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Time & Work

The most frequent topic. Master the 'LCM Method' (Total Work = LCM of Days) and 'MDH Formula' (Man-Days-Hours).

1. The LCM Method (Unit Work)

Forget fractions ( $1/x$ ). Use Units.

Method: Assume Total Work = LCM of given days.
Efficiency (Per Day Work): $\text{Total Work} / \text{Days}$ .

Example:

Q: A takes 10 days, B takes 15 days. Together?

Solution: Total Work = LCM(10,15) = 30 units.
A = 3 u/day. B = 2 u/day. Total = 5 u/day.
Time =

30/5 = 6

days.

2. The MDH Formula (Chain Rule)

Comparing two groups of workers.

Formula: $\frac{M_1 D_1 H_1}{W_1} = \frac{M_2 D_2 H_2}{W_2}$ .
M=Men, D=Days, H=Hours, W=Work (Output).

Example:

Q: 20 men build 10 walls in 5 days. 30 men, 10 days, how many walls?

Solution:

\frac{20 \times 5}{10} = \frac{30 \times 10}{W}

10 = 300/W \implies W = 30

walls.

3. Efficiency Ratio

If A is $n$ times efficient as B.

Eff Ratio: $A:B = n:1$ .
Time Ratio: $A:B = 1:n$ (Inverse).
Combined: Eff $(n+1)$ .

Example:

Q: A is 3 times faster than B. B takes 60 days.

Solution: Ratio Eff A:B = 3:1. Total Eff = 4? No.
Total Work =

B \times 60 = 1 \times 60 = 60

.
Together =

60 / (3+1) = 15

days.

4. Alternate Days (Cycle Method)

Working on turns.

Logic: Find work done in 1 cycle (usually 2 days: A then B).
Divide Total Work by Cycle Work. Handle remainder manually.

Example:

Q: A(10d), B(15d). A starts, alternate days.

Solution: TW=30. A(3), B(2). Cycle(2 days)=5 units.

30/5 = 6

cycles.
Total Days =

6 \times 2 = 12

days.

5. Work & Wages

Money is distributed based on Work Done (not just Time).

Rule: Wage Ratio = Efficiency $\times$ Time.
If Time same, Wage $\propto$ Efficiency.

Example:

Q: A does 1/3 work, B does 2/3. Total Rs 300.

Solution: Ratio 1:2. A gets

1/3 (300) = 100

. B gets 200.

6. Pipes & Cisterns (Negative Work)

Same as Time & Work, but Leaks are Negative.

Inlet: Positive Efficiency (+).
Outlet/Leak: Negative Efficiency (-).

Example:

Q: A fills in 10h, B empties in 15h.

Solution: TW=30. A=+3, B=-2. Net=+1.
Time =

30/1 = 30

hours.

7. The 'Leaving Before' Hack

When someone leaves $n$ days BEFORE completion.

Hack: Do not subtract. ADD their work.
$Total Work + (Person \times Days)$ . Divide by Total Efficiency.

Example:

Q: A(10), B(15). A leaves 2 days before end.

Solution: TW=30. Eff(3,2). A's 2 days work =

3\times2=6

.
New TW =

30+6=36

. Time =

36/5 = 7.2

days.

8. The 'And/Or' Shortcut

For problems like '3 Men OR 5 Women'.

Formula: $\text{Days} = \frac{\text{Given Days}}{ \frac{M_2}{M_1} + \frac{W_2}{W_1} }$ .
Faster than converting Men to Women.

Example:

Q: 3M or 5W take 43 days. 5M and 6W take?

Solution:

D = 43 / (5/3 + 6/5) = 43 / (25+18)/15

D = 43 \times 15 / 43 = 15

days.