SSC CGL Arithmetic Maths Time & Work

Time and Work is one of the most important topics in SSC CGL Quantitative Aptitude. Questions from this chapter frequently appear in the exam and test a candidate’s ability to calculate work rates, efficiency, and task completion time. 

 The topic covers concepts such as individual work, combined work, fractional work, efficiency comparison, and worker replacement. A strong understanding of these concepts helps candidates solve questions quickly and accurately. Learning shortcut methods, efficiency-based approaches, and standard formulas can significantly reduce calculation time and improve overall performance in SSC CGL and other competitive examinations. 

Calculating Work Completion Time with Fractional Work

Calculate work completion time for individuals performing a fraction of total work. First, determine time each person takes for the entire work.

 Example: A completes 2/3 of a work in 10 days. B completes 3/4 of the same work in 15 days. In how many days will 3/5 of the total work be completed by A and B working together?

• A's Time for Full Work: 10 days / (2/3) = 15 days.

• B's Time for Full Work: 15 days / (3/4) = 20 days.

• Combined Time for Full Work: (15 * 20) / (15 + 20) = 300 / 35 = 60/7 days.

• Time for 3/5 of Work: (3/5) * (60/7) = 180 / 35 = 36/7 days or 5 and 1/7 days.

Efficiency and Total Work - Typing Assignment

Total Work and Efficiency are fundamental. Efficiency is work done per unit time.

Example: Ronald types 32 pages in 6 hours. Allan types 40 pages in 5 hours. Total assignment: 110 pages. How long will it take them to complete it together?

• Total Work: 110 pages.

• Ronald's Efficiency: 32 pages / 6 hours = 16/3 pages/hour.

• Allan's Efficiency: 40 pages / 5 hours = 8 pages/hour.

• Combined Efficiency: (16/3) + 8 = 40/3 pages/hour.

• Time Calculation: 110 pages / (40/3 pages/hour) = (110 * 3) / 40 = 33/4 hours.

• Result: 8 hours and 15 minutes.

Efficiency and Days - Inverse Relationship

Efficiency and Days taken have an inverse relationship. Higher efficiency means fewer days.

Example: A is thrice as efficient as B. A takes 20 days less than B. How many days will B take alone?

• Efficiency Ratio (A:B): 3:1.

• Days Ratio (A:B): 1:3 (inverse of efficiency).

• Difference in Days (units): 3 - 1 = 2 units.

• Actual Difference: 20 days.

• Value of One Unit: 2 units = 20 days => 1 unit = 10 days.

• B's Time: 3 * 10 = 30 days.

MDH Formula (Man-Days-Hours) - Application

The MDH formula (Man × Days × Hours / Work) is a constant ratio for a given task.

Formula: (M1 * D1 * H1) / W1 = (M2 * D2 * H2) / W2

Example: 18 persons working 8 hours/day complete 3 units of work in 10 days. How many days will 25 people working 6 hours/day take to complete 5 units of work?

• Scenario 1: M1=18, D1=10, H1=8, W1=3

• Scenario 2: M2=25, D2=?, H2=6, W2=5

• Applying Formula: (18 * 10 * 8) / 3 = (25 * D2 * 6) / 5

• 480 = 30 * D2

• D2 = 16 days.

Work with Men Leaving - Calculation of Initial Number of Men

Handle workforce changes by calculating work done and remaining.

Example: 'n' men complete a work in 15 days. If 5 men leave after 3 days, remaining work is done in 18 days. Find 'n'.

• Total Work (Planned): 15n units.

• Work Done in 3 Days: 3n units (by 'n' men).

• Remaining Work: 15n - 3n = 12n units.

• Workers for Remaining Work: (n - 5) men.

• Equation: (n - 5) * 18 = 12n

• • 18n - 90 = 12n

  • 6n = 90

  • n = 15.

Food Provisioning (MDH Application with Changing Population)

MDH principles apply to resource consumption with changing populations.

Example: Food for 2000 students for 54 days. After 15 days, new students join, and food lasts 20 more days. How many extra students joined?

• Food Remaining (equivalent): For original 2000 students, food would last 54 - 15 = 39 more days. So, 2000 * 39 units.

• New Scenario: Let 'x' be extra students. Total students = (2000 + x). Food lasts 20 days.

• Equation (M1 * D1 = M2 * D2):

• 2000 * 39 = (2000 + x) * 20

• 78000 = (2000 + x) * 20

• 3900 = 2000 + x

• x = 1900 extra students.

Efficiency Ratios and Remaining Work

Calculate total and remaining work from combined and individual efficiency.

Example: A:B:C efficiency ratio is 7:5:4. Together, they finish work in 35 days. If A and B worked for 28 days, how many days will C take for the remaining work?

• Individual Efficiencies: A=7, B=5, C=4 units/day.

• Total Combined Efficiency (A+B+C): 7 + 5 + 4 = 16 units/day.

• Total Work: 16 * 35 = 560 units.

• Work by A and B in 28 Days: (7 + 5) * 28 = 12 * 28 = 336 units.

• Remaining Work: 560 - 336 = 224 units.

• Time for C: 224 / 4 = 56 days.

Complex Efficiency Relationship and Remaining Work

Determine individual efficiencies from intricate relationships to calculate remaining work.

Example: A = B + C. A and B finish in 36 days. C alone takes 60 days. If A and C work 10 days, how many days will B take for remaining work?

• Total Work (LCM of 36, 60): 180 units.

• Efficiency (A+B): 180 / 36 = 5 units/day.

• Efficiency C: 180 / 60 = 3 units/day.

• Individual Efficiencies:

• Given A = B + C. Substitute C=3: A = B + 3.

• From A + B = 5: (B + 3) + B = 5 => 2B = 2 => B's efficiency = 1 unit/day.

• A's efficiency = 1 + 3 = 4 units/day.

• Work by A and C in 10 Days: (4 + 3) * 10 = 7 * 10 = 70 units.

• Remaining Work: 180 - 70 = 110 units.

• Time for B: 110 / 1 = 110 days.

Multi-layered Efficiency Ratios and Remaining Work

Combine percentage efficiency ratios for total and remaining work.

Example: A completes work in 40 days. B is 25% more efficient than A. C is 28% more efficient than B. They work together for 5 days. How many days will B complete the remaining work? 

• Efficiency Ratios:

• • B vs A (25% = 1/4): A:B = 4:5.

  • C vs B (28% = 7/25): B:C = 25:32.

• Combined Ratio A:B:C: Scale A:B by 5 to match B's 25: A:B = 20:25. So, A:B:C = 20:25:32.

• Total Work: A's efficiency = 20 units/day. Total Work = 20 * 40 days = 800 units.

• Work Together in 5 Days: (20 + 25 + 32) * 5 = 77 * 5 = 385 units.

• Remaining Work: 800 - 385 = 415 units.

• Time for B: 415 / 25 = 16 and 3/5 days.

Time Difference and Combined Work - Solving by Options

For time difference problems, work backward from options.

Example: A takes 12 hours less than B. Together, they complete work in 17.5 hours. How much time will B take to complete 50% of the work?

• Given: A = B - 12 hours. Combined time = 17.5 hours (35/2 hours).

• Strategy: Assume an option for B's 50% work time.

• Let B take 21 hours for 50% work, meaning B takes 42 hours for 100% work.

• Then, A takes 42 - 12 = 30 hours for 100% work.

• Verify Combined Time: Work rate A = 1/30. Work rate B = 1/42.

• • Combined rate = (1/30) + (1/42) = (7+5)/210 = 12/210 = 2/35.

  • Combined time = 1 / (2/35) = 35/2 hours = 17.5 hours.

• This matches the given.

• B takes 21 hours to complete 50% of the work.

Group Work - Simplification through Scaling

Simplify group work problems by finding proportional relationships.

Example: 6 men and 8 women complete a work in 10 days. Find the time taken by 15 men and 20 women to complete the same work.

• Given: (6 Men + 8 Women) takes 10 days.

• Goal: Time for (15 Men + 20 Women).

• Scaling Method:

• • Notice (15M + 20W) = 2.5 times (6M + 8W).

  • If workers are 2.5 times more, time taken will be 2.5 times less.

  • Time = 10 days / 2.5 = 4 days.

Man-Boy Efficiency Relationship and Combined Work

Establish efficiency ratios (e.g., men, boys) to calculate total work and combined efficiency.

Example: 10 men complete work in 20 days. 20 boys complete same work in 30 days. How many days will 10 men and 20 boys together complete the work?

• Efficiency Ratio (Man:Boy):

• • Work by Men = 10 * 20 = 200 Man-Days.

  • Work by Boys = 20 * 30 = 600 Boy-Days.

  • 200 Man-Days = 600 Boy-Days => 1 Man = 3 Boys. Ratio = 3:1.

• Total Work: Let Boy's efficiency = 1 unit/day, Man's = 3 units/day.

• • Using Men's data: Total Work = (10 Men * 3 units/man-day) * 20 days = 600 units.

• Combined Efficiency (10 Men + 20 Boys):

• • 10 Men * 3 units/man-day = 30 units/day.

  • 20 Boys * 1 unit/boy-day = 20 units/day.

  • Combined = 30 + 20 = 50 units/day.

• • Time Together: 600 / 50 = 12 days.