Time and Work questions with answers and concepts

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Time and Work problems deal with the relationship between the time taken to complete a task and the efficiency or rate at which workers operate. These problems typically involve calculating how long individuals or groups take to finish jobs, determining work rates, and solving scenarios involving combined efforts, efficiency variations, and partial work completion.

Key formulas and rules

12 rules

Work Rate $= \frac{1}{Time}$ (work done per unit time)

Time $= \frac{Work}{Rate}$

If A can do a work in $n$ days, A's 1 day work $= \frac{1}{n}$

If A's 1 day work $= \frac{1}{n}$ , A can finish the work in $n$ days

Combined rate of A and B $= \frac{1}{a} + \frac{1}{b}$ where $a, b$ are individual times

Time taken by A and B together $= \frac{a \times b}{a + b}$

If A is $x$ times more efficient than B: If B takes $n$ days, A takes $\frac{n}{x}$ days

Man-Days: $M \times D = constant$ for the same work

$M_{1} \times D_{1} = M_{2} \times D_{2}$ (for same work with different men and days)

$M_{1} \times D_{1} \times H_{1} = M_{2} \times D_{2} \times H_{2}$ (including hours per day)

Work done $= \frac{Days worked}{Total days to complete}$

Remaining work $= 1 - Work already done$

Key concepts

Basic Work Rate Concept

The fundamental principle is that if a person can complete a work in 'n' days, their daily work rate is 1/n of the total work. For example, if A completes work in 10 days, A does 1/10 of the work each day. This fraction represents the portion of work completed per day and forms the basis for all time and work calculations.

Combined Work (Working Together)

When multiple people work together, their individual work rates add up. If A does 1/a work per day and B does 1/b work per day, together they do (1/a + 1/b) work per day. The time taken together is the reciprocal: (axb)/(a+b). For three people A, B, C: combined rate = 1/a + 1/b + 1/c, and time = 1/(combined rate).

Efficiency and Comparison

Efficiency refers to how quickly someone completes work relative to another. If A is twice as efficient as B, A takes half the time B takes. If A takes 20 days, B takes 40 days. Efficiency ratios help determine time ratios: Time ratio is inverse of efficiency ratio. Higher efficiency means less time required.

Man-Days (Men and Work)

The concept of man-days states that for a fixed amount of work, the product of workers and days remains constant. M x $D = M$ x D . If 10 men take 20 days, then 20 men take 10 days. When hours per day vary: M x D x $H = M$ x D x H . This principle applies to identical work conditions.

Partial Work and Remaining Work

When workers leave or join partway through a project, calculate work done before the change and remaining work after. If A works for 5 days alone (completing 5/a of work), the remaining (1 - 5/a) work is done by others. Track individual contributions and combine them to verify total work equals 1 (whole work).

Alternating Work (A and B Working on Alternate Days)

When two workers alternate, calculate work done in complete cycles. If A works day 1 and B works day 2: in 2 days they complete (1/a + 1/b) work. Find full cycles needed, then determine who finishes the remaining work. For example, if 2-day cycle completes 1/6 work, after 5 cycles (10 days), 5/6 is done; remaining 1/6 is done by A on day 11.

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Worked examples

Example 1

A can complete a piece of work in 15 days and B can complete the same work in 20 days. If they work together, how many days will they take to finish the work?

Solution

Step 1: Compute 1-day work rates.
- A's 1-day rate: $\frac{1}{15}$
- B's 1-day rate: $\frac{1}{20}$

Step 2: Calculate combined 1-day work rate.

(A + B) ’s 1-day rate = \frac{1}{15} + \frac{1}{20} = \frac{4 + 3}{60} = \frac{7}{60}

Step 3: Calculate required time.

Time = \frac{1}{\frac{7}{60}} = \frac{60}{7} = 8 \frac{4}{7} days \approx 8.57 days

Answer: $8 \frac{4}{7}$ days (or approximately 8.57 days).

Example 2

A and B together can complete a work in 12 days. B and C together can complete it in 15 days. C and A together can complete it in 20 days. How many days will A, B, and C take working together?

Solution

Step 1: Formulate rate equations.

A + B = \frac{1}{12}, B + C = \frac{1}{15}, C + A = \frac{1}{20}

Step 2: Sum all three equations.

2 (A + B + C) = \frac{1}{12} + \frac{1}{15} + \frac{1}{20} = \frac{5 + 4 + 3}{60} = \frac{12}{60} = \frac{1}{5}

A + B + C = \frac{1}{5 \times 2} = \frac{1}{10}

Step 3: Compute combined time.

Time = \frac{1}{\frac{1}{10}} = 10 days

Answer: 10 days.

Example 3

A is twice as efficient as B. Working together, they complete a task in 18 days. In how many days can A alone finish the work?

Solution

Step 1: Express rates in terms of B's rate ( $r$ ).
- B's rate = $r$
- A's rate = $2 r$
- Combined rate = $2 r + r = 3 r$

Step 2: Equate combined rate to given completion time.

3 r = \frac{1}{18} ⟹ r = \frac{1}{54}

Step 3: Determine A's rate and completion time.

A’s rate = 2 r = 2 \times \frac{1}{54} = \frac{1}{27}

Time taken by A alone = \frac{1}{\frac{1}{27}} = 27 days

Answer: 27 days.

Example 4

20 men can complete a project in 30 days. After working for 10 days, 5 additional men join the crew. In how many more days will the work be completed?

Solution

Step 1: Calculate total work in man-days.

Total Work = 20 men \times 30 days = 600 man-days

Step 2: Calculate work completed in first 10 days.

Work Done = 20 men \times 10 days = 200 man-days

Remaining Work = 600 - 200 = 400 man-days

Step 3: Calculate days for expanded crew ( $20 + 5 = 25 men$ ).

Additional Days = \frac{400 man-days}{25 men} = 16 days

Answer: 16 more days.

Representative solved questions

See the kind of question in this topic before opening the full practice set.

Question 1

A and B can do a work in 45 days and 40 days respectively. They began together but A left after some days and B finished in 23 days. When did A leave?

6 days
8 days
9 days
12 days

Answer: C. 9 days

Explanation

Step 1: A's 1 day = 1/45, B's = 1/40.
Step 2: Let A work for x days. Work by $A = x /45$ , work by $B = 23/40$ .
Step 3: Total work: $x /45 + 23/40 = 1$ . Solving: $x /45 = 17/40$ , $x = (17 \times 45) /40 = 765/40 = 19$ .125.
Let A work for x days, then B works (x+23) days total? No, B works entire time.
Let both work x days together, then B works alone for remaining days. Total B works = 23 days means A left when 23 days remained for B alone.
B's 23 days work = 23/40. Remaining = 17/40 done by both at $1/45 + 1/40 = 17/360$ per day.
Time together = (17/40) / (17/360) = 360/40 = 9 days.
Answer: 9 days

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Question 2

A, B, and C can complete a work in 10, 12, and 15 days respectively. They start together but A leaves after 2 days and B leaves 3 days before completion. How many days to complete?

5 days
6 days
7 days
8 days

Answer: C. 7 days

Explanation

Step 1: Let total days = x.
Step 2: A works 2 days, B works (x-3) days, C works x days.
Step 3: $2/10 + (x - 3) /12 + x /15 = 1$ .
Step 4: Multiply by 60: 12 + 5(x-3) + $4 x = 60$ .
Step 5: $12 + 5 x - 15 + 4 x = 60$ . $9 x - 3 = 60$ . $9 x = 63$ . $x = 7$ .
Answer: 7 days

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Question 3

A contractor employs 30 men for 20 days to build a wall. After 12 days, he finds only half the work is done. How many more men must he employ to finish on time?

10 men
12 men
15 men
20 men

Answer: C. 15 men

Explanation

Step 1: Work done by 30 men in 12 days = half the work. So total work = 2 x $(30 \times 12) = 720$ man-days.
Step 2: Remaining work = 360 man-days, to be done in 8 days.
Step 3: Men needed = 360/8 = 45 men. Additional = 45 - 30 = 15 men.
Answer: 15 men

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Common mistakes and useful habits

Always convert 'X can do work in N days' to daily rate = 1/N immediately. This makes calculations easier.
For combined work problems, remember the shortcut: If A takes 'a' days and B takes 'b' days, together they take (axb)/(a+b) days.
When dealing with efficiency ratios, the time ratio is inverse. If A:B efficiency = 3:2, then time ratio A:B = 2:3.
For alternating work problems, calculate work per complete cycle first, then handle the remainder separately.
In man-days problems, keep track of the total work constant. M D = M D is your best friend.
When workers leave or join mid-project, split the problem: calculate work done before change and work remaining after change.
For pipes and cisterns (similar concept), treat filling pipes as positive work and emptying pipes as negative work.
Always verify your answer by checking if the computed rates add up correctly to complete 1 unit of work.

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Time and Work questions with answers and concepts

135 practice questionsPage updated 11 Nov 2026Review metadata pendingPractice questions

Practise this concept

Key formulas and rules

12 rules

Work Rate $= \frac{1}{Time}$ (work done per unit time)

Time $= \frac{Work}{Rate}$

If A can do a work in $n$ days, A's 1 day work $= \frac{1}{n}$

If A's 1 day work $= \frac{1}{n}$ , A can finish the work in $n$ days

Combined rate of A and B $= \frac{1}{a} + \frac{1}{b}$ where $a, b$ are individual times

Time taken by A and B together $= \frac{a \times b}{a + b}$

If A is $x$ times more efficient than B: If B takes $n$ days, A takes $\frac{n}{x}$ days

Man-Days: $M \times D = constant$ for the same work

$M_{1} \times D_{1} = M_{2} \times D_{2}$ (for same work with different men and days)

$M_{1} \times D_{1} \times H_{1} = M_{2} \times D_{2} \times H_{2}$ (including hours per day)

Work done $= \frac{Days worked}{Total days to complete}$

Remaining work $= 1 - Work already done$

Key concepts

Basic Work Rate Concept

Combined Work (Working Together)

Efficiency and Comparison

Man-Days (Men and Work)

Partial Work and Remaining Work

Alternating Work (A and B Working on Alternate Days)

Check the idea while it is fresh.Open a focused exercise without losing this article.Practise this concept

Worked examples

Example 1

A can complete a piece of work in 15 days and B can complete the same work in 20 days. If they work together, how many days will they take to finish the work?

Solution

Step 1: Compute 1-day work rates.
- A's 1-day rate: $\frac{1}{15}$
- B's 1-day rate: $\frac{1}{20}$

Step 2: Calculate combined 1-day work rate.

(A + B) ’s 1-day rate = \frac{1}{15} + \frac{1}{20} = \frac{4 + 3}{60} = \frac{7}{60}

Step 3: Calculate required time.

Time = \frac{1}{\frac{7}{60}} = \frac{60}{7} = 8 \frac{4}{7} days \approx 8.57 days

Answer: $8 \frac{4}{7}$ days (or approximately 8.57 days).

Example 2

A and B together can complete a work in 12 days. B and C together can complete it in 15 days. C and A together can complete it in 20 days. How many days will A, B, and C take working together?

Solution

Step 1: Formulate rate equations.

A + B = \frac{1}{12}, B + C = \frac{1}{15}, C + A = \frac{1}{20}

Step 2: Sum all three equations.

2 (A + B + C) = \frac{1}{12} + \frac{1}{15} + \frac{1}{20} = \frac{5 + 4 + 3}{60} = \frac{12}{60} = \frac{1}{5}

A + B + C = \frac{1}{5 \times 2} = \frac{1}{10}

Step 3: Compute combined time.

Time = \frac{1}{\frac{1}{10}} = 10 days

Answer: 10 days.

Example 3

A is twice as efficient as B. Working together, they complete a task in 18 days. In how many days can A alone finish the work?

Solution

Step 1: Express rates in terms of B's rate ( $r$ ).
- B's rate = $r$
- A's rate = $2 r$
- Combined rate = $2 r + r = 3 r$

Step 2: Equate combined rate to given completion time.

3 r = \frac{1}{18} ⟹ r = \frac{1}{54}

Step 3: Determine A's rate and completion time.

A’s rate = 2 r = 2 \times \frac{1}{54} = \frac{1}{27}

Time taken by A alone = \frac{1}{\frac{1}{27}} = 27 days

Answer: 27 days.

Example 4

20 men can complete a project in 30 days. After working for 10 days, 5 additional men join the crew. In how many more days will the work be completed?

Solution

Step 1: Calculate total work in man-days.

Total Work = 20 men \times 30 days = 600 man-days

Step 2: Calculate work completed in first 10 days.

Work Done = 20 men \times 10 days = 200 man-days

Remaining Work = 600 - 200 = 400 man-days

Step 3: Calculate days for expanded crew ( $20 + 5 = 25 men$ ).

Additional Days = \frac{400 man-days}{25 men} = 16 days

Answer: 16 more days.

Representative solved questions

See the kind of question in this topic before opening the full practice set.

Question 1

A and B can do a work in 45 days and 40 days respectively. They began together but A left after some days and B finished in 23 days. When did A leave?

6 days
8 days
9 days
12 days

Answer: C. 9 days

Explanation

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Question 2

A, B, and C can complete a work in 10, 12, and 15 days respectively. They start together but A leaves after 2 days and B leaves 3 days before completion. How many days to complete?

5 days
6 days
7 days
8 days

Answer: C. 7 days

Explanation

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Question 3

A contractor employs 30 men for 20 days to build a wall. After 12 days, he finds only half the work is done. How many more men must he employ to finish on time?

10 men
12 men
15 men
20 men

Answer: C. 15 men

Explanation

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Common mistakes and useful habits

Always convert 'X can do work in N days' to daily rate = 1/N immediately. This makes calculations easier.
For combined work problems, remember the shortcut: If A takes 'a' days and B takes 'b' days, together they take (axb)/(a+b) days.
When dealing with efficiency ratios, the time ratio is inverse. If A:B efficiency = 3:2, then time ratio A:B = 2:3.
For alternating work problems, calculate work per complete cycle first, then handle the remainder separately.
In man-days problems, keep track of the total work constant. M D = M D is your best friend.
When workers leave or join mid-project, split the problem: calculate work done before change and work remaining after change.
For pipes and cisterns (similar concept), treat filling pipes as positive work and emptying pipes as negative work.
Always verify your answer by checking if the computed rates add up correctly to complete 1 unit of work.

Ready to test your understanding?

Work through 135 questions with explanations after each answer.

Start practice

Next topic: Pipes and Cisterns

Key formulas and rules

Key concepts

Basic Work Rate Concept

Combined Work (Working Together)

Efficiency and Comparison

Man-Days (Men and Work)

Partial Work and Remaining Work

Alternating Work (A and B Working on Alternate Days)

Worked examples

Example 1

Solution

Example 2

Solution

Example 3

Solution

Example 4

Solution

Representative solved questions

Question 1

Question 2

Question 3

Common mistakes and useful habits

Ready to test your understanding?

Keep building the same skill family.

Key formulas and rules

Key concepts

Basic Work Rate Concept

Combined Work (Working Together)

Efficiency and Comparison

Man-Days (Men and Work)

Partial Work and Remaining Work

Alternating Work (A and B Working on Alternate Days)

Worked examples

Example 1

Solution

Example 2

Solution

Example 3

Solution

Example 4

Solution

Representative solved questions

Question 1

Question 2

Question 3

Common mistakes and useful habits

Ready to test your understanding?

Keep building the same skill family.