Question

A sonata watch is sold for Rs. 440 cash or for Rs. 200 cash down payment together with Rs. 244 to be paid after one month. Find the rate of interest charged in the instalment scheme : (a) \(10 \%\) (b) \(15 \%\) (c) \(20 \%\) (d) \(25 \%\)

Short answer

a) \(10 \%\) b) \(15 \%\) c) \(20 \%\) d) \(25 \%\) Answer: (c) \(20 \%\).

Step-by-step solution

Calculate the principal amount and interest

After the down payment of Rs. 200, the remaining amount is the principal for the loan. So, the principal amount (P) will be: Principal Amount (P) = Total cost of watch - Down payment = 440 - 200 = 240 Rs. Next, we need to calculate the interest (I), which is the extra amount the customer must pay after one month (244 Rs) minus the remaining principal amount (240 Rs): Interest (I) = Amount to pay after one month - Principal Amount = 244 - 240 = 4 Rs.

Convert the time period to years

Since the installment is made after one month, we need to convert this time period to years, as our simple interest formula requires the time in years. Time Period (T) = 1 month = 1/12 years

Substitute values into the simple interest formula and solve for R

Now, we plug the values P, I, and T into the simple interest formula and solve for R. I = (P * R * T) / 100 4 = (240 * R * (1/12)) / 100 Now we solve for R: 4 * 12 * 100 = 240 * R 4800 = 240 * R R = 4800 / 240 R = 20 So, the rate of interest charged in the installment scheme is 20%. Answer: (c) \(20 \%\).

Key concepts

Interest Calculation

Interest calculation is a fundamental concept that helps determine the additional amount paid on borrowed money. In simple terms, interest is the cost of borrowing money or the gain from investing it. The formula for simple interest is:\[ I = \frac{P \times R \times T}{100} \]Where:

• \( I \) = Interest
• \( P \) = Principal
• \( R \) = Rate of interest
• \( T \) = Time period in years

In the case of the sonata watch, the interest calculation was based on the extra payment over the principal amount during one month. By calculating it as shown in the formula, we determine the additional payment required beyond the principal amount, in this instance, Rs. 4.

Installment Payment

Installment payments allow customers to spread the cost of a purchase over time. This is particularly useful for expensive items. In our exercise, the installment payment method consists of an initial cash down payment followed by a later payment. This kind of plan helps make large purchases more manageable by breaking them into smaller payments. For the sonata watch, an initial amount of Rs. 200 was required, with the remaining balance of Rs. 244 due after one month. Essentially, the buyer makes a smaller upfront payment and pays the balance monthly, which is where interest is typically applied. The installment scheme in this problem implies the consumer benefits from immediate ownership, while the seller uses the interest as a charge for deferred payment.

Time Conversion

Understanding time conversion in interest calculations is crucial because interest rates often reference annual periods, yet the actual time frame may differ. When dealing with periods shorter than a year, like a month or a fortnight, you must convert these to a year fraction. The formula requires the time in years to ensure consistent and accurate calculations. For one month, the conversion would be:\[ T = \frac{1}{12}\, \text{years} \]This conversion allows us to use the simple interest formula correctly, translating the month into a proportionate segment of the year. In calculating the interest on the remaining amount after paying the down payment, the time frame of one month was converted to \( \frac{1}{12} \) year to facilitate accurate interest assessment.

Rate of Interest

The rate of interest is key to assessing the cost of credit or the benefit from investments. It's usually expressed as a percentage of the principal amount. With simple interest, it's often calculated annually, even when the period may be shorter.To find the rate of interest, we rearrange the simple interest formula:\[ R = \frac{100 \times I}{P \times T} \]In the exercise, solving the equation resulted in a calculation of a 20% rate of interest. This rate reflects the additional cost over the principal that the consumer must pay for using the installment plan. Calculating the correct rate of interest is important for buyers to understand how much more they are paying over the original price and for sellers to ensure they get a fair return on the deferred payment option.