The problem states:

The principal and the amount for a given period of time are in the ratio 4:5 at a given rate of simple interest. After 3 years, the ratio of the principal to the amount becomes 5:7 at the same rate of interest. We need to determine the rate of interest.

Solution:

1. Let the principal be $P$, and the amount after the first time period be $A$.

   • According to the first condition: $\frac{P}{A} = \frac{4}{5} \implies A = \frac{5}{4}P$

2. The simple interest (SI) after the first period: $\text{SI} = A - P = \frac{5}{4}P - P = \frac{P}{4}$

3. After 3 years, the ratio of the principal to the amount becomes $5:7$. Let $A'$ be the amount after 3 years.

   • According to the second condition: $\frac{P}{A'} = \frac{5}{7} \implies A' = \frac{7}{5}P$

4. The simple interest for 3 years: $\text{SI (3 years)} = A' - P = \frac{7}{5}P - P = \frac{2}{5}P$

5. Simple interest is given by: $\text{SI} = \frac{P \cdot R \cdot T}{100}$

6. Equate the SI for 3 years: $\frac{2}{5}P = \frac{P \cdot R \cdot 3}{100}$

7. Simplify to find $R$: $\frac{2}{5} = \frac{3R}{100} \implies 3R = \frac{200}{5} \implies 3R = 40 \implies R = \frac{40}{3} \approx 13.33\%$

Final Answer:

The rate of interest is approximately 13.33%.

Would you like further explanations or examples to clarify this process?

Related Questions:

1. What is the formula for simple interest and its components?
2. How can ratios be used to find financial relationships?
3. What is the significance of the time period in calculating simple interest?
4. How would the result change if the ratios were reversed (e.g., 5:4 and 7:5)?
5. Can this problem be solved graphically? If yes, how?

Tip:

Always cross-verify the units (time, rate, principal