A man invests a certain sum of money at 6% p.a. simple interest and another sum at 7% p.a. simple interest. His income from interest after 2 years was ₹ 354. One-fourth of the first sum is equal to one-fifth of the second sum. The total sum invested was:

### Concept & Proportions in Investments When a ratio between two principal sums is given, we can express both sums in terms of a single variable ($x$). We then use the total interest formula to solve for $x$ and determine the total investment. Total $SI$ = $SI_1 + SI_2$ ### Step-by-Step Solution * **Given:** * Rate 1 = $6\%$, Rate 2 = $7\%$, Time ($T$) = 2 years. Total Interest = ₹ 354. * Relation: $\frac{1}{4}$ of First Sum ($P_1$) = $\frac{1}{5}$ of Second Sum ($P_2$). * **Calculation:** Let the sums be $P_1$ and $P_2$. * $\frac{P_1}{4} = \frac{P_2}{5}$ implies that $P_1 = 4x$ and $P_2 = 5x$. * The total sum invested = $P_1 + P_2 = 4x + 5x = 9x$. * Express the total interest equation: * $\frac{4x \times 6 \times 2}{100} + \frac{5x \times 7 \times 2}{100} = 354$ * Simplify the numerators: * $\frac{48x}{100} + \frac{70x}{100} = 354$ * Combine the terms: * $\frac{118x}{100} = 354$ * Solve for $x$: * $118x = 35400 \implies x = \frac{35400}{118} = 300$. * Calculate total sum: * Total Sum = $9x = 9 \times 300 = 2700$. ### Exam Strategy & Shortcut Once you establish the ratio $P_1:P_2 = 4:5$, you know the total sum must be a multiple of $4+5=9$. Check the options: (a) 2600 is not divisible by 9. (b) 2700 is divisible by 9. (c) 2880 is divisible by 9. (d) 2900 is not divisible by 9. Test 2700: $P_1 = 1200$, $P_2 = 1500$. Interest = $(1200 \times 0.06 \times 2) + (1500 \times 0.07 \times 2) = 144 + 210 = 354$. It matches perfectly. ### Common Pitfall A common mistake is reading "one-fourth of the first sum is equal to one-fifth of the second sum" and incorrectly assigning the ratio as $P_1:P_2 = 5:4$ instead of the correct $4:5$. ### Final Answer Therefore, the correct answer is **₹ 2700**.