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Time and Work

Practice MCQs

None

Quantitative Aptitude Time and Work Competitive Exams

Time and Work is an important quantitative aptitude topic based on the relationship between work, time, efficiency, work rate, combined work, alternate work, wages, and pipes or cisterns. These questions are commonly asked in SSC, banking, railways, police exams, entrance tests, and aptitude exams.

What is Time and Work?

Time and Work questions deal with how much work a person, machine, or group can complete in a given time. The main idea is that if a person takes fewer days to complete the same work, that person is more efficient.

Work problems are usually solved by finding the work done per day. If A can complete a work in 10 days, then A completes \(\frac{1}{10}\) of the work in one day.

Quick idea: Total work is often assumed as 1 unit or as the LCM of given days. LCM method makes calculation faster in competitive exams.
Term Meaning Example
Work Total task to be completed Building a wall
Time Duration taken to complete work 10 days
Efficiency Work done per unit time \(\frac{1}{10}\) work per day
Combined Work Work done by two or more persons together A and B working together
Wages Payment divided according to work done A gets more if A works more

“In time and work questions, convert days into work rate before solving.”

Aptitude Tip
Key points
  • If A takes \(x\) days, one-day work is \(\frac{1}{x}\).
  • Less time means more efficiency.
  • Combined work means adding work rates.
  • For same work, time and efficiency are inversely proportional.
  • Use LCM of days as total work for faster solving.
  • Wages are divided according to work done.
work time efficiency combined work wages

Visual Understanding

These diagrams show the relationship between work, time, efficiency, and combined effort.

One-Day Work
Complete Work = 1 If work is completed in 6 days, one-day work is 1/6
\[ \text{One-day work}=\frac{1}{\text{Total days}} \]

If a person completes work in \(x\) days, daily work is \(\frac{1}{x}\).

Efficiency Comparison
Same work completed in different days A 10 days B 5 days B is twice as efficient as A
\[ \text{Efficiency ratio}=\frac{1}{T_1}:\frac{1}{T_2} \]

Time and efficiency are inversely proportional for the same work.

Combined Work
A works + B works Faster completion
\[ \text{Combined one-day work}=\frac{1}{A}+\frac{1}{B} \]

When people work together, their individual work rates are added.

LCM Method
A = 6 days, B = 8 days LCM 24 units Efficiency A=4, B=3 Together: 4 + 3 = 7 units/day
\[ \text{Time together}=\frac{\text{Total work}}{\text{Combined efficiency}} \]

LCM method avoids fractions and is very useful in exam calculations.

Important Formulas and Rules

One-Day Work
\[ \text{One-day work}=\frac{1}{x} \]

If a person completes work in \(x\) days.

Combined Work
\[ \frac{1}{A}+\frac{1}{B} \]

Add one-day work of both persons.

Time Together
\[ T=\frac{1}{\frac{1}{A}+\frac{1}{B}} \]

Used when A and B work together.

Two Persons Shortcut
\[ T=\frac{AB}{A+B} \]

Time taken by A and B together.

Efficiency
\[ \text{Efficiency}=\frac{\text{Work}}{\text{Time}} \]

Work completed per unit time.

Work Done
\[ \text{Work}=\text{Efficiency}\times\text{Time} \]

Useful in partial work questions.

Time Ratio
\[ T_1:T_2=E_2:E_1 \]

Time and efficiency are inversely proportional.

Wage Ratio
\[ \text{Wages ratio}=\text{Work done ratio} \]

Payment depends on contribution.

A, B, C Together
\[ \frac{1}{A}+\frac{1}{B}+\frac{1}{C} \]

Add all individual one-day works.

Remaining Work
\[ \text{Remaining work}=1-\text{Work done} \]

Useful when work is done partly.

LCM Method
\[ \text{Total work}=\text{LCM of days} \]

Converts fractional work into units.

Men and Days
\[ M_1D_1=M_2D_2 \]

Used when efficiency is same.

Rule: If a person takes more time, efficiency is less. If a person takes less time, efficiency is more. For the same work, time and efficiency are inversely proportional.

Common Types of Questions

Individual Work

Find one-day work, total work, or time taken by one person.

  • One-day work
  • Work completed in days
  • Remaining work
  • Efficiency comparison
Combined Work

Find time taken by two or more persons working together.

  • A and B together
  • A, B, C together
  • One leaves midway
  • One joins later
Alternate Work

Persons work on alternate days or in a repeated pattern.

  • A works first
  • B works first
  • Cycle of days
  • Remaining work calculation
Wages and Work

Divide wages according to work contribution.

  • Wage ratio
  • Efficiency ratio
  • Days worked
  • Payment division
Exam approach: If fractions become difficult, use LCM of given days as total work. Then calculate each person’s daily efficiency in units.
Method Bank
One-Day Work

A completes work in 12 days.

\[ \text{A's one-day work}=\frac{1}{12} \]
Together Work

A = 10 days, B = 15 days.

\[ T=\frac{10\times15}{10+15}=6\text{ days} \]
LCM Method

A = 6 days, B = 8 days.

\[ \text{Total work}=24,\quad A=4,\quad B=3 \]
Efficiency Ratio

A takes 5 days, B takes 10 days.

\[ E_A:E_B=10:5=2:1 \]

Tip: In time and work, LCM method is usually faster than fraction method for exam questions.

Time and Work Solving Flow
Find Days Find Efficiency Calculate Work Final Answer Days → efficiency → combined work → time
This flow helps solve individual, combined, and partial work problems.
\[ \text{Time}=\frac{\text{Total work}}{\text{Efficiency per day}} \]
\[ \text{Work done}=\text{Efficiency}\times\text{Time} \]

Step-by-Step Solving Method

Step Fraction Method LCM Method
Step 1 Write one-day work of each person. Take LCM of given days as total work.
Step 2 Add work rates if they work together. Find each person’s daily efficiency.
Step 3 Find total time using reciprocal of combined rate. Add efficiencies for combined work.
Step 4 Calculate work done or remaining work if needed. Use total work divided by combined efficiency.
Step 5 Write final answer in days or required unit. Check remaining work in partial work questions.
Important: For alternate-day work, calculate work completed in one full cycle first.

Solved Examples

Question Method Answer
A can complete a work in 10 days. What is A's one-day work? Use:
\[ \text{One-day work}=\frac{1}{10} \]
\(\frac{1}{10}\)
A can complete a work in 12 days and B can complete it in 18 days. In how many days can they complete it together? One-day work:
\[ A=\frac{1}{12},\quad B=\frac{1}{18} \]
Together:
\[ \frac{1}{12}+\frac{1}{18}=\frac{3+2}{36}=\frac{5}{36} \]
Time:
\[ \frac{36}{5}=7.2\text{ days} \]
7.2 days
A can do a work in 10 days and B can do it in 15 days. Find time taken together. Shortcut:
\[ T=\frac{AB}{A+B} \]
\[ T=\frac{10\times15}{10+15}=\frac{150}{25}=6 \]
6 days
A can complete a work in 6 days and B in 8 days. Find time together using LCM method. Total work:
\[ \text{LCM of }6\text{ and }8=24 \]
Efficiency:
\[ A=4,\quad B=3 \]
Together:
\[ 4+3=7\text{ units/day} \]
Time:
\[ \frac{24}{7}=3\frac{3}{7}\text{ days} \]
\(3\frac{3}{7}\) days
A is twice as efficient as B. If B completes work in 20 days, in how many days will A complete it? A is twice as efficient, so A takes half the time.
\[ \text{A's time}=\frac{20}{2}=10 \]
10 days
A and B can complete a work in 12 days. A alone can complete it in 20 days. Find B alone. Together one-day work:
\[ \frac{1}{12} \]
A's one-day work:
\[ \frac{1}{20} \]
B's one-day work:
\[ \frac{1}{12}-\frac{1}{20}=\frac{5-3}{60}=\frac{2}{60}=\frac{1}{30} \]
30 days
A can complete a work in 15 days. After working for 5 days, how much work is left? Work done in 5 days:
\[ 5\times\frac{1}{15}=\frac{1}{3} \]
Remaining work:
\[ 1-\frac{1}{3}=\frac{2}{3} \]
\(\frac{2}{3}\)
A and B earn ₹900 for a work. A worked for 5 days at 4 units/day and B worked for 6 days at 5 units/day. Find their shares. Work done:
\[ A=5\times4=20,\quad B=6\times5=30 \]
Ratio:
\[ A:B=20:30=2:3 \]
Shares:
\[ A=\frac{2}{5}\times900=360,\quad B=\frac{3}{5}\times900=540 \]
A = ₹360, B = ₹540

Note: In combined work questions, add work rates, not the number of days.

Common Traps and Shortcuts

Common Traps
  • Adding days directly instead of adding one-day work.
  • Forgetting that efficiency and time are inverse.
  • Confusing work done with remaining work.
  • Ignoring the person who joins or leaves midway.
  • Dividing wages equally instead of according to work done.
  • Not calculating the cycle correctly in alternate-day work.
Useful Shortcuts
  • If A takes \(x\) days, A's one-day work is \(\frac{1}{x}\).
  • For A and B together, use \(\frac{AB}{A+B}\).
  • Use LCM of days as total work.
  • Efficiency = Total work ÷ Days.
  • Wages are divided in the ratio of work done.
  • For alternate work, calculate one cycle first.
Exam approach: Avoid direct addition of days. Convert every person’s time into daily efficiency.

Practice

A) Multiple Choice Questions
  1. A can complete a work in 20 days. What is A's one-day work?
    \(\frac{1}{10}\) \(\frac{1}{15}\) \(\frac{1}{20}\) \(\frac{1}{25}\)
  2. A can do a work in 10 days and B in 15 days. Together they can complete it in:
    5 days 6 days 8 days 12 days
  3. A is twice as efficient as B. If B takes 30 days, A takes:
    10 days 15 days 20 days 60 days
  4. A and B together complete work in 12 days. A alone takes 20 days. B alone takes:
    24 days 30 days 36 days 40 days
  5. If work done ratio is \(2:3\), wages should be divided in the ratio:
    3 : 2 2 : 3 1 : 1 4 : 9
B) Solve the Higher-Order Problems
  1. A can complete a work in 16 days and B can complete it in 24 days. Find the time taken together. (Hint: Use \(\frac{AB}{A+B}\).)
  2. A can complete a work in 12 days. After working for 4 days, how much work is left? (Hint: Work done \(=4\times\frac{1}{12}\).)
  3. A and B together complete a work in 8 days. A alone can complete it in 12 days. Find B alone. (Hint: B's work = Together work - A's work.)
  4. A takes 6 days and B takes 9 days to complete the same work. Find their efficiency ratio. (Hint: Efficiency ratio is inverse of time ratio.)
  5. A and B earn ₹1200. A worked 4 days at 5 units/day and B worked 5 days at 4 units/day. Divide the wages. (Hint: Find work done by each.)
C) Match the Concept with the Correct Meaning
Concept Correct Meaning
One-Day Work Part of work completed in one day
Efficiency Work done per unit time
Combined Work Sum of individual work rates
Remaining Work Total work minus completed work
LCM Method Assuming total work as LCM of given days
Wage Division Payment according to work contribution
Aptitude Reminder

Time and Work questions are based on work rate. Convert time into one-day work or efficiency, then add or subtract work rates as required. Use LCM method for faster calculations.

Task: Create five questions using one-day work, combined work, efficiency ratio, remaining work, and wage division.

Show Suggested Answers
Multiple Choice
  1. \(\frac{1}{20}\)
    If A completes work in 20 days:
    \[ \text{One-day work}=\frac{1}{20} \]
  2. 6 days
    \[ T=\frac{10\times15}{10+15}=\frac{150}{25}=6 \]
  3. 15 days
    A is twice as efficient as B, so A takes half the time.
    \[ \frac{30}{2}=15 \]
  4. 30 days
    \[ \frac{1}{B}=\frac{1}{12}-\frac{1}{20} = \frac{5-3}{60} = \frac{1}{30} \]
  5. 2 : 3
    Wages are divided according to work done.
Higher-Order Problems
  1. A \(=16\) days, B \(=24\) days.
    \[ T=\frac{16\times24}{16+24} = \frac{384}{40} = 9.6 \]
    Answer = 9.6 days.
  2. A completes work in 12 days. Work done in 4 days:
    \[ 4\times\frac{1}{12}=\frac{1}{3} \]
    Remaining work:
    \[ 1-\frac{1}{3}=\frac{2}{3} \]
    Answer = \(\frac{2}{3}\).
  3. Together work:
    \[ \frac{1}{8} \]
    A's work:
    \[ \frac{1}{12} \]
    B's work:
    \[ \frac{1}{8}-\frac{1}{12} = \frac{3-2}{24} = \frac{1}{24} \]
    Answer = 24 days.
  4. Time ratio:
    \[ A:B=6:9=2:3 \]
    Efficiency ratio is inverse:
    \[ E_A:E_B=3:2 \]
    Answer = 3 : 2.
  5. Work done by A:
    \[ 4\times5=20 \]
    Work done by B:
    \[ 5\times4=20 \]
    Ratio:
    \[ A:B=20:20=1:1 \]
    Answer = A = ₹600, B = ₹600.
Concept Matching
  1. One-Day Work → Part of work completed in one day
  2. Efficiency → Work done per unit time
  3. Combined Work → Sum of individual work rates
  4. Remaining Work → Total work minus completed work
  5. LCM Method → Assuming total work as LCM of given days
  6. Wage Division → Payment according to work contribution
Clue Explanation

Time and Work is based on work rate. If a person completes work in \(x\) days, one-day work is \(\frac{1}{x}\). When people work together, add their one-day works.

Exam tips
  • Convert days into one-day work.
  • Add work rates for combined work.
  • Use LCM method for faster calculation.
  • Efficiency and time are inversely proportional.
  • For remaining work, subtract work done from total work.
  • Divide wages according to actual work contribution.
Practice MCQs