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
Anil lent a sum of Rs. 5,000 on simple interest for 10 years in such a way that the rate of interest is 6% per annum for the first 2 years, 8% per anmum for the next 2 years and 10% per annum beyond 4 years. How much interest (in Rs.) will he earn at the end of 10 years?
What will be the simple interest on a sum of Rs. 12000 at the rate of 15 percent per annum for three years ?
If in 13 years fixed sum doubles at simple interest, what will be the interest rate per year? (correct to two decimal places)
On simple interest a sum of Rs. 640 becomes Rs. 832 in 2 years. What will Rs. 860 become in 4 years at the same rate of simple interest?
A certain sum amounts to Rs. 81840 in 3 years and to Rs. 92400 in 5 years at x% p.a. under simple interest. If the rate of interest is becomes (x + 2)%, then in how many years will the same sum double itself?
