A person borrowed ₹9,000 at 7%, ₹12,000 at 8% and ₹15,000 at 9% simple interest per annum. He had to pay ₹50,700 at the end of n years. What is the value of n?
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Calculating Time Period for Simple Interest Loans
This problem involves calculating the time period for which a person borrowed different amounts at different simple interest rates, given the total amount paid back at the end of the period. We need to find the value of 'n', which represents the number of years.
The formula for simple interest (SI) is:
\(SI = \frac{P \times R \times T}{100}\)
Where:
- P is the Principal amount
- R is the Rate of interest per annum
- T is the Time period in years
The total amount paid back is the sum of the total principal borrowed and the total simple interest accrued over the time period 'n'.
Let's break down the calculation for each borrowed amount:
- Loan 1: Principal (P1) = ₹9,000, Rate (R1) = 7%, Time (T) = n years
- Loan 2: Principal (P2) = ₹12,000, Rate (R2) = 8%, Time (T) = n years
- Loan 3: Principal (P3) = ₹15,000, Rate (R3) = 9%, Time (T) = n years
Calculating Simple Interest for Each Loan
Using the simple interest formula, we calculate the interest for each loan in terms of 'n':
Simple Interest for Loan 1 (SI1):
\(SI_1 = \frac{9000 \times 7 \times n}{100} = \frac{63000n}{100} = 630n\)
Simple Interest for Loan 2 (SI2):
\(SI_2 = \frac{12000 \times 8 \times n}{100} = \frac{96000n}{100} = 960n\)
Simple Interest for Loan 3 (SI3):
\(SI_3 = \frac{15000 \times 9 \times n}{100} = \frac{135000n}{100} = 1350n\)
Calculating Total Simple Interest and Total Principal
The total simple interest (Total SI) paid is the sum of the simple interest from all three loans:
\(\text{Total SI} = SI_1 + SI_2 + SI_3 = 630n + 960n + 1350n\)
\(\text{Total SI} = (630 + 960 + 1350)n = 2940n\)
The total principal (Total P) borrowed is the sum of the principal amounts of all three loans:
\(\text{Total P} = P_1 + P_2 + P_3 = 9000 + 12000 + 15000\)
\(\text{Total P} = 36000\)
Setting Up and Solving the Equation for Total Amount
The total amount paid back is given as ₹50,700. This total amount is the sum of the total principal and the total simple interest:
\(\text{Total Amount} = \text{Total P} + \text{Total SI}\)
Substitute the values we know:
\(50700 = 36000 + 2940n\)
Now, we need to solve this equation for 'n'. First, subtract the total principal from the total amount:
\(50700 - 36000 = 2940n\)
\(14700 = 2940n\)
To find 'n', divide the total simple interest by the coefficient of 'n':
\(n = \frac{14700}{2940}\)
Simplify the fraction:
\(n = \frac{1470}{294}\)
Dividing 1470 by 294 gives:
\(n = 5\)
So, the value of n is 5 years.
After 5 years, the total simple interest would be \(2940 \times 5 = ₹14,700\). The total amount paid back would be the total principal ₹36,000 plus the total interest ₹14,700, which equals ₹50,700. This matches the amount given in the problem statement.
Revision Table: Simple Interest Concepts
| Concept | Formula / Description |
|---|---|
| Simple Interest (SI) | \(SI = \frac{P \times R \times T}{100}\) |
| Total Amount | Total Amount = Principal + Simple Interest |
| Principal (P) | The initial amount borrowed or invested. |
| Rate (R) | The percentage of interest charged or earned per year. |
| Time (T) | The duration for which the money is borrowed or invested, usually in years. |
Additional Information on Simple Interest Problems
Simple interest is a basic concept in finance. It is calculated only on the principal amount. Unlike compound interest, simple interest does not add the accumulated interest from previous periods to the principal for calculating the next period's interest.
Problems involving simple interest often require calculating one of the variables (Principal, Rate, Time, or Interest Amount) when others are given. When multiple loans are involved, as in this case, the total simple interest is the sum of the interest calculated for each loan separately.
Understanding how to combine information from different loans and set up a single equation is crucial for solving such problems. Always ensure that the time period and interest rate correspond to the same unit of time (usually per annum).
Similar Questions
If a sum of money at a certain rate of simple interest per year doubles in 5 years and at a different rate of simple interest per year becomes three times in 12 years, then the difference in the two rates of simple interest per year is
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Two equal amounts were borrowed at 5% and 4% simple interest. The total interest after 4 years amounted to Rs. 405. What was the total amount borrowed?
A lent Rs. 25000 to B and at the same time lent some amount to C at the same 7% simple interest. After 4 years a received Rs. 11200 as interest from B and C. How much did A lend to C?
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A person borrows Rs. 5000 at 5% rate of interest per annum and immediately lent it at 5.5%. After two years he collected the amount and settled his loan. What is the amount gained by him this transaction?
A person divided a sum of Rs. 17, 200 into three parts and invested at 5%, 6% and 9% per annum simple interest. At the end of two years, he got the same interest on each part of money. What is the money invested at 9%?
The simple interest on a certain sum is one-fourth of the sum. If the number of years and the rate of annual interest are numerically equal, then the number of years is
Important Questions from Simple Interest
At what rate percent per annum will the simple interest on a sum of money be 2/5 of the principal in 10 years?
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?
