A cycle can be purchased on cash payment of Rs. 1500 but the same cycle can also be purchased on the cash down payment of Rs. 350 and rest can be paid in 3 equal annual installment of Rs. 400 for next 3 months. Find rate of S.I.?

Options

A. a. `23 (9)/17%`
B. b `17(9)/23%`
C. c. `(80)/3%`
D. d. none of these

Answer

To solve the problem of finding the rate of simple interest (SI) for the cycle purchase, we can follow these steps:

### Step 1: Determine the total cost of the cycle
The cycle can be purchased for a cash payment of Rs. 1500.

**Hint:** This is the total amount that needs to be compared with the installment payment plan.

### Step 2: Calculate the down payment and remaining amount
The down payment is Rs. 350. Therefore, the remaining amount to be paid after the down payment is:
\[ \text{Remaining Amount} = \text{Total Cost} - \text{Down Payment} = 1500 - 350 = 1150 \]

**Hint:** Subtract the down payment from the total cost to find out how much is left to pay.

### Step 3: Calculate the total amount paid through installments
The remaining amount of Rs. 1150 can be paid in 3 equal annual installments of Rs. 400 each. The total amount paid through installments is:
\[ \text{Total Installments} = 3 \times 400 = 1200 \]

**Hint:** Multiply the number of installments by the amount of each installment to find the total paid through installments.

### Step 4: Calculate the total amount paid with down payment
Now, add the down payment to the total installments to find the total amount paid:
\[ \text{Total Amount Paid} = \text{Down Payment} + \text{Total Installments} = 350 + 1200 = 1550 \]

**Hint:** Combine the down payment with the total installments to get the overall payment.

### Step 5: Calculate the extra amount paid
Now, we can find the extra amount paid compared to the cash price:
\[ \text{Extra Amount} = \text{Total Amount Paid} - \text{Total Cost} = 1550 - 1500 = 50 \]

**Hint:** Subtract the total cost from the total amount paid to find out how much extra was paid.

### Step 6: Calculate the principal amount for interest calculation
The principal amount (P) for the calculation of interest is the remaining amount after the down payment, which is Rs. 1150.

**Hint:** The principal is the amount that was financed through the installments.

### Step 7: Determine the time period in years
The time period (T) for the installments is 3 months, which can be converted into years:
\[ T = \frac{3}{12} = \frac{1}{4} \text{ years} \]

**Hint:** Convert months into years by dividing by 12.

### Step 8: Use the simple interest formula
The formula for simple interest is:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- SI = Simple Interest (Rs. 50)
- P = Principal (Rs. 1150)
- R = Rate of interest (unknown)
- T = Time in years (1/4)

### Step 9: Set up the equation
Substituting the known values into the formula:
\[ 50 = \frac{1150 \times R \times \frac{1}{4}}{100} \]

**Hint:** Rearranging the formula will help isolate the rate (R).

### Step 10: Solve for R
Rearranging gives:
\[ 50 = \frac{1150R}{400} \]
Multiplying both sides by 400:
\[ 20000 = 1150R \]
Now, divide by 1150:
\[ R = \frac{20000}{1150} \approx 17.39\% \]

**Hint:** Divide the total by the principal amount to find the rate.

### Final Answer
The rate of simple interest (R) is approximately **17.39%**.

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