DEFINITION
One-Line Definition
The Chain Rule states that total work output is directly proportional to Men ($M$), Days ($D$), Hours per day ($H$), and Efficiency ($E$). Wages earned are divided strictly in direct proportion to the amount of work completed by each individual.
Golden Ratio: $\text{Wage Share} \propto \text{Work Done} = \text{Efficiency} \times \text{Time}$
CORE INTUITION ⚡
The Man-Hour Energy Model
Think of work as total Man-Hours of human effort required:
- • 10 Men working 8 hrs/day for 5 days = $10 \times 8 \times 5 = \mathbf{400\text{ Man-Hours}}$
- • If paid ₹40,000 total wage $\implies 1\text{ Man-Hour} = \mathbf{₹100}$
- • Worker contributing 50 Man-Hours gets $50 \times 100 = \mathbf{₹5,000}$
Wages are paid for WORK COMPLETED, never merely for spending time on site!
💡 WHY THIS CONCEPT MATTERS & REAL-LIFE APPLICATIONS
Man-Days and Wage Distribution problems combine Time & Work with Ratio & Proportion. They appear frequently in CAT, CET, SNAP, and NMAT. Click below to explore connected Quant Vault topics:
Where Is This Used in Real Life & Business?
🚜 Construction Cost Estimation
💼 Freelance & Agency Payroll
🏭 Factory Shift Management
🤝 Joint Project Contracts
📐 KEY FORMULAS & CHAIN RULE EQUATIONS
MASTER CHAIN RULE FORMULA
$$\frac{M_1 \cdot D_1 \cdot H_1 \cdot E_1}{W_1} = \frac{M_2 \cdot D_2 \cdot H_2 \cdot E_2}{W_2}$$
$M$ = Workers | $D$ = Days | $H$ = Hours/Day | $E$ = Efficiency | $W$ = Work Output
💰 Wage Distribution Rules
Rule 1: Equal Time Worked
If workers $A, B, C$ work for the same number of days:
Wage Ratio ($W_A : W_B : W_C$) = Efficiency Ratio ($E_A : E_B : E_C$)
Rule 2: Different Times Worked
If workers $A, B, C$ work for different days ($T_A, T_B, T_C$):
Wage Ratio = Work Contribution Ratio = $(E_A \cdot T_A) : (E_B \cdot T_B) : (E_C \cdot T_C)$
Rule 3: Group Work Equivalence (Men, Women, Children)
If $x$ Men = $y$ Women = $z$ Children, find individual efficiency ratio:
$$E_M : E_W : E_C = \frac{1}{x} : \frac{1}{y} : \frac{1}{z}$$
🧮 INTERACTIVE WAGE & CHAIN RULE CALCULATOR
Calculate proportional wage shares by choosing to enter either Solo Completion Days or Direct Worker Efficiencies & Days Worked:
📝 SOLVED EXAMPLES (LEVEL 0 TO ADVANCED)
EASY • EXAMPLE 1
15 workers complete a project in 12 days working 8 hours a day. How many days will 20 workers take working 6 hours a day for the same work?
Solution (Chain Rule Formula):
Since work output is identical ($W_1 = W_2$):
$$M_1 \times D_1 \times H_1 = M_2 \times D_2 \times H_2$$
$$15 \times 12 \times 8 = 20 \times D_2 \times 6$$
$$1440 = 120 \times D_2 \implies D_2 = \frac{1440}{120} = \mathbf{12\text{ days}}$$
Answer: 12 days
MEDIUM • EXAMPLE 2
4 men or 6 women can complete a work in 20 days. How many days will 2 men and 9 women take working together?
Solution (Efficiency Equivalence):
Work done by $4\text{ Men} = 6\text{ Women} \implies 1\text{ Man} = 1.5\text{ Women}$.
Convert combined team into Women equivalence:
$$2\text{ Men} + 9\text{ Women} = 2(1.5\text{ Women}) + 9\text{ Women} = 3 + 9 = \mathbf{12\text{ Women}}$$
Now apply Chain Rule ($M_1 \times D_1 = M_2 \times D_2$):
$$6\text{ Women} \times 20\text{ days} = 12\text{ Women} \times D_2$$
$$120 = 12 D_2 \implies D_2 = \mathbf{10\text{ days}}$$
Answer: 10 days
HARD • EXAMPLE 3
A and B contract a job for ₹15,000. A alone can do it in 10 days, and B alone in 15 days. With the help of C, all three complete the job together in 4 days. What is C's share of the wage?
Solution (LCM & Efficiency Split):
Total Work = LCM(10, 15, 4) = 60 units
• A's Rate = $60 / 10 = \mathbf{6\text{ units/day}}$
• B's Rate = $60 / 15 = \mathbf{4\text{ units/day}}$
• Combined Rate (A + B + C) = $60 / 4 = \mathbf{15\text{ units/day}}$
• C's Rate = $15 - (6 + 4) = \mathbf{5\text{ units/day}}$
Since all three worked for the same 4 days, wage ratio equals efficiency ratio:
$$A : B : C = 6 : 4 : 5 \quad (\text{Total Parts} = 6 + 4 + 5 = 15)$$
$$\text{C's Wage Share} = \frac{5}{15} \times ₹15,000 = \mathbf{₹5,000}$$
Answer: ₹5,000
⚠️ COMMON MISTAKES TO AVOID
❌ Mistake 1: Splitting wages inversely based on solo days taken
Never divide total money by individual days taken directly! Wages are proportional to efficiency (or work done), which is the reciprocal of solo days taken.
❌ Mistake 2: Forgetting to account for different working periods
If Worker A works for 10 days while Worker B leaves after 3 days, you MUST multiply their daily efficiency by their actual days worked before finding the wage ratio.
❌ Mistake 3: Putting Work ($W$) in numerator of Chain Rule
Work done goes in the denominator ($\frac{M_1 D_1 H_1}{W_1} = \frac{M_2 D_2 H_2}{W_2}$). Remember: More work requires MORE man-hours directly!
🚀 CAT & MBA CET SPEED SHORTCUTS
⚡ Shortcut 1: The "OR" vs "AND" Group Trick
For questions like "a Men OR b Women do work in D days, find time for c Men AND d Women":
$$\text{Days Taken} = \frac{D}{\frac{c}{a} + \frac{d}{b}}$$
Example: 4 Men OR 6 Women in 20 days. For 2 Men AND 9 Women $\implies \frac{20}{\frac{2}{4} + \frac{9}{6}} = \frac{20}{0.5 + 1.5} = \frac{20}{2} = 10\text{ days}$ (Done in 5 seconds!).
⚡ Shortcut 2: Fractional Wage Fraction
$$\text{Individual Wage} = \left(\frac{\text{Work Done by Individual}}{\text{Total Work Completed}}\right) \times \text{Total Contract Amount}$$
🎯 PRACTICE QUESTIONS (DIFFICULTY LEVEL-WISE)
EASY • LEVEL 0
CHAIN RULE BASIC
If 18 contractors build a 120m wall in 10 days working 6 hours a day, how many days will 24 contractors take to build a 160m wall working 8 hours a day?
Chain Rule Formula:
$$\frac{M_1 \cdot D_1 \cdot H_1}{W_1} = \frac{M_2 \cdot D_2 \cdot H_2}{W_2}$$
Substitute values:
$$\frac{18 \times 10 \times 6}{120} = \frac{24 \times D_2 \times 8}{160}$$
$$\frac{1080}{120} = \frac{192 \cdot D_2}{160} \implies 9 = 1.2 \cdot D_2 \implies D_2 = \frac{9}{1.2} = \mathbf{7.5\text{ days}}$$
Correct Answer: 7.5 days
MODERATE • LEVEL 1
GROUP EQUIVALENCE
4 men or 6 women can complete a work in 20 days. How many days will 2 men and 9 women take working together?
Solution Steps:
Work done by $4\text{ Men} = 6\text{ Women} \implies 1\text{ Man} = 1.5\text{ Women}$.
Convert combined team into Women equivalence:
$$2\text{ Men} + 9\text{ Women} = 2(1.5\text{ Women}) + 9\text{ Women} = 3 + 9 = \mathbf{12\text{ Women}}$$
Apply Chain Rule ($M_1 \times D_1 = M_2 \times D_2$):
$$6\text{ Women} \times 20\text{ days} = 12\text{ Women} \times D_2$$
$$120 = 12 D_2 \implies D_2 = \mathbf{10\text{ days}}$$
Correct Answer: 10 days
HARD • LEVEL 2
WAGE SPLIT & EARLY DEPARTURE
A can complete a project in 12 days and B in 18 days. They worked together for 4 days, after which A left and B finished the remaining work alone. If the total contract wage is ₹9,000, how much money should A and B receive respectively?
Solution Steps:
Total Work = LCM(12, 18) = 36 units.
A's Efficiency = $36 / 12 = \mathbf{3\text{ units/day}}$.
B's Efficiency = $36 / 18 = \mathbf{2\text{ units/day}}$.
A worked for 4 days $\implies \text{A's Work} = 3 \times 4 = \mathbf{12\text{ units}}$.
Remaining Work = $36 - 12 = \mathbf{24\text{ units}}$ (completed entirely by B).
$$\text{A's Share} = \left(\frac{12}{36}\right) \times ₹9,000 = \mathbf{₹3,000}$$
$$\text{B's Share} = \left(\frac{24}{36}\right) \times ₹9,000 = \mathbf{₹6,000}$$
Correct Answer: A = ₹3,000 | B = ₹6,000
❓ FREQUENTLY ASKED QUESTIONS
Q: What is the difference between Man-Days and Man-Hours in Chain Rule?
Man-Days represents total human effort measured in days ($M \times D$), assuming a constant daily work shift. When daily shift hours ($H$) vary, we use Man-Hours ($M \times D \times H$) to measure total energy input accurately.
Q: Why are wages divided by Work Done and NOT by individual days taken?
Individual solo days are inversely proportional to efficiency. A faster worker takes FEWER days because they do more work per unit time. Therefore, contract wages must be split according to actual work units completed ($\text{Efficiency} \times \text{Days Worked}$), never by dividing money directly by individual days.
Q: How do you solve group equivalence problems (e.g., 4 Men OR 6 Women)?
First equate their work output ($4M = 6W \implies 1M = 1.5W$). Then convert the combined target group (e.g., $2M + 9W$) into a single worker type ($3W + 9W = 12W$) before applying the $M_1 D_1 = M_2 D_2$ formula.
Q: How are wages split if a worker leaves before project completion?
Calculate the total units of work in the job (via LCM of solo times). Determine the exact number of units completed by the departing worker while present. Their wage share equals $\left(\frac{\text{Units Completed}}{\text{Total Job Units}}\right) \times \text{Total Contract Amount}$.