Time and Work – Complete Short Notes for Competitive Exams

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1. Basic Concept

👉 Work, Time, and Efficiency are directly related:

$$\boxed{\text{Work} = \text{Rate (Efficiency)} × \text{Time}}$$ $$\boxed{{\displaystyle \text{Rate (Efficiency)}=\frac{\text{Work}}{\text{Time<span class="vlist-s"></span>}}}}$$$$\boxed{\text{Time} = \frac{\text{Work}}{\text{Rate}}}$$

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2. Key Relationship Between Time & Work

• If A can complete a work in n days,
  → A’s 1 day work = 1/n

• If A’s 1 day work = 1/n,
  → Time taken = n days

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Example:

If A can complete work in 10 days → 1 day work = 1/10
If B can complete in 15 days → 1 day work = 1/15

Together in 1 day → (1/10 + 1/15) = 1/6 → Work done in 6 days

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3. Work Efficiency Concept

If efficiency ratio of A : B = 2 : 3
→ Time ratio = 3 : 2
(B is faster because more efficient → takes less time)

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4. Formula Summary

Concept	Formula
A’s 1-day work	1 / Time taken
Combined work	(A’s + B’s work)
Work done in x days	x × (1/Total time)
Remaining work	1 − (Work done)
Total work (in units)	LCM of days taken by each person
Efficiency	Work units / Days

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5. Combined Work

If A can do a work in x days, and B in y days,

$$\text{Together they finish in }\frac{xy}{x+y}\text{ days}$$

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Example:

A = 10 days, B = 15 days
Together = (10×15)/(10 + 15) = 150/25 = 6 days

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6. Partial Work Concept

If A & B together can finish work in 6 days, and A alone in 10 days:

A’s 1 day = 1/10,
A+B = 1/6,
So B’s 1 day = 1/6 − 1/10 = 1/15 → B alone can finish in 15 days

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7. Work Done or Remaining

Work done = (Days worked / Total days to finish work)
Work left = 1 − Work done

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Example:

A completes work in 12 days.
In 8 days → work done = 8/12 = 2/3, work left = 1/3.

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8. When A works alone and then with B

If A works alone for x days, then A & B work together for y days:

$$\text{Total work done}=x(1\mathrm{/}a)+y(1\mathrm{/}a+1\mathrm{/}b)$$

If total work = 1, substitute and solve for unknown.

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Example:

A = 10 days, B = 15 days.
A works 2 days alone, then both work together.
Work done in 2 days = 2/10 = 1/5
Remaining = 1 − 1/5 = 4/5
(A + B) in 1 day = 1/6 → 4/5 ÷ 1/6 = 4.8 days
Total = 2 + 4.8 = 6.8 days

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9. More Than Two Persons

If A, B, and C can complete a work in a, b, and c days respectively:

$$\text{1 day’s work} = \frac{1}{a} + \frac{1}{b} + \frac{1}{c}$$ $$\text{Total time} = \frac{1}{(1/a + 1/b + 1/c)}$$

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10. Efficiency Ratio

Efficiency ∝ 1 / Time

If A : B :: Time = 3 : 4
Efficiency = 4 : 3

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11. Work & Wages Relationship

Wages ∝ Work Done

If A : B = 2 : 3 (work done),
then wages distributed = 2 : 3.

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Example:

A, B complete work together and get ₹500.
A’s efficiency : B’s = 3 : 2
Then A : B = 3 : 2 → A = ₹300, B = ₹200

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12. Pipes and Cisterns (Application)

Inlet Pipe: Fills the tank (positive work)

Outlet Pipe: Empties the tank (negative work)

Case	Formula
Fill pipe alone	1/x
Empty pipe alone	−1/y
Together	1/x − 1/y
Time to fill	1 / (1/x − 1/y)

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Example:

A pipe fills tank in 6 hr, outlet empties in 9 hr
Net 1 hr work = (1/6 − 1/9) = 1/18
So tank filled in 18 hours

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13. Leakage Case

If a tank should fill in x hrs but takes t hrs more due to leakage:

$$\text{Leak work in 1 hr} = \frac{1}{x} - \frac{1}{x + t}$$

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14. Alternate Days Work

If A and B work on alternate days:

• Find 2-day work = A + B

• Divide remaining work accordingly

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Example:

A = 5 days, B = 10 days
2-day work = (1/5 + 1/10) = 3/10
After 3 such pairs (6 days) → 9/10 done, 1/10 left → next A does 1 day = 1/5 → finish 2 more days → 7 days total

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15. Chain Rule in Time & Work

$$\text{More Men → Less Time (Inverse)}$$
$$\text{More Days → More Work (Direct)}$$

Use the man-days concept:

$$\text{Men} × \text{Days} × \text{Hours} = \text{Constant (Work)}$$

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Example:

10 men finish work in 12 days → 10×12 = 120 man-days
If 15 men work, time = 120/15 = 8 days

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16. Important Ratios

Type	Ratio Relation
Time	∝ 1 / Efficiency
Work	∝ Efficiency × Time
Wage	∝ Work done

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17. Shortcut Table

Type	Formula
Together time (A & B)	(A×B)/(A + B)
A’s work in x days	x/A
Remaining work	1 − x/A
Work done by A & B	(x/A + x/B)
Wages ratio	Work ratio
Pipe & Cistern	1/x ± 1/y

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18. Practice Questions

Q1. A can finish work in 10 days, B in 15 days. Together → ?
= (10×15)/(10 + 15) = 6 days

Q2. A works 2 days alone, then B joins; total 5 days → complete.
Find B’s time if A alone in 10 days.
Let total work = 30 units → A = 3/day, work done in 2 days = 6
Remaining 24 in 3 days → B = (24 − 9)/3 = 5 units/day → B alone = 30/5 = 6 days

Q3. Two pipes fill tank in 10 & 15 hr; third empties in 30 hr.
Effective = (1/10 + 1/15 − 1/30) = 1/6 → Tank filled in 6 hours

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19. Quick Revision Chart

Concept	Formula
Work formula	Work = Rate × Time
A’s 1-day work	1/n
Combined work	(1/x + 1/y)
Together time	xy/(x + y)
Efficiency	1/Time
Wages ratio	Work ratio
Pipe & Cistern	1/x ± 1/y

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20. Common Tricks

✅ Always assume total work = LCM of given days.
✅ Efficiency = Work ÷ Time.
✅ Wages distributed by work done.
✅ Pipes filling = +ve, emptying = −ve.
✅ Use “1-day work” method for simplicity.

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21. Real-life Concept Links

• Construction: man-days

• Pumping, storage, leak detection problems

• Work allocation & project management

• Productivity and wage calculation

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✅ In One Line:

“Work is done faster by higher efficiency or teamwork — Time × Rate = Work always holds true!”