Time & Work

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Efficiency, 'Twice as Good' & Ratio Cases

⏱ 5 min read🧩 5 question types🎯 13 practice Q

The idea in one minute

Efficiency is rate: the share of a job a person finishes per day. Efficiency and time are inversely proportional — twice as efficient means half the days.

If A is k times as efficient as B and together they take T days, the job is $(k+1)T$ units: B alone needs $(k+1)T$ days and A alone $\dfrac{(k+1)T}{k}$ days.

Convert every 'twice as good' or '50% more efficient' statement into a rate ratio before doing anything else.

Efficiency is rate

'Efficient' here simply means faster. Efficiency is another word for rate, and it moves opposite to time:

\frac{\text{efficiency of A}}{\text{efficiency of B}} = \frac{\text{days of B}}{\text{days of A}}

Twice as efficient means half the days. Three times as efficient means one-third of the days. Write every comparison as a rate ratio before touching the numbers.

Rule: Rate × days = one whole job. Multiply the rate by k and the days must divide by k.

The unit trick

A is k times as good as B. Let B do 1 unit a day and A do k units a day — together $(k+1)$ units a day. If they finish in T days, the whole job is $(k+1)T$ units:

weaker worker alone: $(k+1)T$ days
stronger worker alone: $\dfrac{(k+1)T}{k}$ days

A is twice as good as B, and together they take 18 days → job $= 3 \times 18 = 54$ units → B alone 54 days, A alone 27. The weaker worker takes the longer time, always.

Watch: If the stronger worker got the bigger number of days, the ratio was inverted.

Ratios and per cents

'A and B are in efficiency ratio 3 : 2' is the same idea with both rates written at once: rates $3u$ and $2u$ , together $5u$ . If B alone takes 30 days, $2u = \dfrac{1}{30}$ , so $u = \dfrac{1}{60}$ and together $5u = \dfrac{1}{12}$ → 12 days.

Per cents become ratios at sight. 50% more efficient → 3 : 2. 25% more → 5 : 4. 20% less → 4 : 5. Then continue exactly as with a plain ratio.

Tip: '25% more efficient' is 125 : 100 = 5 : 4, never 25 : 100.

Days-difference questions

'A is twice as fast as B and takes 12 days less.' Efficiency ratio 2 : 1 makes the times 1x and 2x. The gap $2x - x = 12$ gives x = 12: A takes 12 days, B takes 24. Together they need $\dfrac{12 \times 24}{36} = 8$ days.

In general, with efficiency ratio a : b, the times run b : a. Call them bx and ax, put the difference equal to the given gap, and solve for x.

Difference of their times

Exams often ask for the difference of the two times instead of one of them. Price the job first, then subtract.

A is 3 times as efficient as B, and together they take 12 days → job $= 4 \times 12 = 48$ units. B alone 48 days, A alone 16 days, difference 32 days. One subtraction after the pricing step.

Keep the direction honest

The strongest worker always takes the fewest days — and later, in wage questions, earns the bigger share. Run that check on every answer before choosing the option.

Recompute one full chain to stay safe: A is twice as good as B, together 9 days → job = 27 units → B alone 27 days, A alone 13.5. The weaker worker's days equal the job size in units — no accident, since the weaker worker sets the unit at 1 a day.

Note: '30% more efficient' means 13 : 10, not 30 : 100. Convert the wording to a ratio before any arithmetic.

Question types you will see

Each type: how to recognise it, the method step by step, and one question to try.

Type 1very common3 practice Q

'k times as efficient', together time given

How to spot it:

'A is twice/thrice as efficient as B and together they finish in T days' — the individual times are asked.

T_B = (k+1)T, \qquad T_A = \frac{(k+1)T}{k}

Method

Let the weaker worker do 1 unit/day, the stronger k units/day.
Together $(k+1)$ units/day → job $= (k+1) \times T$ units.
Divide the job by each worker's own rate for his days.

Why it works:

The together time prices the job in units; each worker then spends those units at his own speed.

Try this

A is twice as efficient as B and together they complete a work in 18 days. B alone can complete it in:

Show solution

Rates: A $= 2$ , B $= 1$ → $3$ /day.
Job $= 3 \times 18 = 54$ units.
B alone $= 54 \div 1 = 54$ days (A alone $27$ ).

Answer

54 days

Type 2common3 practice Q

Efficiency ratio given as a : b

How to spot it:

An explicit ratio like 3 : 2 is given for the efficiencies, with either the together time or one person's time.

\text{job} = (a + b) \times T \text{ units}, \qquad T_{\text{weak}} = \frac{(a+b)T}{b}

Method

Fix rates $au$ and $bu$ per day.
From the given time, price the job in units.
Divide the job by the rate whose time is asked.

Why it works:

A ratio is only a k-times statement written for both workers at once.

Try this

The efficiencies of A and B are in the ratio 3 : 2. B alone can finish the work in 30 days. Working together, they will finish it in:

Show solution

B: $2u = \dfrac{1}{30}$ → $u = \dfrac{1}{60}$ .
Together $5u = \dfrac{5}{60} = \dfrac{1}{12}$ per day.
$12$ days.

Answer

12 days

Type 3common3 practice Q

Efficiency ratio plus a gap of days

How to spot it:

'A is twice as fast as B and takes 12 days less' — find either individual time or the together time.

bx - ax = \text{gap}, \text{ where the times are } bx \text{ and } ax

Method

With efficiency ratio $a : b$ , the times are in $b : a$ — call them $bx$ and $ax$ .
Put the difference equal to the given gap and solve for x.
Combine the two times with $\dfrac{ab}{a+b}$ if the together time is asked.

Why it works:

The gap is a difference of two numbers in a known ratio, so one bracket pins both.

Try this

A is twice as fast a worker as B and takes 12 days less than B to finish a piece of work. Working together, they will finish it in:

Show solution

Times $x$ and $2x$ : $2x - x = 12$ → A $= 12$ days, B $= 24$ days.
Together $= \dfrac{12 \times 24}{36} = 8$ days.

Answer

8 days

Type 4occasional2 practice Q

Percent more / less efficient

How to spot it:

'A is 50% more efficient than B', 'A works 20% faster' — percentages of efficiency instead of a ratio.

a\% \text{ more efficient} \Rightarrow E_A : E_B = (100 + a) : 100

Method

Convert to a ratio: 50% more → 3 : 2; 25% more → 5 : 4; 20% less → 4 : 5.
Continue with the unit method — price the job, then divide.
Keep fractions; answers like 10.8 days are legitimate.

Why it works:

'a% more efficient' means the rate is multiplied by one plus a hundredth.

Try this

A is 50% more efficient than B. B alone can finish a work in 27 days. Working together, they will finish it in:

Show solution

A's rate $= \dfrac{3}{2} \times \dfrac{1}{27} = \dfrac{1}{18}$ .
Sum $= \dfrac{1}{18} + \dfrac{1}{27} = \dfrac{5}{54}$ per day.
$\dfrac{54}{5} = 10.8$ days.

Answer

10.8 days

Type 5common

Difference of the two individual times

How to spot it:

'A is k times as efficient as B and together they take T days' — the difference between their individual times is asked.

T_B - T_A = (k+1)T - \frac{(k+1)T}{k}

Method

Let the weaker worker do 1 unit a day, the stronger k units.
Job $= (k+1)T$ units; weaker alone $(k+1)T$ days, stronger $\dfrac{(k+1)T}{k}$ days.
Subtract the two times.

Why it works:

Both times come from the same priced job, so one subtraction finishes the question.

Try this

A is 3 times as efficient as B and together they complete a work in 12 days. Find the difference between the times A and B alone would take.

Show solution

Job $= 4 \times 12 = 48$ units.
B alone $= 48$ days; A alone $= 48 \div 3 = 16$ days.
Difference $= 48 - 16 = 32$ days.

Answer

32 days

Formula sheet

Efficiency and time

\frac{E_A}{E_B} = \frac{T_B}{T_A}

k-times worker

T_B = (k+1)T, \quad T_A = \frac{(k+1)T}{k}

Times from an efficiency ratio

E_A : E_B = a : b \Rightarrow T_A : T_B = b : a

Per cent more efficient

a\% \text{ more} \Rightarrow E_A : E_B = (100 + a) : 100

Shortcuts that save time

⚡ Rate units from the efficiency ratio

Let B = 1 unit/day, A = k units/day; together (k+1) per day prices the job.

Example

A is twice as efficient as B, and together they finish a job in 12 days. B alone would take:

Show solution

Units: $2 + 1 = 3$ /day → job $= 3 \times 12 = 36$ units.
B alone $= 36 \div 1 = 36$ days (A: $18$ ).

Answer

36 days

⚡ Invert, never scale

'A is 3 times as good' means A's days = B's days ÷ 3.

Example

A is 3 times as efficient as B and together they complete the work in 12 days. A alone takes:

Show solution

Job $= 4 \times 12 = 48$ units.
A $= 48 \div 3 = 16$ days (B: $48$ ).

Answer

16 days

⚡ Days-difference cases

'B takes 24 days more than A' plus an efficiency ratio pins both times.

Example

A is 3 times as fast as B and takes 24 days less than B. Together they would finish the work in:

Show solution

Times $x$ and $3x$ with $3x - x = 24$ → $x = 12$ : A $= 12$ , B $= 36$ days.
Together $= \dfrac{12 \times 36}{48} = 9$ days.

Answer

9 days

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Reading 'twice as efficient' as twice the days.

Efficiency up means days down — half the days, not double.

Mistake 02

Giving the k-times worker $(k+1)T$ days.

That is the weaker worker's time. The stronger takes (k+1)T/k.

Mistake 03

Treating '25% more efficient' as 25 : 100.

It is 125 : 100 = 5 : 4. Add the per cent to 100 first.

Mistake 04

Adding efficiencies as if they were days.

Efficiencies are rates; invert to times (or price in units) before combining.

Mistake 05

Mixing up whose time is longer in days-difference questions.

The weaker worker takes the longer time. Check the direction before answering.

Quick revision

Read this the night before the exam.

Efficiency ratio = rate ratio = inverse time ratio.
k-times worker, together T days: weaker (k+1)T, stronger (k+1)T/k.
50% / 25% / 30% more efficient → 3:2 / 5:4 / 13:10.
Efficiency a : b makes times b : a; gap = (b − a)x.
Difference of times: price the job in units, then subtract.
Sanity check: faster worker, fewer days — always.

Practice: 13 questions

Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.

Topic test · 13 questions

Suggested time 9 min · wrong answers go to your mistake notebook automatically.

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