1. The Work-as-Rate Model
The entire chapter starts from one convention: treat the whole job as 1 unit of work, and describe every worker by how much of that 1 unit they complete per day.
If A can finish a job in 10 days:
A's rate = 1/10 of the job per day
If B can finish the same job in 15 days:
B's rate = 1/15 of the job per day
Working TOGETHER, rates simply ADD:
Combined rate = 1/10 + 1/15 = 3/30 + 2/30 = 5/30 = 1/6
Combined time = 1 / (combined rate) = 1 / (1/6) = 6 daysEvery quantity in this chapter — "days to finish alone," "days to finish together," "fraction of work done" — is just this rate model read in a different direction. Once a problem is translated into rates, the arithmetic is almost always simple fraction addition.
2. Combined Work & Efficiency Ratios
"A is twice as efficient as B" is a ratio statement about rates, not times — and converting it correctly (rather than accidentally inverting it) is the main place students slip.
"A is twice as efficient as B" means:
A's RATE = 2 x B's RATE
A's TIME = (1/2) x B's TIME -- twice as fast means HALF
the time, not double
Worked: A is twice as efficient as B, and together they
finish a job in 6 days. Find how long B alone would take.
Let B's rate = x, then A's rate = 2x
Combined rate = 2x + x = 3x
3x = 1/6 (job per day)
x = 1/18
B's rate = 1/18 -> B alone takes 18 days
A's rate = 2/18 = 1/9 -> A alone takes 9 days
Check: 1/9 + 1/18 = 2/18+1/18 = 3/18 = 1/6 -- matchesA common error is treating "A is twice as efficient" as "A takes twice as long" — exactly backward. Efficiency and rate move together (higher efficiency = higher rate = LESS time), which is the opposite relationship from efficiency and time. When in doubt, work entirely in rates (as above) and only convert to time as the very last step.
3. Wage-Sharing
When multiple workers share a job and get paid a total wage, the fair split is proportional to each worker's contribution — which is their rate, not the raw time they were involved for.
A can do a job in 6 days, B can do it in 8 days. They work
together and are paid Rs. 700 total. Find each share.
A's rate = 1/6, B's rate = 1/8
Ratio of contribution = 1/6 : 1/8 = 4:3 (multiply both by 24)
A's share = 700 x 4/7 = 400
B's share = 700 x 3/7 = 300This is Week 6's single-multiplier ratio technique yet again, applied to work rates instead of raw quantities — the ratio of contributions (not the ratio of individual completion times) determines how the payment splits, since payment should track how much of the actual job each person did.
4. Pipes & Cisterns
Identical to Sections 1-2's rate model, with one addition: a pipe that empties the tank contributes a negative rate, working against the filling pipes.
Pipe A fills a tank in 12 hours.
Pipe B fills the same tank in 15 hours.
Pipe C (a leak/outlet) EMPTIES the full tank in 20 hours.
All three are opened together. Find the time to fill the tank.
Combined rate = 1/12 + 1/15 - 1/20 (C is SUBTRACTED)
= 5/60 + 4/60 - 3/60
= 6/60
= 1/10
Time to fill = 1 / (1/10) = 10 hoursThe sign is the only new idea in this section — everything else (rates adding, converting between rate and time) is identical to Sections 1-3. A useful sanity check: if the combined rate comes out negative or zero, the tank never fills (or never empties) at all, which is worth stating as the answer rather than reporting a nonsensical negative time.
5. Practice Problems
30 problems, no calculator — Easy, Medium & Tough
Convert every quantity to a rate before doing any arithmetic. Easy problems should take under 30 seconds each; Medium under 60 seconds; Tough under 90 seconds.
Easy (1–10)
- A can do a job in 10 days. Find A's one-day work.
- If A's one-day work is 1/8, find the number of days A takes.
- A can finish a job in 6 days, B in 12 days. Find their combined one-day work.
- A and B together can do a job in 4 days. Find their combined rate.
- A can do a job in 15 days. How much work does A do in 5 days?
- A pipe fills a tank in 20 hours. Find its rate per hour.
- A can complete a task in 12 days. If A's efficiency doubles, find the new time.
- B can do a job in 9 days. Find B's work done in 3 days.
- A and B can each do a job in 8 days alone. Find the time taken together.
- A pipe empties a tank in 10 hours. Find its rate per hour.
1) 1/10. 2) 8 days. 3) 1/6+1/12=1/4. 4) 1/4. 5) 5/15=1/3. 6) 1/20. 7) Double efficiency = half time = 6 days. 8) 3/9=1/3. 9) 1/8+1/8=1/4 → 4 days. 10) 1/10.
Medium (11–20)
- A can do a job in 12 days, B in 18 days. Find the time taken together.
- A is twice as efficient as B. If B takes 20 days to finish a job, find A's time.
- A, B and C can do a job in 10, 15 and 30 days respectively. Find the time taken together.
- Pipe A fills a tank in 15 hours, pipe B empties it in 20 hours. Find the time to fill the tank if both are open.
- A can do a job in 20 days. He works for 4 days and then leaves. Find the fraction of work completed.
- A and B can do a job in 8 days together. A alone can do it in 12 days. Find B's time alone.
- Two pipes can fill a tank in 10 and 15 hours respectively. Find the time to fill the tank if both are opened together.
- A does a job in 6 days and B does the same job in 3 days. Find the ratio of their efficiencies.
- 6 men can complete a job in 8 days. Find the number of days for 12 men.
- A can do 1/3 of a job in 4 days. Find the time to complete the whole job.
11) 1/12+1/18=5/36 → 36/5=7.2 days. 12) A takes half of 20=10 days. 13) 1/10+1/15+1/30=1/5 → 5 days. 14) 1/15-1/20=1/60 → 60 hours. 15) 4/20=1/5. 16) 1/8-1/12=1/24 → 24 days. 17) 1/10+1/15=1/6 → 6 hours. 18) A rate=1/6, B rate=1/3 → ratio 1:2. 19) 6×8=48=12×d → d=4 days. 20) 4×3=12 days.
Tough (21–30)
- A can do a job in 18 days and B in 24 days. They work together for 4 days, then A leaves. Find the time B takes to finish the remaining work.
- A, B and C together can complete a job in 6 days. A alone can do it in 12 days, B alone in 18 days. Find C's time alone.
- 12 men can complete a job in 15 days. After 5 days, 4 more men join. Find the total time to complete the job.
- A tap can fill a tank in 12 hours. Due to a leak, it takes 16 hours to fill. Find the time the leak alone would take to empty the full tank.
- A can do a job in 20 days and B is 25% more efficient than A. Find B's time alone.
- A and B can complete a job together in 12 days. They worked together for 3 days, after which B left. A completed the remaining work in 18 days. Find A's time alone.
- Two pipes A and B can fill a tank in 20 and 30 minutes respectively. Both are opened together, but pipe A is closed after 10 minutes. Find the total time to fill the tank.
- 15 men can complete a job in 20 days working 8 hours a day. Find the number of days for 20 men working 6 hours a day.
- A does half as much work as B in three-fourths of the time B takes. If together they take 18 days to complete the work, find A's time alone.
- A cistern has two inlet pipes that fill it in 10 and 15 hours respectively, and one outlet pipe that empties it in 6 hours. Find the time to fill the cistern if all three are opened together.
21) Combined rate=1/18+1/24=7/72; 4 days=7/18 done; remaining=11/18; B's time=(11/18)÷(1/24)=14⅔ days. 22) Combined=1/6; A+B=1/12+1/18=5/36; C=1/6-5/36=1/36 → 36 days. 23) 12×15=180 man-days total; 5 days×12=60 done, 120 remain; 16 men → 120/16=7.5 days; total=12.5 days. 24) Leak rate=1/12-1/16=1/48 → 48 hours. 25) B's rate=1.25×1/20=1/16 → 16 days. 26) 3 days=1/4 done; remaining=3/4; A's rate=(3/4)/18=1/24 → A's time=24 days. 27) 10 min both: 10×(1/20+1/30)=5/6 done; remaining=1/6; B alone: (1/6)/(1/30)=5 min; total=15 minutes. 28) Man-hours=15×20×8=2400; for 20 men, 6hrs/day: 2400/120=20 days. 29) Let B's time=t; A's rate=2/(3t); combined=1/t+2/(3t)=5/(3t)=1/18 → t=30 (B's time); A's rate=2/90=1/45 → A's time=45 days. 30) Inlet rate=1/10+1/15=1/6, exactly equal to outlet rate 1/6 → net rate=0, the cistern never fills.
6. Knowledge Check
Four quick questions. Expand each to check your answer.
Q1
Why do individual work rates simply ADD when two people work together, but their TIMES don't simply combine the same way?
Why do individual work rates simply ADD when two people work together, but their TIMES don't simply combine the same way?
Rates measure "fraction of the job completed per unit time," and two people working simultaneously genuinely complete their fractions in parallel, so their contributions per day literally sum. Time is the inverse of a rate, and the inverse of a sum isn't the sum of the inverses — which is exactly why times can't be added directly the way rates can.
Q2
If A is three times as efficient as B, why does A take one-third the time, rather than three times the time?
If A is three times as efficient as B, why does A take one-third the time, rather than three times the time?
Efficiency describes rate — doing work faster — and rate and time are inversely related. A worker with three times the rate completes the same fixed amount of work in one-third the time, since time = work / rate, and tripling the denominator divides the result by three.
Q3
Why is a joint wage split based on each worker's rate ratio, rather than splitting the payment equally?
Why is a joint wage split based on each worker's rate ratio, rather than splitting the payment equally?
A fair wage split should reflect how much of the actual job each person completed, and each worker's rate directly measures exactly that — a faster worker (higher rate) contributed a larger fraction of the total finished work in the same time period, and should be compensated proportionally more than an equal split would give them.
Q4
Why does an emptying pipe's rate get SUBTRACTED from the combined rate, rather than added like the filling pipes?
Why does an emptying pipe's rate get SUBTRACTED from the combined rate, rather than added like the filling pipes?
The emptying pipe is undoing progress toward the goal (a full tank) rather than contributing toward it, the opposite direction of the filling pipes' effect. Representing it as a negative rate correctly captures that it works against the other pipes, reducing the net combined rate rather than increasing it.