Work and Time Concepts

1. Basic Work Rate

• Formula:

$$\text{1-day work}=\frac1{\text{Number of days}}$$

If A completes a work in (x) days:

$$R_A=\frac1x$$

Combined rate:

$$R_{\text{total}}=R_A+R_B+\cdots$$

• Example: A can complete a work in 12 days and B in 18 days. How long will they take together? Solution:

$$R_A=\frac1{12},\qquad R_B=\frac1{18}$$ $$R_{\text{total}}=\frac1{12}+\frac1{18} =\frac5{36}$$

Therefore:

$$T=\frac1{5/36} =\boxed{\frac{36}{5}=7.2\text{ days}}$$

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2. Efficiency & Substitution

• Formula:

$$\text{Efficiency}\propto\frac1{\text{Time}}$$

If efficiency ratio is (a:b), time ratio is:

$$b:a$$

• Example: A is 50% more efficient than B. B takes 30 days alone. How many days will they take together? Solution: A’s efficiency:

$$=150$$

Therefore time ratio:

$$A:B=2:3$$

B takes 30 days, so A takes:

$$30\times\frac23=20\text{ days}$$

Together:

$$T=\frac{20\times30}{20+30} =\boxed{12\text{ days}}$$

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3. Work Done in a Given Number of Days

• Formula:

$$\text{Work done}=\text{Rate}\times\text{Time}$$

If A completes work in (x) days, work done in (d) days:

$$\frac d x$$

• Example: A completes a work in 15 days. What fraction of the work does A complete in 6 days? Solution:

$$\text{Work done}=\frac6{15} =\boxed{\frac25}$$

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4. Remaining Work After Partial Completion

• Formula:

$$\text{Remaining work}=1-\text{Work completed}$$ $$\text{Remaining time} =\frac{\text{Remaining work}}{\text{New rate}}$$

• Example: A completes a work in 12 days. After working alone for 4 days, B joins and they finish the remaining work in 3 days. Find B’s individual time. Solution: A’s rate:

$$\frac1{12}$$

Work done by A in 4 days:

$$\frac4{12}=\frac13$$

Remaining:

$$1-\frac13=\frac23$$

A and B together complete (\frac23) in 3 days:

$$R_{A+B}=\frac{2/3}{3}=\frac29$$

Therefore B’s rate:

$$\frac29-\frac1{12} =\frac8{36}-\frac3{36} =\frac5{36}$$

B’s time:

$$\boxed{\frac{36}{5}=7.2\text{ days}}$$

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5. Finding Individual Time from Combined Time

• Formula: If A and B take (x) and (y) days:

$$\frac1T=\frac1x+\frac1y$$

Hence:

$$T=\frac{xy}{x+y}$$

• Example: A takes 8 days and A+B together take 4.8 days. Find B’s time. Solution:

$$\frac1B=\frac1{4.8}-\frac18$$ $$=\frac5{24}-\frac3{24} =\frac2{24}=\frac1{12}$$

Therefore:

$$\boxed{B=12\text{ days}}$$

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6. Alternate Working Days

• Formula:

$$\text{Work in 2 days}=\frac1x+\frac1y$$

If A starts, calculate complete 2-day cycles first and then check the final day.

• Example: A takes 10 days and B takes 15 days. They work on alternate days, starting with A. When is the work completed? Solution: Work in 2 days:

$$\frac1{10}+\frac1{15} =\frac16$$

After 10 days:

$$5\times\frac16=\frac56$$

Remaining:

$$\frac16$$

Day 11 is A’s turn:

$$\frac1{10}$$

Remaining:

$$\frac16-\frac1{10} =\frac1{15}$$

Day 12 is B’s turn, and B completes:

$$\frac1{15}$$

Therefore work finishes on:

$$\boxed{12\text{th day}}$$

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7. Alternate Working with Unequal Final Day

• Formula: Calculate work completed in complete cycles, then:

$$\text{Final time}= \text{Complete cycles}+\frac{\text{Remaining work}}{\text{Rate of next worker}}$$

• Example: A completes a work in 6 days and B in 8 days. They work alternately starting with A. Find the completion time. Solution: Work in 2 days:

$$\frac16+\frac18=\frac7{24}$$

After 6 days (3 cycles):

$$3\times\frac7{24}=\frac78$$

Remaining:

$$\frac18$$

Day 7 is A’s turn. A’s rate:

$$\frac16$$

Time required:

$$\frac{1/8}{1/6}=\frac34\text{ day}$$

Total:

$$\boxed{6\frac34\text{ days}}$$

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8. Men-Days / Workforce Problems

• Formula: For the same work:

$$M_1D_1=M_2D_2$$

More generally:

$$\text{Work}\propto \text{Men}\times\text{Days}$$

when all workers have equal efficiency.

• Example: 8 men complete a work in 15 days. How many days will 12 men take? Solution:

$$8\times15=12\times D$$ $$D=\frac{120}{12} =\boxed{10\text{ days}}$$

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9. Men-Women-Children Workforce Equations

• Formula: Convert different workers into equivalent units using their efficiency ratio. If 1 woman does the work of (k) men:

$$\text{Equivalent men}= M+kW$$

Then use:

$$\text{Equivalent workers}\times\text{Days}=\text{Constant}$$

• Example: 2 women can do the work of 1 man. If 6 men complete a work in 10 days, how many days will 4 men and 4 women take? Solution: 4 women are equivalent to:

$$4\times\frac12=2\text{ men}$$

So total equivalent workers:

$$4+2=6$$

Since 6 men take 10 days:

$$\boxed{10\text{ days}}$$

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10. Workforce Change During a Project

• Formula:

$$\text{Total work}=\text{Workers}\times\text{Days}$$

Calculate completed and remaining work separately when workers join or leave.

• Example: 8 men complete a work in 15 days. After 5 days, 4 more men join. How many additional days are required? Solution: Total work:

$$8\times15=120\text{ man-days}$$

Work completed in 5 days:

$$8\times5=40$$

Remaining:

$$120-40=80$$

New workforce:

$$8+4=12$$

Required days:

$$\frac{80}{12} =\boxed{6\frac23\text{ days}}$$

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11. Workers Leaving During a Project

• Formula:

$$\text{Remaining work} =\text{New workforce}\times\text{Remaining days}$$

• Example: 12 men can complete a work in 20 days. After 8 days, 4 men leave. How many more days are required? Solution: Total work:

$$12\times20=240$$

Work completed:

$$12\times8=96$$

Remaining:

$$240-96=144$$

Remaining workers:

$$12-4=8$$

Days required:

$$\frac{144}{8} =\boxed{18\text{ days}}$$

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12. Wages Based on Work Done

• Formula:

$$\text{Individual wage} ====================== \frac{\text{Individual work}}{\text{Total work}} \times\text{Total wage}$$

If workers have equal efficiency:

$$\text{Wage ratio}=\text{Work ratio}$$

• Example: A, B and C work for 6, 4 and 5 days respectively at equal efficiency. If the total wage is ₹3,600, find A’s share. Solution: Work ratio:

$$6:4:5$$

Total:

$$15$$

A’s share:

$$3600\times\frac6{15} =\boxed{₹1440}$$

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13. Wages Based on Efficiency × Time

• Formula:

$$\text{Wage ratio} =E_1T_1:E_2T_2$$

• Example: A works 6 days at efficiency 2 units/day, while B works 4 days at efficiency 3 units/day. If ₹2,400 is distributed according to work, find their shares. Solution: A’s work:

$$2\times6=12$$

B’s work:

$$3\times4=12$$

Ratio:

$$1:1$$

Each gets:

$$\boxed{₹1200}$$

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14. Efficiency Ratio and Time Ratio

• Formula:

$$E_A:E_B=T_B:T_A$$

If A is (x%) more efficient than B:

$$E_A:E_B=(100+x):100$$

• Example: A is 25% more efficient than B. If B takes 20 days, how many days does A take? Solution:

$$E_A:E_B=125:100=5:4$$

Therefore:

$$T_A:T_B=4:5$$ $$T_A=20\times\frac45 =\boxed{16\text{ days}}$$

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15. Work Equivalence from Efficiency

• Formula:

$$\text{Work done}\propto\text{Efficiency}\times\text{Time}$$

Therefore:

$$E_1T_1=E_2T_2$$

for the same amount of work.

• Example: A is twice as efficient as B. If A works for 6 days, how many days must B work to complete the same amount of work? Solution:

$$E_A=2E_B$$

Thus:

$$2E_B\times6=E_B\times T$$ $$T=12$$

Therefore:

$$\boxed{12\text{ days}}$$

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Advanced Variants

16. A, B and C Working Together

• Formula:

$$R_{A+B+C}=R_A+R_B+R_C$$

Hence:

$$T=\frac1{R_A+R_B+R_C}$$

• Example: A, B and C can complete a work in 12, 18 and 36 days respectively. How long will they take together? Solution:

$$R=\frac1{12}+\frac1{18}+\frac1{36}$$ $$=\frac3{36}+\frac2{36}+\frac1{36} =\frac6{36}=\frac16$$

Therefore:

$$\boxed{6\text{ days}}$$

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17. One Worker Joins or Leaves at a Specific Time

• Formula:

$$\text{Total work} =\text{Work in first phase}+\text{Work in second phase}$$

For each phase:

$$W=R\times T$$

• Example: A completes a work in 12 days and B in 18 days. A works alone for 3 days, then B joins. Find the total completion time. Solution: A’s rate:

$$\frac1{12}$$

Work in 3 days:

$$\frac3{12}=\frac14$$

Remaining:

$$\frac34$$

Combined rate:

$$\frac1{12}+\frac1{18}=\frac5{36}$$

Time for remaining work:

$$\frac{3/4}{5/36} =\frac34\times\frac{36}{5} =\frac{27}{5}=5.4$$

Total:

$$3+5.4=\boxed{8.4\text{ days}}$$

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18. Pipes and Workers Combined

• Formula: Treat every entity as a work rate:

$$R_{\text{net}}=R_{\text{workers}}+R_{\text{filling}}-R_{\text{emptying}}$$

• Example: A worker completes a job in 10 days. A machine completes it in 15 days. A faulty process removes (\frac1{30}) of the work per day. How long do they take together? Solution:

$$R=\frac1{10}+\frac1{15}-\frac1{30}$$ $$=\frac3{30}+\frac2{30}-\frac1{30} =\frac4{30}=\frac2{15}$$

Therefore:

$$T=\frac{15}{2} =\boxed{7.5\text{ days}}$$

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19. Work Completed by Alternating Groups

• Formula: Calculate the work of each cycle:

$$W_{\text{cycle}}=R_1t_1+R_2t_2$$

Then determine complete cycles and the remaining work.

• Example: A can finish a work in 8 days and B in 12 days. A works for 2 days, B for 1 day repeatedly. How long will the work take? Solution: Work in one 3-day cycle:

$$2\left(\frac18\right)+1\left(\frac1{12}\right) =\frac14+\frac1{12} =\frac13$$

After 2 cycles (6 days):

$$\frac23$$

Remaining:

$$\frac13$$

A needs:

$$\frac{1/3}{1/8}=\frac83$$

days.

Total:

$$6+\frac83 =\boxed{8\frac23\text{ days}}$$

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20. Work and Wages with Different Efficiencies

• Formula:

$$\text{Wage ratio} =E_1T_1:E_2T_2$$

Do not use only the number of days when efficiencies differ.

• Example: A works for 5 days at efficiency 3 units/day and B works for 6 days at efficiency 2 units/day. Total wage is ₹2,100. Find A’s share. Solution: A’s work:

$$5\times3=15$$

B’s work:

$$6\times2=12$$

Ratio:

$$15:12=5:4$$

A’s share:

$$2100\times\frac59 =\boxed{₹1166.67}$$

B’s share:

$$\boxed{₹933.33}$$