A sum of Rs.1200 becomes Rs.1560 at a rate of simple interest in 3 years. In how many years will the sum of Rs.800 amount to Rs.1120 at the same rate of simple interest?

Question

Prepp Team · Accepted Answer

Understanding Simple Interest Problems

This problem involves calculating simple interest in two parts. First, we need to find the rate of simple interest using the information given about the first sum of money. Then, we use this rate to find the time period required for a second sum of money to reach a specific amount.

Step-by-Step Calculation of Simple Interest Rate

Let's analyze the first scenario:

Principal (P₁) = Rs. 1200
Amount (A₁) = Rs. 1560
Time (T₁) = 3 years

The simple interest (SI₁) earned is the difference between the amount and the principal.

$\text{SI}_1 = \text{A}_1 - \text{P}_1$

$\text{SI}_1 = 1560 - 1200 = \text{Rs. } 360$

Now, we use the simple interest formula to find the rate (R). The formula is:

$\text{SI} = \frac{\text{P} \times \text{R} \times \text{T}}{100}$

Rearranging the formula to find R:

$\text{R} = \frac{\text{SI} \times 100}{\text{P} \times \text{T}}$

Substituting the values from the first scenario:

$\text{R} = \frac{360 \times 100}{1200 \times 3}$

$\text{R} = \frac{36000}{3600}$

$\text{R} = 10$

So, the rate of simple interest is 10% per annum.

Calculating Time Period for the Second Scenario

Now, let's analyze the second scenario using the rate we just found:

Principal (P₂) = Rs. 800
Amount (A₂) = Rs. 1120
Rate (R) = 10% per annum (same rate)

The simple interest (SI₂) earned in the second scenario is:

$\text{SI}_2 = \text{A}_2 - \text{P}_2$

$\text{SI}_2 = 1120 - 800 = \text{Rs. } 320$

We use the simple interest formula again, this time to find the time period (T₂):

$\text{SI} = \frac{\text{P} \times \text{R} \times \text{T}}{100}$

Rearranging the formula to find T:

$\text{T} = \frac{\text{SI} \times 100}{\text{P} \times \text{R}}$

Substituting the values from the second scenario:

$\text{T}_2 = \frac{320 \times 100}{800 \times 10}$

$\text{T}_2 = \frac{32000}{8000}$

$\text{T}_2 = 4$

Therefore, it will take 4 years for the sum of Rs. 800 to amount to Rs. 1120 at the same rate of simple interest.

Conclusion

The time required for Rs. 800 to amount to Rs. 1120 at a simple interest rate of 10% per annum is 4 years.

Revision Table: Simple Interest Concepts

Term	Definition	Formula Relation
Principal (P)	The initial sum of money invested or borrowed.	Base value for interest calculation.
Amount (A)	The total sum after adding simple interest to the principal.	A = P + SI
Simple Interest (SI)	Interest calculated only on the principal amount.	$\text{SI} = \frac{\text{P} \times \text{R} \times \text{T}}{100}$
Rate of Interest (R)	The percentage at which interest is calculated per period (usually per annum).	$\text{R} = \frac{\text{SI} \times 100}{\text{P} \times \text{T}}$
Time (T)	The duration for which the money is invested or borrowed.	$\text{T} = \frac{\text{SI} \times 100}{\text{P} \times \text{R}}$

Additional Information on Simple vs. Compound Interest

It's important to distinguish between simple interest and compound interest. Simple interest is calculated only on the original principal amount. Compound interest, however, is calculated on the principal amount and also on the accumulated interest of previous periods. This means compound interest grows faster than simple interest over time for the same principal and rate.

In this problem, the term "simple interest" is specified, so we strictly use the simple interest formulas. Problems involving compound interest require different formulas and calculations.