  • Understand what compound proportion means.
  • Solve problems where several ratios affect a quantity.
  • Use rates of work (workers and time) to find how long tasks take together.

1. What is Compound Proportion?

Compound proportion is when more than one proportional change affects a quantity. Each factor may be directly proportional (increase leads to increase) or inversely proportional (increase leads to decrease).

Rule (simple):
If quantity Q changes by several factors, apply each change one after another:
Q_new = Q ร— (factor1) ร— (factor2) ร— ...
Use factor = new/old. For inverse proportion (e.g., time โˆ 1/workers), use factor = old/new.

Example of compound proportion (items & machines)

"3 machines make 120 bricks in 4 hours. How many machines are needed to make 200 bricks in 5 hours?"

Step-by-step:

  1. Find rate per machine per hour:
    3 machines โ†’ 120 bricks in 4 h โ‡’ 1 machine in 4 h makes 120 รท 3 = 40 bricks โ‡’ 1 machine in 1 h makes 40 รท 4 = 10 bricks
  2. Say x machines are needed for 200 bricks in 5 h. Total bricks = (machines ร— rate per machine per hour ร— hours):
  3. So: x ร— 10 ร— 5 = 200 โ‡’ x = 200 รท 50 = 4 machines.

2. Direct and Inverse Proportion in Compound Problems

- Direct proportion: if A doubles, B doubles (same direction). Use factor = new/old.
- Inverse proportion: if A doubles, B halves (opposite direction). Use factor = old/new.

Visual: if Q_old = 100 and two changes happen: one direct ร—1.5 and one inverse ร—(1/2), then Q_new = 100 ร— 1.5 ร— 0.5 = 75.

3. Rates of Work (Workers and Time)

Basic idea: Work = Rate ร— Time. If a worker does a job in T days, rate = 1/T job per day. When workers work together, rates add.

Important formula (two workers):
If A does job in a days and B in b days, together time t = 1 / (1/a + 1/b) = ab / (a + b)

Example 1 โ€” Two workers

"James can paint a fence in 6 hours. Wanjiru can paint the same fence in 4 hours. How long if they work together?"

Solution:

  1. James' rate = 1/6 fence per hour. Wanjiru's rate = 1/4 fence per hour.
  2. Combined rate = 1/6 + 1/4 = (2 + 3)/12 = 5/12 fence per hour.
  3. Time = 1 รท (5/12) = 12/5 = 2.4 hours = 2 hours 24 minutes.

Example 2 โ€” Different start times

"A can finish a job in 8 days. B can finish it in 12 days. A starts and works alone for 3 days, then B joins. How many more days until finish?"

Solution:

  1. A's daily rate = 1/8. In 3 days A does 3 ร— 1/8 = 3/8 of the job.
  2. Remaining = 1 โˆ’ 3/8 = 5/8.
  3. Together rate = 1/8 + 1/12 = (3 + 2)/24 = 5/24 per day.
  4. Time to finish = (5/8) รท (5/24) = (5/8) ร— (24/5) = 3 days.
  5. So total = 3 + 3 = 6 days.

4. More Compound Proportion Examples

Example 3 โ€” Multiple factors

"A contractor finds that with 4 masons, a wall is built in 15 days using 10 sacks of cement per 5 mยฒ. If he wants to build a wall twice as big in 10 days, how many masons and how many sacks are needed?"

Approach:

  1. Area doubles โ†’ twice the work (direct ร—2).
  2. Time reduces from 15 to 10 days โ†’ work per day must increase (inverse ร—15/10 = ร—3/2).
  3. So overall factor to daily workforce = 2 ร— 15/10 = 3.
  4. Needed masons = 4 ร— 3 = 12 masons.
  5. Cement: original used 10 sacks for 5 mยฒ. Twice area โ‡’ ร—2 โ‡’ 20 sacks. (Assuming cement needed proportional to area.)

5. Quick Methods and Tips

  • Turn each proportional change into a simple multiplying factor.
  • For direct: factor = new/old. For inverse: factor = old/new.
  • Multiply all factors to get net change. Apply net change to original quantity.
  • For work problems, convert times to rates (1/time) before adding.

6. Practice Questions (Try these)

  1. Five workers build a hut in 12 days. How many workers are needed to build the same hut in 8 days?
  2. A machine fills 500 bottles in 4 hours. How many hours will 3 such machines take to fill 1500 bottles?
  3. Jane alone takes 10 days to do a job; Peter takes 15 days. If they start together but Jane leaves after 4 days, how many more days will Peter need?
  4. 2 pumps can empty a tank in 6 hours. How long will 5 pumps take (same pump rate)?

7. Answers (check your work)

  1. Workers โˆ 1/time (inverse). Factor = 12/8 = 3/2 โ†’ workers = 5 ร— 3/2 = 7.5 โ‡’ 8 workers (must be whole). If fractional allowed, 7.5.
  2. One machine rate = 500/4 = 125 bottles/hour. Three machines rate = 375 bottles/hour. Time = 1500 รท 375 = 4 hours.
  3. Combined rate = 1/10 + 1/15 = (3+2)/30 = 1/6 job/day. Jane + Peter do 4 ร— 1/6 = 2/3 job. Remaining = 1/3. Peter alone rate = 1/15 โ†’ time = (1/3) รท (1/15) = 5 days.
  4. 2 pumps โ†’ 6 h, so 1 pump โ†’ 12 h (inverse). 5 pumps โ†’ 12 รท 5 = 2.4 h = 2 h 24 min.

Final tip: practice making a small table of factors (direct and inverse) for each change. This makes compound proportion problems much easier.

Note: These notes use simple methods suitable for Kenyan secondary school learners (around Form 2/3 level). For exam-style practice, try varying numbers and mixing direct/inverse factors.
๐Ÿ“ Practice Quiz

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