If 80 lamps can be lighted, 5 hours per day for 10 days for ₹ 21.25, then the number of lamps, which can be lighted 4 hours daily for 30 days, for ₹ 76.50, is

### Concept & Extended Chain Rule This problem uses the extended chain rule where cost represents the "work done" or output of the system. The total cost is directly proportional to the number of lamps, days, and hours per day. We equate the ratio of total effort to total cost for both scenarios. $$ \frac{M_1 D_1 H_1}{W_1} = \frac{M_2 D_2 H_2}{W_2} $$ ### Step-by-Step Solution * Identify the variables for the first scenario: $M_1 = 80 \text{ lamps}$, $H_1 = 5 \text{ hours/day}$, $D_1 = 10 \text{ days}$, $W_1 = \text{₹ } 21.25$. * Identify the variables for the second scenario: $M_2 = x \text{ lamps}$, $H_2 = 4 \text{ hours/day}$, $D_2 = 30 \text{ days}$, $W_2 = \text{₹ } 76.50$. * Set up the chain rule equation: $$ \frac{80 \times 10 \times 5}{21.25} = \frac{x \times 30 \times 4}{76.50} $$ * Simplify the numerators first: $\frac{4000}{21.25} = \frac{120x}{76.50}$. * Isolate $x$ by cross-multiplying: $x = \frac{4000 \times 76.50}{120 \times 21.25}$. * Simplify the expression. Divide $4000$ by $120$: $\frac{4000}{120} = \frac{100}{3}$. * The equation becomes $x = \frac{100 \times 76.50}{3 \times 21.25} = \frac{7650}{63.75}$. * Divide to find the final value: $7650 \div 63.75 = 120$. ### Exam Strategy & Shortcut To handle the decimals quickly, multiply the denominator and numerator by 100 or a factor that makes them whole numbers. For instance, notice that $21.25 \times 4 = 85$ and $76.50 \times 2 = 153$. Utilizing fractions like $\frac{85}{4}$ and $\frac{153}{2}$ instead of decimals often speeds up cancellation significantly without needing a calculator. ### Common Pitfall Treating the cost (₹) as just another effort multiplier in the numerator instead of correctly placing it in the denominator as the "work" or "output" equivalent. ### Final Answer Therefore, the correct answer is **120**.