P took Rs. 7500 loan at a certain rate of simple interest per annum and Rs. 4500 at 1% more. After three years he paid 1575 as simple interest. Find at what rate did he take Rs 4500 loan ?

Options

A. `6(1)/(4)%`
B. `6(2)/(3)%`
C. `%`
D. `5.5%`

Answer

To solve the problem step by step, we will first break down the information given and then apply the formula for Simple Interest.

### Step 1: Identify the amounts and rates
- P took a loan of Rs. 7500 at a certain rate of interest, which we will denote as R%.
- P also took another loan of Rs. 4500 at a rate of (R + 1)%.

### Step 2: Write the formula for Simple Interest
The formula for Simple Interest (SI) 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

### Step 3: Calculate the Simple Interest for both loans
1. For the first loan of Rs. 7500 at R% for 3 years:
\[ SI_1 = \frac{7500 \times R \times 3}{100} = \frac{22500R}{100} = 225R \]

2. For the second loan of Rs. 4500 at (R + 1)% for 3 years:
\[ SI_2 = \frac{4500 \times (R + 1) \times 3}{100} = \frac{13500(R + 1)}{100} = 135R + 135 \]

### Step 4: Total Simple Interest paid
According to the problem, the total Simple Interest paid after 3 years is Rs. 1575. Therefore, we can write:
\[ SI_1 + SI_2 = 1575 \]
Substituting the values we calculated:
\[ 225R + (135R + 135) = 1575 \]

### Step 5: Combine like terms
Combine the terms:
\[ 360R + 135 = 1575 \]

### Step 6: Solve for R
Now, isolate R:
\[ 360R = 1575 - 135 \]
\[ 360R = 1440 \]
\[ R = \frac{1440}{360} = 4 \]

### Step 7: Find the rate for the second loan
The rate for the second loan is:
\[ R + 1 = 4 + 1 = 5 \]

### Final Answer
The rate at which P took the Rs. 4500 loan is **5%**.

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