In a business, B invests half the amount invested by A. After 6 months from the start of the business, C joins the business with an amount equal to twice of B's investment. After 8 months from the start of the business B withdraws completely from the business. If at the end of the years, C's share in the profit was ₹ 2460, what was the total profit received that year?

### Concept & Profit Sharing Ratio When investment amounts are given relative to one another, establish a base variable (e.g., let B's investment be $x$) to express all investments. Calculate equivalent capital by multiplying these expressions by their respective time periods. $$ \text{Profit Ratio} = (I_A \times T_A) : (I_B \times T_B) : (I_C \times T_C) $$ ### Step-by-Step Solution * **Given:** * Let A's investment be $2x$. Then B invests half of A, so B's investment = $x$. * C joins with twice of B's investment, so C's investment = $2x$. * A remains for the full 12 months. * C joins after 6 months, so C is invested for $12 - 6 = 6$ months. * B withdraws after 8 months, so B is invested for 8 months. * **Calculation:** * Equivalent Capital of A = $2x \times 12 = 24x$ * Equivalent Capital of B = $x \times 8 = 8x$ * Equivalent Capital of C = $2x \times 6 = 12x$ * Profit Ratio A : B : C = $24x : 8x : 12x = 6 : 2 : 3$ * Total Ratio Parts = $6 + 2 + 3 = 11$ parts. * C's share represents 3 parts out of 11. * We are given C's share is ₹2460. * $\frac{3}{11} \times \text{Total Profit} = 2460$ * $\text{Total Profit} = \frac{2460 \times 11}{3} = 820 \times 11 = 9020$ ### Exam Strategy & Shortcut Instead of algebraic variables ($x$), you can simply assume an arbitrary base number that is easy to halve and double. Let A = 2, then B = 1, and C = 2. Directly multiply by months: A(2×12=24), B(1×8=8), C(2×6=12). Ratio is 24:8:12 → 6:2:3. ### Common Pitfall A common error is confusing the joining time with the duration of investment. If C joins *after* 6 months in a 1-year business, the duration is $12 - 6 = 6$ months, not 12. ### Final Answer Therefore, the correct answer is **₹ 9020**.