If the simple interest at the same interest rate on ₹500 for 4 years and ₹700 for 2 years, combined together, is ₹280, then what is the rate of interest?
Calculating the Simple Interest Rate
This problem asks us to find the rate of interest given the total simple interest earned from two different investments over different periods. We are given the principal amounts, the time periods, and the combined simple interest. We will use the simple interest formula to solve this.
Understanding Simple Interest
Simple interest is calculated only on the principal amount. The formula for simple interest is:
\( \text{Simple Interest (SI)} = \frac{\text{Principal (P)} \times \text{Rate (R)} \times \text{Time (T)}}{100} \)
Where:
- P is the principal amount (the initial money).
- R is the annual rate of interest (in percent).
- T is the time the money is invested or borrowed for (in years).
Analyzing the Given Information
We have two separate cases of simple interest calculation:
| Case | Principal (P) | Time (T) | Simple Interest (SI) |
|---|---|---|---|
| 1 | ₹500 | 4 years | SI1 |
| 2 | ₹700 | 2 years | SI2 |
| Combined | - | - | SI1 + SI2 = ₹280 |
Let the rate of interest be R% per annum for both cases, as the problem states the interest rate is the same.
Calculating Simple Interest for Each Case
Using the simple interest formula, we can find the simple interest for each case in terms of R.
Case 1: Simple Interest on ₹500 for 4 years
Principal (P1) = ₹500
Time (T1) = 4 years
Rate (R1) = R %
Simple Interest (SI1) = \( \frac{P_1 \times R_1 \times T_1}{100} = \frac{500 \times R \times 4}{100} \)
\( SI_1 = \frac{2000R}{100} = 20R \)
Case 2: Simple Interest on ₹700 for 2 years
Principal (P2) = ₹700
Time (T2) = 2 years
Rate (R2) = R %
Simple Interest (SI2) = \( \frac{P_2 \times R_2 \times T_2}{100} = \frac{700 \times R \times 2}{100} \)
\( SI_2 = \frac{1400R}{100} = 14R \)
Using the Combined Simple Interest
The problem states that the combined simple interest from both cases is ₹280.
\( SI_1 + SI_2 = 280 \)
Substitute the expressions for SI1 and SI2 that we found:
\( 20R + 14R = 280 \)
Combine the terms with R:
\( 34R = 280 \)
Finding the Rate of Interest (R)
Now, we need to solve for R by dividing both sides of the equation by 34:
\( R = \frac{280}{34} \)
We can simplify the fraction by dividing the numerator and the denominator by their greatest common divisor, which is 2:
\( R = \frac{280 \div 2}{34 \div 2} = \frac{140}{17} \)
To express this as a mixed number, we divide 140 by 17:
- 17 goes into 140 eight times (\(17 \times 8 = 136\)).
- The remainder is \(140 - 136 = 4\).
So, \( \frac{140}{17} \) as a mixed number is \( 8 \frac{4}{17} \).
Therefore, the rate of interest is \( 8 \frac{4}{17} \% \).
Checking the Options
Let's compare our calculated rate with the given options:
- Option 1: \( 9 \frac{3}{17} \% \)
- Option 2: \( 6 \frac{5}{17} \% \)
- Option 3: \( 8 \frac{4}{17} \% \)
- Option 4: \( 7 \frac{3}{17} \% \)
Our calculated rate \( 8 \frac{4}{17} \% \) matches Option 3.
Summary of Steps
- Identify the given information for both investments: principals, times, and the total simple interest.
- Recall the simple interest formula: \( SI = \frac{P \times R \times T}{100} \).
- Express the simple interest for each investment in terms of the unknown rate R.
- Set up an equation where the sum of the individual simple interests equals the total given simple interest.
- Solve the equation for R.
- Convert the resulting improper fraction to a mixed number if necessary.
- Compare the result with the given options.
Revision Table: Simple Interest Calculation
| Concept | Formula | Explanation |
|---|---|---|
| Simple Interest (SI) | \( SI = \frac{P \times R \times T}{100} \) | Interest calculated only on the principal amount. P = Principal, R = Rate (%), T = Time (Years). |
| Principal (P) | - | The initial amount of money borrowed or invested. |
| Rate (R) | - | The percentage at which interest is charged or earned per year. |
| Time (T) | - | The duration for which the money is borrowed or invested, usually in years. |
Additional Information: Simple vs. Compound Interest
It's important to distinguish simple interest from compound interest. This problem deals only with simple interest.
- Simple Interest: Interest is calculated solely on the initial principal. The interest earned does not get added back to the principal for future calculations.
- Compound Interest: Interest is calculated on the initial principal AND on the accumulated interest from previous periods. This means the principal grows over time, leading to higher interest earnings compared to simple interest for the same rate and time (for T > 1 year). The formula for compound interest is more complex, involving exponential growth.
Understanding the difference is crucial for solving interest-related problems correctly.
Similar Questions
A man invests a total sum of Rs. 10,000 in a company. A part of the sum was invested at 10% simple interest per annum and the remaining part at 15% simple interest per annum. If the total interest accrued to him in two years equals Rs. 2,400, the sum invested at 15% simple interest per annum is:
If Rs. 72 amounts to Rs. 104.4 in 3 years, what will Rs. 120 amount to in 5 years at the same rate percent per annum?
A person deposited Rs. 500 for 3 years, Rs. 650 for 5 years, and Rs. 1,250 for 7 years. He received a total simple interest of Rs. 1,620. The rate of interest per annum is:
A sum of money invested at a certain rate of simple interest per annum amounts to Rs. 14,522 in seven years and to Rs. 18,906 in eleven years. Find the sum invested (in Rs.).
A person took a loan at 5% per annum simple interest during the first year and with an increase of 0.5% simple interest every year from the second year onwards. After 4 years, he paid Rs. 4,600 as a total interest to settle the loan completely. How much was the loan?
In how many years will a sum of Rs. 9,500 amount to Rs. 11,780 at the rate of 8% per annum at simple interest?
A sum of money at a fixed rate of simple interest amounts to Rs. 1,630 in 3 years and to Rs. 1,708 in 4 years. Find the sum (in Rs.).
A sum of money becomes \( \frac{8}{7} \) of itself in 2 years at a certain rate of simple interest. The rate per annum is:
A sum of money earns a simple interest at 7.25% per annum for the first eight years, at 8.5% for the next six years, and at 6.5% for the final four years. If the total interest earned during these eighteen years was Rs. 35,100, what was the original sum invested (in Rs.)?
A certain amount is lent at x% p.a. simple interest for 3 years. Instead, if the amount was lent at 3x% p.a. simple interest for 'y' more years, then the simple interest would have been seven times the earlier interest. What is the value of y?
Important Questions from Simple Interest
If ₹12,800 is invested in a bank for 5 years at the rate of 9% per annum simple interest. what amount is returned by the bank?
Somu has borrowed ₹10,000 from a money lender with simple interest at a rate of 7% half yearly. How much amount will he pay to the money lender after 3 years?
Find the Simple interest on Rs. 2,400 from 20 March 2019 to 31 may 2019 at \(6{1 \over 4}\) % rate?
If the simple interest for five years is equal is 35% of the principal, that rate of interest is:
A sum fetched a simple interest of Rs. 3,040 at the rate of 8% p.a in 5 years. what is the sum?
