Question

A sum was put at Suu 1 Had it been put at \(4 \%\) higher rate, it would have fetched Rs. 112 more. The sum is : (a) 1120 (b) 1400 (c) 1200 (d) 8000

Short answer

Answer: The sum invested is 1400 Rs.

Step-by-step solution

Define the variables

Let's denote the invested sum as P, original rate as R, and time as T.

Write down the given information

If the original rate was R, the increased rate would be R+4%. It is given that at this increased rate, the interest earned would be 112 Rs more.

Write the simple interest formulas for original and increased rates

Using the simple interest formula, we can write the interest earned at the original rate as: Interest1 = P × R × T Similarly, at the increased rate: Interest2 = P × (R + 0.04) × T

Create the equation based on given information

Since the difference in interests is 112 Rs, we can write: Interest2 - Interest1 = 112 Substituting the simple interest formulas, we get: (P × (R + 0.04) × T) - (P × R × T) = 112

Simplify the equation and solve for P

Now, we can simplify the equation: P × (R + 0.04) × T - P × R × T = 112 P × RT + P × 0.04T - P × RT = 112 P × 0.04T = 112 To find P, we can isolate it by dividing by 0.04T: P = 112 / (0.04T) Notice that we cannot find the exact value for P and T as we need more information. However, we can use the given options to find the correct answer.

Check the options

Let's plug each option into the equation: (a) P = 1120 1120 = 112 / (0.04T) 1120 × 0.04T = 112 44.8T = 112 T = 112 / 44.8 T ≈ 2.5 (not a whole number) (b) P = 1400 1400 = 112 / (0.04T) 1400 × 0.04T = 112 56T = 112 T = 112 / 56 T = 2 (a whole number) (c) P = 1200 1200 = 112 / (0.04T) 1200 × 0.04T = 112 48T = 112 T ≈ 2.33 (not a whole number) (d) P = 8000 8000 = 112 / (0.04T) 8000 × 0.04T = 112 320T = 112 T ≈ 0.35 (not a whole number) The only option that results in a whole number for T is (b) P = 1400. So, the sum invested is 1400 Rs.

Key concepts

Simple Interest Formula

When it comes to understanding investments and loans, the simple interest formula is a fundamental concept that everyone should know. Simple interest is a quick way of calculating the interest charge on a loan. Unlike compound interest, where interest is calculated on top of interest over time, simple interest is calculated only on the original amount, also known as the principal.

The formula for calculating simple interest is given by:
\( I = P \times R \times T \)
where \( I \) represents the interest earned or paid, \( P \) is the principal amount, \( R \) is the rate of interest per period, and \( T \) is the time the money is invested or borrowed for, usually given in years.

To give an educational improvement to the existing step by step solution for the given exercise, let's consider the following points. First and foremost, a clear understanding that the principal \( P \) is the main subject of our investigation. In our problem, we wish to find \( P \), given that with an interest rate 4% higher, the simple interest earned on \( P \) increases by Rs. 112. This scenario perfectly illustrates how sensitive simple interest is to changes in rate and time.

Interest Rate Calculation

The heart of any simple interest problem lies in calculating the interest rate. It's essential to distinguish between the nominal interest rate and the real interest rate. The nominal interest rate is the percentage increase in money you pay the lender for the use of the money you borrowed, or in the case of investments, it's the percentage increase in money the bank pays you for saving your money with them.

In the interest rate calculation for the original exercise, we're told that increasing the rate by 4% yields Rs. 112 more in interest. From this information, we can build an equation to solve for the principal. The steps provided in the solution involve substituting the simple interest formula into a difference of interest equation. However, we lacked the exact time period \( T \), which could have given us the exact principal \( P \) without the need to test the multiple-choice options.

Enhancing the original solution, it's important to clarify how we deal with percentages. The rate needs to be expressed as a decimal when used in formulas, which is why the 4% is written as 0.04 in the calculations. This common point of confusion can greatly affect the accuracy of students' calculations, and it’s crucial to emphasize the correct usage and conversion from percentage to decimals in interest calculations.