P borrowed an amount from Q at a simple interest of 10% p.a. At the end of the year, P paid back Rs 99000 which was 90% of what she owed. How much money did P borrow?
Rs. 100000
Calculating the Principal Amount with Simple Interest
This problem involves understanding simple interest and working with percentages to find the original amount borrowed (the principal).
Understanding the Problem
We are given that P borrowed an amount from Q at a simple interest rate of 10% per annum. At the end of one year, P paid back Rs 99000. This amount, Rs 99000, represents 90% of the total amount P owed to Q at that time. The total amount owed includes the original borrowed amount (principal) plus the simple interest earned over one year. We need to find the original amount that P borrowed.
Key Information
- Simple Interest Rate (R) = 10% p.a.
- Time (T) = 1 year
- Amount Paid = Rs 99000
- Amount Paid = 90% of the Total Amount Owed
Step-by-Step Solution
Step 1: Find the Total Amount Owed
P paid Rs 99000, which is 90% of the total amount owed. Let the Total Amount Owed at the end of the year be \(A\). We can set up the equation:
\[90\% \text{ of } A = 99000\] \[\frac{90}{100} \times A = 99000\] \[0.90 \times A = 99000\]To find the Total Amount Owed (\(A\)), we divide the amount paid by the percentage it represents (as a decimal):
\[A = \frac{99000}{0.90}\] \[A = \frac{99000}{\frac{9}{10}}\] \[A = 99000 \times \frac{10}{9}\] \[A = 11000 \times 10\] \[A = 110000\]So, the Total Amount Owed at the end of the year was Rs 110000.
Step 2: Relate Total Amount Owed to Principal and Simple Interest
The Total Amount Owed is the sum of the Principal (\(P\)) and the Simple Interest (\(SI\)) earned over the year.
\[A = P + SI\]Step 3: Calculate Simple Interest in terms of Principal
The formula for Simple Interest is:
\[SI = \frac{P \times R \times T}{100}\]Substitute the given values R = 10% and T = 1 year:
\[SI = \frac{P \times 10 \times 1}{100}\] \[SI = \frac{10P}{100}\] \[SI = \frac{P}{10}\]Step 4: Set up an Equation and Solve for Principal
Now substitute the value of \(A\) (Rs 110000) and the expression for \(SI\) (\(\frac{P}{10}\)) into the equation from Step 2:
\[A = P + SI\] \[110000 = P + \frac{P}{10}\]Combine the terms on the right side by finding a common denominator:
\[110000 = \frac{10P}{10} + \frac{P}{10}\] \[110000 = \frac{10P + P}{10}\] \[110000 = \frac{11P}{10}\]Now, solve for \(P\) by multiplying both sides by 10 and dividing by 11:
\[110000 \times 10 = 11P\] \[1100000 = 11P\] \[P = \frac{1100000}{11}\] \[P = 100000\]The original amount P borrowed was Rs 100000.
Verification
- If P borrowed Rs 100000 at 10% simple interest for 1 year, the simple interest would be \( \frac{100000 \times 10 \times 1}{100} = 10000 \).
- The total amount owed would be Principal + Simple Interest = \(100000 + 10000 = 110000\).
- 90% of the total amount owed is \(0.90 \times 110000 = 99000\). This matches the amount P paid.
The calculation is correct.
Final Answer
The amount P borrowed was Rs 100000.
| Description | Value |
|---|---|
| Amount Paid | Rs 99000 |
| Amount Paid as % of Total Owed | 90% |
| Total Amount Owed (\(A\)) | Rs 110000 |
| Interest Rate (R) | 10% p.a. |
| Time (T) | 1 year |
| Simple Interest (\(SI\)) | \(\frac{P}{10}\) |
| Relation: \(A = P + SI\) | \(110000 = P + \frac{P}{10}\) |
| Principal (\(P\)) | Rs 100000 |
Revision Table: Simple Interest Concepts
| Term | Definition | Formula |
|---|---|---|
| Principal (P) | The initial amount of money borrowed or invested. | - |
| Rate (R) | The percentage of the principal charged as interest per period, usually per year. | - |
| Time (T) | The duration for which the money is borrowed or invested, usually in years. | - |
| Simple Interest (SI) | Interest calculated only on the principal amount. | \(SI = \frac{P \times R \times T}{100}\) |
| Amount (A) | The total sum after adding the interest to the principal. This is the amount repaid at the end of the term. | \(A = P + SI\) or \(A = P(1 + \frac{R \times T}{100})\) |
Additional Information: Percentage Calculations
Understanding percentages is crucial for solving many quantitative problems, including those involving finance like simple interest.
- A percentage is a number or ratio expressed as a fraction of 100. For example, 90% means \(\frac{90}{100}\).
- To find 'X% of a number Y', you calculate \(\frac{X}{100} \times Y\).
- If a number Z is X% of Y, then \(Z = \frac{X}{100} \times Y\). To find Y, you rearrange the formula: \(Y = \frac{Z}{\frac{X}{100}} = Z \times \frac{100}{X}\). This is what we did to find the Total Amount Owed when we knew 90% of it was Rs 99000.
In this problem, 90% of the Total Amount Owed was Rs 99000. We used the percentage concept to first find the Total Amount Owed before proceeding with the simple interest calculation to find the principal.
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?
