Question

Akul lent Rs. 6000 to Bakul for 2 years and Rs. 1500 to camlin for 4 years and received altogether from both is. 900 as simple interest. The rate of interest is : (a) \(4 \%\) (b) \(8 \%\) (c) \(10 \%\) (d) \(5 \%\)

Short answer

Answer: 5%

Step-by-step solution

Identify the knowns and unknown variables

We need to find the rate of interest (R). We know the following information: - Akul lent Rs. 6000 to Bakul for 2 years (Principal 1: P1 = 6000, Time 1: T1 = 2) - Akul lent Rs. 1500 to camlin for 4 years (Principal 2: P2 = 1500, Time 2: T2 = 4) - Total interest received from both loans is Rs. 900 (Total Interest: I_total = 900)

Write the equation for simple interest for both loans

Simple Interest Formula: Interest = Principal x Rate x Time For Bakul's loan: Interest 1 (I1) = P1 x R x T1 For camlin's loan: Interest 2 (I2) = P2 x R x T2

Write the equation for the total interest

Total Interest = Interest from Bakul's loan + Interest from camlin's loan I_total = I1 + I2 We know the total interest is Rs. 900. So the equation becomes: 900 = P1 x R x T1 + P2 x R x T2

Substitute the known values into the equation

Now, we can substitute the values of P1, T1, P2, and T2 into the equation from Step 3: 900 = 6000 x R x 2 + 1500 x R x 4

Simplify and solve for the rate of interest (R)

We can simplify the equation: 900 = 12000R + 6000R Combine like terms: 900 = 18000R Now, solve for R: R = 900 / 18000 R = 0.05 To express R as a percentage, multiply by 100: Rate of interest = R x 100 = 0.05 x 100 = 5% So, the rate of interest is 5% (Answer: (d) 5%).

Key concepts

Rate of Interest Calculation

The rate of interest is essentially the percentage that determines how much interest one earns or owes on a sum of money over a period. It acts like the price of borrowing money or the reward for lending it. Calculating the rate of interest involves understanding the relationship between the interest earned, the principal amount, and the time period.
In the exercise above, the interest rate was determined by analyzing two separate loans with different principals and times. Each loan contributes to the total interest earned, and looking at them both gives us clues about the rate of interest.

• First, identify all known variables such as principal amounts and time periods for each loan.
• Determine the total interest accumulated from both loans combined.
• Use the information to solve for the rate, by making a combined equation as shown in the solution step.

This calculated rate gives you an annual percentage, showing you the annual cost or gain of borrowing or lending money.

Principal and Time in Loans

When dealing with loans, two of the key factors to consider are the principal and the time period. The principal is the initial amount of money that is either borrowed or lent. Time, usually measured in years, is how long the money is borrowed or lent for.
In this exercise, there were two principal amounts involved in different loans: Rs. 6000 loaned for 2 years, and Rs. 1500 loaned for 4 years.

• The principal amount directly affects the interest calculation: larger principals tend to generate more interest.
• Time influences how interest accumulates; longer durations can lead to higher interest amounts over time.

Understanding these helps to comprehend how they interact in calculating overall interest. The larger each factor is, the larger the interest amount will be, assuming a constant rate of interest.

Interest Formula

The simple interest formula is a straightforward way to calculate interest based on the linear relationship between the principal, rate, and time. The formula is:
\[I = P \times R \times T\]
Where:

• \( I \) is the interest earned or paid.
• \( P \) is the principal or the amount initially borrowed or lent.
• \( R \) is the rate of interest expressed as a decimal.
• \( T \) is the time period for which the money is borrowed or lent, in years.

For example, if you know the principal, rate, and time, you can easily calculate the interest. Conversely, if you know the total interest and the other two variables, as in this exercise, you can solve for the missing rate.
Using the formula correctly helps you to easily anticipate or verify any interest payments or earnings across different financial scenarios.