Pipes and Cisterns Model 4: Faster, Slower and Efficiency Ratios

Pipes and Cisterns · 25 min

Aptitude · Pipes and Cisterns · Model 4

Thrice as fast, and still forty hours alone

Every “twice as fast”, “60% more efficient” and “takes 40 minutes less” question is the same two lines. Split the work into parts per hour, then divide the tank by each pipe’s share — and watch the times come out in the reverse ratio.

Split the parts and read off the times →
Words like “thrice as fast” describe rate, never time. Rates add; times invert. Own time = joint time × total parts / own parts.

01 The idea

Efficiency adds up, and time does not

“Tap A is thrice as fast as tap B” is a statement about how much water moves per hour. It is not a statement about hours. Write it as a ratio of rates — A : B = 3 : 1 — and you have converted the only difficult part of the question, the English, into arithmetic.

Once the rates are in parts, everything else is division. Two pipes at 3 and 1 parts an hour deliver 4 parts an hour together. If that pair fills the tank in 30 hours, the tank is 4 × 30 = 120 parts. A moves 3 of them an hour, so A alone needs 120/3 = 40 hours; B moves 1, so B alone needs 120 hours.

Look at what happened to the times. The rates were 3 : 1 and the times came out 40 : 120, which is 1 : 3 — the ratio turned over. That inversion is the entire content of this model, and it is why you can never add or average times: 30 × 3 = 90 and 30 ÷ 3 = 10 are both wrong, and A’s real 40 hours sits between them.

Percentages are the same idea wearing a disguise. “B is 60% more efficient than A” means A : B = 100 : 160, which is 5 : 8. “The capacity of B is 80% more than A” means 5 : 9. Convert to a clean ratio before you do anything else, and never let a percentage stay a percentage past the first line of working.

One ratio, one sum, two divisions. The pipe with more parts always ends up with the smaller time — if yours does not, you have inverted the ratio.
EfficiencyWork done per unit time — the rate. This is what “faster”, “more efficient” and “greater capacity” all measure, and it is the only quantity in the chapter that adds when pipes run together.
PartsThe whole numbers you write a ratio in. A : B = 3 : 1 means A does 3 parts an hour and B does 1, so the pair does 4. The tank is then measured in those same parts.
Reverse ratio of timesIf the rates are a : b, the individual times are b : a. It follows from time = work / rate with the work held fixed, and it is the check that catches nearly every wrong answer here.

02 Worked example

Thrice as fast, thirty hours together

One pair runs the whole lesson. Tap A is thrice as fast as tap B. Opened together they fill the cistern in 30 hours. How long does A alone take, and how long does B?

1
Write the words as parts per hour“Thrice as fast” is about rate, so A gets three parts for B’s one.A : B = 3 : 1 parts of work per hour
2
Together the parts addBoth open at once, so the rates go into one sum. This is the only sum in the question.3 + 1 = 4 parts per hour
3
Size the tank from the joint time4 parts an hour for 30 hours empties the whole job, so that product is the tank. Measuring the tank in parts keeps both later divisions whole.tank = 4 × 30 = 120 parts
4
A aloneA moves 3 of the 120 parts every hour.120 / 3 = 40 hours
5
B alone, then checkB moves 1 part an hour. Adding the two one-hour works must give the joint rate back.120 / 1 = 120 hours  ·  1/40 + 1/120 = 3/120 + 1/120 = 1/30 ✓

The rates were 3 : 1 and the times came out 40 : 120 = 1 : 3, exactly reversed. Notice what the two tempting shortcuts would have given: 30 × 3 = 90 hours and 30 ÷ 3 = 10 hours, neither of them near 40. The reason is that 30 hours is the pair’s time, and scaling a pair’s time by one member’s ratio has no meaning. Scale the parts instead: A owns 3 of 4, so A needs 4/3 of 30.

03 The method

One formula, and the version for a stated gap

Most of these questions hand you the joint time. A large minority hand you the difference between the two individual times instead, which needs one extra line and no new idea.

Joint time given: own time = joint time × total parts / own parts. So at 3 : 1 and 30 hours, A = 30 × 4/3 = 40 and B = 30 × 4/1 = 120.
Gap given instead: the times are in the reverse ratio, so at 3 : 1 the times are 1 : 3 — a gap of 2 time-parts. “A is thrice as fast as B and takes 40 hours less” makes 2 parts = 40, so one part is 20 and the times are 20 and 60 hours, with a joint time of 15. Watch the wording too: exams write “3 times faster” when they mean “3 times as fast” and solve it as 3 : 1. Read literally it would be 4 : 1. Write the ratio down explicitly, and solve the ratio you wrote.
What the question saysRatio A : B of ratesWhat it does NOT mean
A is thrice as fast as B3 : 1A’s time is 3 × B’s
B is twice as fast as A1 : 2B’s time is 2 × A’s
B is 60% more efficient than A100 : 160 = 5 : 8add 60% to a time
Capacity of B is 80% more than A100 : 180 = 5 : 9B is 80% quicker
C twice B, B twice A1 : 2 : 41 : 2 : 3
A is 3 times faster than B3 : 1 by convention4 : 1 if read strictly
Average the two individual timesnever valid40 and 120 give 30, not 80

05 Cheat sheet

Efficiency questions on one page

Every row is one line of the method or one way the wording hides it. The example column is the lesson pair — A : B = 3 : 1 with a joint time of 30 hours.

StepRuleOn A : B = 3 : 1, together 30 h
Read the wordsratio of rates, not timesA : B = 3 : 1
Joint rateadd the parts3 + 1 = 4 parts / hour
Tanktotal parts × joint time4 × 30 = 120 parts
Own timetank / own partsA 40 h, B 120 h
Times ratioreverse of the rates40 : 120 = 1 : 3
Gap given insteadgap / (difference in parts)40 h gap → 20 h and 60 h
Scaling the joint timenever by one pipe’s rationot 90 h, not 10 h
Three pipes work the same way“C is twice as fast as B, and B twice as fast as A” is 1 : 2 : 4, not 1 : 2 : 3. Seven parts filling in 5 hours makes the tank 35 parts, so A alone takes 35 hours and C alone takes 35/4 = 8¾.
A gap without a ratio needs a quadratic“Together 12 hours, and one pipe is 10 hours faster than the other” gives you no ratio at all. Set 1/x + 1/(x−10) = 1/12, which becomes x² − 34x + 120 = 0 and factors as (x−30)(x−4). Take 30, because x = 4 would make the other pipe −6 hours.
The same method with a minus signIf an outlet is described by efficiency — “the inlet is twice as efficient as the outlet” — nothing changes except that the outlet’s parts are subtracted. An inlet at 2 parts filling alone in 20 hours makes the tank 40 parts, and a net of 2 − 1 = 1 part gives 40 hours with both open.

06 Where & why

Where this shows up

This is the most heavily set model in the module, because the arithmetic is trivial and the reading is not. Almost every mark lost here is lost in the first line.

TCS NQT · Wipro
“A is thrice as fast as B”

One ratio and one division. The whole question exists to check whether you attach the “thrice” to the rate or to the time.

SSC CGL · RRB
“60% more efficient”

The percentage becomes 100 : 160 = 5 : 8 and the rest is the standard division. Candidates who add 60% to a time get a number that is always in the option list.

Bank PO · IBPS
“Takes 40 minutes less than”

A stated gap instead of a joint time. Convert to time-parts — the gap in parts is the difference of the rate parts — and the extra line costs five seconds.

Capacity planning
Two workers, two machines, two servers

Any “this one is 40% faster” statement about throughput is this model. The reverse ratio of times is why doubling a machine’s speed halves its own runtime but does far less to the pair’s.

The transferable habit is small and worth a lot: when a sentence compares two things, decide first which quantity it compares. Rate, time and work all get described with the word “faster”, and only one of them adds.

07 Interview questions

What gets asked

Ten, from the wording of the first line to the quadratic that turns up when no ratio is given at all.

Does “thrice as fast” describe the rate or the time?
The rate. A is thrice as fast as B means A does three times as much work per hour, so A : B = 3 : 1 in efficiency. The times then come out in the reverse ratio 1 : 3, which is a consequence, not the starting point.
Tap A is thrice as fast as tap B and together they fill the cistern in 30 hours. How long for A alone?
40 hours, and B alone takes 120. The parts are 3 and 1, so the pair does 4 an hour and the tank is 4 × 30 = 120 parts. A moves 3 of them, so 120/3 = 40. Check: 1/40 + 1/120 = 1/30.
Why is scaling the joint time by the ratio wrong?
Because 30 hours belongs to the pair, not to either pipe. Multiplying it by 3 gives 90 and dividing by 3 gives 10, and A’s real answer of 40 hours is between them. What you scale is the share of the parts: A owns 3 of the 4, so A needs 4/3 of the pair’s time.
How do you convert “B is 60% more efficient than A”?
To A : B = 100 : 160, which reduces to 5 : 8. If they fill the tank in 20 minutes together, the tank is 13 × 20 = 260 parts and A alone needs 260/5 = 52 minutes. The percentage never touches a time — it only builds the ratio.
A is thrice as fast as B and takes 40 minutes less. When do they finish together?
15 minutes. The rates are 3 : 1 so the times are 1 : 3, a gap of 2 time-parts. Two parts equal 40 minutes, so a part is 20 and the times are 20 and 60. Together: 20 × 60 / 80 = 15 minutes.
C is twice as fast as B and B twice as fast as A, and together they fill it in 5 hours. A alone?
35 hours. The chain gives A : B : C = 1 : 2 : 4, so seven parts an hour clear the tank in 5 hours and the tank is 35 parts. A moves one part, so 35 hours. The common slip is writing 1 : 2 : 3 — doubling twice is four, not three.
Is “3 times faster” the same as “3 times as fast”?
Strictly no — three times faster than B would be B plus three more B, or 4 : 1. In practice aptitude papers use the two phrases interchangeably and expect 3 : 1, which is what the standard keys assume. Write the ratio down explicitly as your first line so the reading you chose is visible, then solve that.
Two pipes fill a reservoir in 12 hours together and one is 10 hours faster than the other. How long for the slower one?
30 hours. There is no ratio here, so you need the equation: 1/x + 1/(x−10) = 1/12 gives x^2 − 34x + 120 = 0, which factors as (x−30)(x−4). Only x = 30 survives — x = 4 would make the faster pipe −6 hours.
When would you use parts rather than the LCM-of-times method?
Use parts when the question gives you a ratio and one real number, which is what this model always does — you have no individual times to take an LCM of. Use LCM-of-times when the question hands you each pipe’s own time. They are the same idea; the only difference is which end you are given.
When would you actually use this?
Whenever throughput is described in relative terms, which is most of the time in real work: a machine 40% faster than another, a server twice the capacity, a colleague who codes at half your pace. The useful takeaway is the reverse ratio — making one worker twice as quick halves that worker’s own time but moves the joint time much less.

08 Practice problems

Six on parts and reverse ratios

Write the ratio of rates as your first line every single time, even when the question already looks like a ratio. Every wrong answer in this model starts before the arithmetic does.

One pipe’s own time is given

Easy
Tap B is twice as fast as tap A. Tap B alone fills the tank in 15 minutes. How long do the two take together?
Follow-up
You are given an individual time rather than the joint one, so the tank has to be sized from B instead. The ratio still does the work; only the anchor moved.
Show the hint
B : A = 2 : 1, and B’s 2 parts clear the tank in 15 minutes.

A percentage in the first line

Easy
The efficiency of pipe B is 60% more than that of pipe A. Opened together they fill the tank in 20 minutes. How long would pipe A alone take?
Follow-up
The percentage has to become a clean ratio before anything else happens, and it is a ratio of rates — adding 60% to a time gives an answer that will be sitting in the options.
Show the hint
100 : 160 reduces to 5 : 8, so the pair does 13 parts a minute.

A chain of comparisons

Medium
A tank is filled in 5 hours by three pipes A, B and C running together. Pipe C is twice as fast as pipe B, and pipe B is twice as fast as pipe A. How long would pipe A alone take?
Follow-up
Two comparisons stacked, and the second one is relative to the first. Doubling twice is not the same as adding twice, which is exactly what the wrong answer assumes.
Show the hint
Build A : B : C from the slowest end and add the parts before you touch the 5 hours.

A gap instead of a joint time

Medium
One fill pipe A is three times as fast as a second fill pipe B, and A takes 32 minutes less than B to fill the tank. When will the cistern be full if both pipes are opened together?
Follow-up
Nothing here is the joint time, so the tank cannot be sized directly. The gap has to be converted into time-parts first, and the number of time-parts in the gap is not 3.
Show the hint
Rates 3 : 1 means times 1 : 3, so the gap is 2 time-parts.

An outlet described by efficiency

Medium
An inlet pipe is twice as efficient as an outlet pipe fitted to the same tank. The inlet alone fills the empty tank in 20 minutes. Both are opened on an empty tank — how long until it is full?
Follow-up
The outlet’s parts carry a minus, so the joint rate is a difference rather than a sum. The answer is not 20 minutes and not 40 minutes for the reason you might first guess — work out the net parts before dividing.
Show the hint
Inlet 2 parts, outlet 1 part, and the inlet’s own 20 minutes fixes the size of the tank.

No ratio at all

Hard
If two pipes work at the same time, a reservoir is filled in 12 hours. One pipe fills the reservoir 10 hours faster than the other. How many hours does the slower pipe take on its own?
Follow-up
There is no efficiency ratio to write down, so the parts method has nothing to stand on and you have to build an equation. It produces two roots and only one of them describes a real pipe — saying why the other is rejected is part of the answer.
Show the hint
Let the slower pipe take x hours and write 1/x + 1/(x − 10) = 1/12.