A person invests money in three different schemes for $6$ years, $10$ years and $12$ years at $10$ percent, $12$ percent and $15$ percent simple interest respectively. At the completion of each scheme, he gets the same interest. The ratio of his investments is

### Concept & Equating Multiple Interests When the simple interest earned from different investments is identical, the product of Principle, Rate, and Time for each investment is a constant. This inversely links the principal to the product of Rate and Time. ### Step-by-Step Solution - Let the investments (principals) in the three schemes be $P_1$, $P_2$, and $P_3$. - The problem states that the simple interest earned in all three cases is the same. - $SI_1 = SI_2 = SI_3$ - $\frac{P_1 \times 10 \times 6}{100} = \frac{P_2 \times 12 \times 10}{100} = \frac{P_3 \times 15 \times 12}{100}$ - $60P_1 = 120P_2 = 180P_3$ - Simplify by dividing all terms by $60$: - $1P_1 = 2P_2 = 3P_3$ - To find the continuous ratio, divide by the LCM of coefficients (LCM of $1, 2, 3$ is $6$): - $\frac{P_1}{6} = \frac{2P_2}{6} = \frac{3P_3}{6}$ - $\frac{P_1}{6} = \frac{P_2}{3} = \frac{P_3}{2}$ - The ratio of the investments is $6 : 3 : 2$. ### Exam Strategy & Shortcut If $x \cdot P_1 = y \cdot P_2 = z \cdot P_3$, the ratio $P_1 : P_2 : P_3$ is quickly found as $\frac{1}{x} : \frac{1}{y} : \frac{1}{z}$. Here, the products are $60, 120, 180$. The ratio is $\frac{1}{60} : \frac{1}{120} : \frac{1}{180} = 6 : 3 : 2$. ### Common Pitfall A frequent error is writing the ratio directly as the coefficients (e.g., $1:2:3$), which is mathematically incorrect when setting expressions to an equal constant. ### Final Answer Therefore, the correct answer is **$6 : 3 : 2$**.