Sam for 7 years invested Rs. 55000 in PSB. He also invested Rs. 35000 in HDFC for 4 years. At the end he received Rs. 5000 from both banks as simple interest. What must have been rate of interest?

To solve this problem, we need to find the rate of interest that Sam received from both PSB and HDFC banks. Since the total simple interest received from both banks is given as Rs. 5000, we can set up equations for each investment and solve for the interest rates.

### Given:
1. Investment in PSB: \( P_1 = 55000 \) for \( T_1 = 7 \) years.
2. Investment in HDFC: \( P_2 = 35000 \) for \( T_2 = 4 \) years.
3. Total interest received: \( SI_1 + SI_2 = 5000 \).

### Formula for Simple Interest:
The formula for calculating simple interest is:

\[
SI = \frac{P \times R \times T}{100}
\]

where:
- \( SI \) = Simple Interest
- \( P \) = Principal amount
- \( R \) = Rate of interest (per annum)
- \( T \) = Time (in years)

Let's denote the interest rate by \( R_1 \) for PSB and \( R_2 \) for HDFC.

### Calculating the Simple Interest from each investment:

1. **For PSB:**
   \[
   SI_1 = \frac{55000 \times R_1 \times 7}{100} = \frac{385000 R_1}{100}
   \]
   \[
   SI_1 = 3850 R_1
   \]

2. **For HDFC:**
   \[
   SI_2 = \frac{35000 \times R_2 \times 4}{100} = \frac{140000 R_2}{100}
   \]
   \[
   SI_2 = 1400 R_2
   \]

### Total Simple Interest:
\[
SI_1 + SI_2 = 5000
\]
\[
3850 R_1 + 1400 R_2 = 5000
\]

### Solving the Equation:
To find the rate of interest, let's assume \( R_1 = R_2 = R \) (since the exact distribution between the two banks isn’t specified):

\[
3850 R + 1400 R = 5000
\]
\[
5250 R = 5000
\]
\[
R = \frac{5000}{5250}
\]
\[
R \approx 0.952\% \text{ per annum}
\]

So, the approximate rate of interest from both banks must have been **0.952% per annum** if the rates are assumed to be equal.

### Final Answer:
**The rate of interest must have been approximately 0.952% per annum.**

Would you like more details on any part of this calculation or have any other questions?

### Related Questions:
1. How would the rate of interest change if different rates were offered by each bank?
2. What would be the total interest if Sam had invested Rs. 75000 in PSB instead?
3. How would the calculation change if the investments were compounded annually instead of being simple interest?
4. What would be the interest earned from HDFC alone at a rate of 1% per annum?
5. How much would Sam need to invest in HDFC to earn Rs. 5000 in interest over 4 years at 0.952%?

### Tip:
When dealing with simple interest problems, breaking down each investment's interest helps isolate variables and find solutions efficiently.

Learn how to calculate the rate of interest for investments of Rs. 55000 in PSB for 7 years and Rs. 35000 in HDFC for 4 years, with a total simple interest of Rs. 5000. This problem uses the simple interest formula and algebraic equation-solving techniques. Suitable for students in Grades 9-10.

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