A = Rs. 6000 is invested at simple interest for 5 years and annual rate of interest is 15 percent. B = Rs. 5000 is invested at simple interest for 2 years and annual rate of interest is 20 percent. What is the ratio of interests earned in A and B respectively?
9 ∶ 4
Calculating Simple Interest and Ratio
This problem involves calculating the simple interest earned on two different investments and then finding the ratio of these interests. Simple interest is calculated using the formula:
$$ \text{Simple Interest (SI)} = \frac{\text{Principal (P)} \times \text{Rate (R)} \times \text{Time (T)}}{100} $$
Where:
- P is the principal amount (the initial investment).
- R is the annual rate of interest in percent.
- T is the time period in years.
Simple Interest Calculation for Scenario A
In scenario A:
- Principal (P\(_A\)) = Rs. 6000
- Annual Rate (R\(_A\)) = 15 percent
- Time (T\(_A\)) = 5 years
Using the simple interest formula:
$$ \text{SI}_A = \frac{P_A \times R_A \times T_A}{100} $$
$$ \text{SI}_A = \frac{6000 \times 15 \times 5}{100} $$
$$ \text{SI}_A = \frac{6000 \times 75}{100} $$
$$ \text{SI}_A = 60 \times 75 $$
$$ \text{SI}_A = 4500 $$
So, the simple interest earned in scenario A is Rs. 4500.
Simple Interest Calculation for Scenario B
In scenario B:
- Principal (P\(_B\)) = Rs. 5000
- Annual Rate (R\(_B\)) = 20 percent
- Time (T\(_B\)) = 2 years
Using the simple interest formula:
$$ \text{SI}_B = \frac{P_B \times R_B \times T_B}{100} $$
$$ \text{SI}_B = \frac{5000 \times 20 \times 2}{100} $$
$$ \text{SI}_B = \frac{5000 \times 40}{100} $$
$$ \text{SI}_B = 50 \times 40 $$
$$ \text{SI}_B = 2000 $$
So, the simple interest earned in scenario B is Rs. 2000.
Finding the Ratio of Interests Earned
We need to find the ratio of the interests earned in A and B respectively, which is SI\(_A\) : SI\(_B\).
$$ \text{Ratio} = \text{SI}_A : \text{SI}_B $$
$$ \text{Ratio} = 4500 : 2000 $$
To simplify the ratio, we can divide both numbers by their greatest common divisor. Both numbers can be divided by 100:
$$ 4500 \div 100 = 45 $$
$$ 2000 \div 100 = 20 $$
The ratio becomes 45 : 20. Both 45 and 20 are divisible by 5:
$$ 45 \div 5 = 9 $$
$$ 20 \div 5 = 4 $$
The simplified ratio is 9 : 4.
Therefore, the ratio of interests earned in A and B respectively is 9 ∶ 4.
Revision Table: Simple Interest Calculation Summary
| Scenario | Principal (P) | Rate (R) | Time (T) | Simple Interest (SI) |
|---|---|---|---|---|
| A | Rs. 6000 | 15% | 5 years | Rs. 4500 |
| B | Rs. 5000 | 20% | 2 years | Rs. 2000 |
Additional Information: Understanding Simple Interest
Simple interest is a basic and quick method of calculating the interest charge on a loan or investment. It is calculated only on the principal amount, not on any accumulated interest. This is in contrast to compound interest, where interest is calculated on the initial principal and also on the accumulated interest from previous periods. Simple interest is often used for short-term loans or specific types of investments where the interest is paid out periodically rather than reinvested.
- Principal: The initial sum borrowed or invested.
- Interest Rate: The percentage charged or earned on the principal per period (usually annually).
- Time: The duration for which the principal is borrowed or invested.
- Simple Interest Amount: The total interest earned or paid over the time period.
- Total Amount: The sum of the principal and the simple interest earned. Total Amount = P + SI.
Understanding simple interest is fundamental before moving on to more complex concepts like compound interest or annuities.
Important Questions from Simple Interest
How much time will it take for an amount of Rs. 450 to yield Rs. 81 as interest at 4.5% per annum of simple interest ?
Nirav and Mehul borrowed Rs.4000 and Rs.5000 respectively for 2.5 years at the rate of x% per annum. Mehul paid Rs 125 more interest than Nirav. Find x.
If the interest on a sum of Rs.1200 is more than the interest on Rs.1000 by Rs.120 in three years, then what is the rate of interest per annum?.
The difference between the simple interest received from two banks on Rs. 500 for two years is Rs. 2.50. What is the difference between their rates?
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?
