Aptitude · Time and Work · Model 6

The ratio splits the daily output, never the time

“A is three times as efficient as B.” That fixes their rates in the ratio 3 : 1 — and therefore their times in the ratio 1 : 3, the other way round. Applying the ratio to the wrong quantity is the one error this model exists to catch.

Set the together time and the efficiency ratio →

Efficiency ratio 3 : 1 means time ratio 1 : 3. Efficiencies and times are always inverted, because one is work per day and the other is days per work.

01 The idea

Rates split, times invert

A and B together finish a job in 9 days, and A does three times as much as B in a given time. How long does A take alone? The ratio 3 : 1 is a ratio of efficiencies, so it splits what they produce each day — not the days themselves.

Take the job as 36 units so the pair does 36/9 = 4 units a day. Split those 4 units in the ratio 3 : 1 and A supplies 3 a day while B supplies 1. So A alone needs 36/3 = 12 days and B needs 36/1 = 36 days.

Notice the times came out 12 and 36, which is a ratio of 1 : 3 — the inverse of the efficiency ratio. That inversion is not a coincidence and it is the whole content of the model: efficiency is work per day, time is days per work, so they are reciprocals.

Percentage phrasings are the same idea with an extra step. “A is 30% more efficient than B” means the rates are in the ratio 130 : 100, which simplifies to 13 : 10. So the times are in the ratio 10 : 13. Convert the percentage to a clean ratio first and the question becomes the one you already know how to do.

Convert every efficiency statement into a clean ratio, split the daily output by it, and remember the times come out inverted.

EfficiencyWork done per day. A ratio of efficiencies splits the combined daily output, which is why it applies to rates and not to times.

Inverse relationtime ∝ 1/efficiency. An efficiency ratio of a : b gives a time ratio of b : a.

Percentage more efficient“p% more efficient than B” means the rates are (100 + p) : 100. So 30% more gives 13 : 10, and 25% more gives 5 : 4.

02 Worked example

Together in 9 days, A three times B

The source question. A and B together can do a piece of work in 9 days. If A does thrice as much work as B in a given time, how long does A alone take to do the work?

Read the ratio as efficienciesThree times as much work in the same time means three times the rate. So the efficiencies are 3 : 1, giving 4 parts between them.A : B = 3 : 1 ⇒ 4 parts per day

Size the job from the together timeThey supply 4 parts a day and finish in 9 days, so the job is 36 parts.W = 9 × 4 = 36 parts

Give A three of the four partsA supplies three parts of every four produced.A = 3 parts/day

Divide the job by A’s rateThirty-six parts at three a day.36 / 3 = 12 days for A alone

Check B, and check the inversionB supplies one part a day, so B needs 36 days — and the times are in the ratio 1 : 3.B = 36/1 = 36 days ⇒ times 12 : 36 = 1 : 3 ✓

The inversion check in the last step is worth making a habit. You were given efficiencies 3 : 1; if your two times do not come out in the ratio 1 : 3, you have applied the ratio to the wrong quantity. Note also the shortcut this reveals: A alone takes the together time multiplied by (total parts / A’s parts) — here 9 × 4/3 = 12 days, in one line.

03 The method

The phrasings, and the one that is a trap

Every version of this model is a phrase to be converted into a ratio. One of the phrases is set specifically to be misread.

time = W / efficiency, so an efficiency ratio a : b gives a time ratio b : a. From a together time t, one worker’s solo time is t × (total parts) / (their parts).

The trap: “three times as much” and “three times more” are read differently. The first means a ratio of 3 : 1. The second, read strictly, means three times more than B, so 4 : 1. Indian competitive exams almost always intend 3 : 1 for both, but a careful question will say “as much as” when it means 3 : 1. If a question is ambiguous, take the ratio that makes the arithmetic come out clean — setters choose numbers that work.

The question says	Efficiency ratio	Time ratio
A is twice as efficient as B	`2 : 1`	`1 : 2`
A does thrice the work of B	`3 : 1`	`1 : 3`
A is 30% more efficient	`13 : 10`	`10 : 13`
A is 25% more efficient	`5 : 4`	`4 : 5`
A is 20% less efficient	`4 : 5`	`5 : 4`
B is 50% as efficient as A	`2 : 1`	`1 : 2`
A is three times MORE than B	`4 : 1 strictly`	usually intended as 3 : 1

05 Cheat sheet

Model 6 on one page

The conversions, the shortcut, and the two ways the ratio gets misapplied.

Case	Route	On together 9, ratio 3 : 1
Split the output	`parts of the daily total`	4 parts/day, A gets 3
Size the job	`together time × total parts`	9 × 4 = 36
One solo time	`t × total/their parts`	9 × 4/3 = 12 days
Time ratio	`inverse of the efficiency ratio`	1 : 3
p% more efficient	`(100+p) : 100`	30% → 13 : 10
Splitting the TIME by the ratio	`wrong`	gives 6.75, not 12
'Times more' read strictly	`4 : 1`	usually intended as 3 : 1

Times come out invertedEfficiency 3 : 1 gives times 1 : 3. If your two answers are not in the inverse ratio, you applied the ratio to the wrong quantity.

Convert percentages to clean ratios first30% more efficient is 13 : 10, not 1.3 to be used as a decimal. Clean ratios keep every later number whole.

One-line shortcutA solo time is the together time times total parts over that worker's parts. Here 9 × 4/3 = 12 days with no job size needed.

06 Where & why

Where Model 6 shows up

Efficiency phrasing is how a paper avoids giving you the times directly, and it appears everywhere in this chapter and the next.

Bank PO · SSC CGL

“A is twice as efficient as B”

The standard form, usually with a together time given. The one-line shortcut answers it immediately.

TCS NQT · Infosys

Percentage efficiency

“30% more efficient” and “20% less efficient”. Converting to 13 : 10 and 4 : 5 is the whole difficulty.

Pipes and cisterns

“A is thrice as fast as B”

One of the four question sets there is exactly this model with taps. The conversion and inversion transfer unchanged.

Illness and slowdown variants

“A worked at 60% efficiency for 3 days”

Write the actual rate for the actual days rather than trying to adjust an average. It is a stage question with a modified rate.

The single discipline for this model: write the efficiency ratio down as two whole numbers before anything else, then check at the end that your times came out inverted. Those two habits cover every question in it.

07 Interview questions

What gets asked

Ten, and the first two are the model in its entirety.

A and B together take 9 days and A does three times B’s work. How long does A take alone?

Twelve days. The efficiencies are 3 : 1, so 4 parts a day between them and the job is 9 × 4 = 36 parts. A supplies 3 parts a day, so A alone needs 36/3 = 12 days. B needs 36 days.

If the efficiency ratio is 3 : 1, what is the time ratio?

1 : 3 — the inverse. Efficiency is work per day and time is days per work, so they are reciprocals. Here the times are 12 and 36 days, which is 1 : 3. Checking this inversion at the end catches the model’s main error.

What goes wrong if you split the time by the efficiency ratio?

You get the wrong answer without any warning. Splitting 9 days as 3 : 1 gives 6.75 days for A, which is less than the pair’s combined time and therefore impossible. The ratio applies to the daily output, not the duration.

How do you convert “30% more efficient” into a ratio?

As 130 : 100, which simplifies to 13 : 10. Similarly 25% more is 5 : 4 and 20% less is 4 : 5. Converting to clean whole numbers first keeps every subsequent figure whole, which is why it is worth doing before anything else.

A is 30% more efficient than B and together they take 10 days. How long does B take alone?

Twenty-three days. The ratio is 13 : 10, giving 23 parts a day, so the job is 230 parts. B supplies 10 a day, so B needs 23 days. The pleasingly clean answer is why setters like 30%.

Give the one-line shortcut for a solo time.

Solo time = together time × (total parts) / (that worker’s parts). For 9 days with a 3 : 1 ratio, A takes 9 × 4/3 = 12 days. No job size is needed, which makes it the fastest route in this model.

What is the difference between “three times as much as” and “three times more than”?

Strictly, the first means a ratio of 3 : 1 and the second means 4 : 1 — three times more than B is B plus three more B’s. In practice Indian competitive exams intend 3 : 1 for both. If a question is ambiguous, take the reading that makes the numbers come out clean, because setters choose them to.

B is 50% as efficient as A. What is the ratio?

2 : 1 for A to B. Fifty per cent as efficient means half the rate, so B is 1 to A’s 2. Note this is a different phrasing from “50% more efficient”, which would be 3 : 2 — read whether the percentage is of or more than.

How do you handle a worker who falls ill and works at reduced efficiency?

Hard

A and B can do a job alone in 12 and 18 days. They start together, but from the fourth day onward A works at only 50% efficiency because of illness. (a) Find the total time the job takes. (b) A student solves it by averaging A’s efficiency over the whole job and using a single combined rate. Explain why that is wrong. (c) State the one circumstance in which the student’s method would happen to give the right answer.

Follow-up

Part (c) is the interesting one: averaging the rate weights the two periods equally, so it is correct only if they turn out to be equal in length — which you cannot know before solving. That is exactly why stage-splitting is a method and averaging is a coincidence.

Show the hint

Stage one is the first 3 days at full efficiency; stage two is everything after, with A at half rate.