Understanding the Compound Interest Problem
The question asks us to determine the initial amount of money (the principal) based on the total amounts accumulated after specific periods under compound interest. We are given the following information:
- The sum grows to Rs. 13,380 in 3 years.
- The sum grows to Rs. 20,070 in 6 years.
Our goal is to find the original sum (Principal, P).
Compound Interest Formula Basics
The standard formula for calculating the future amount (A) under compound interest is:
$ A = P \left(1 + \frac{r}{n}\right)^{nt} $Where:
- A = the future value of the investment/loan, including interest
- P = the principal investment amount (the initial deposit or loan amount)
- r = the annual interest rate (as a decimal)
- n = the number of times that interest is compounded per year
- t = the number of years the money is invested or borrowed for
In this problem, it's assumed that interest is compounded annually, so $n=1$. The formula simplifies to:
$ A = P(1 + r)^t $Step-by-Step Solution
Step 1: Formulate Equations Based on Given Data
Let P represent the principal sum and r represent the annual rate of compound interest. We can set up two equations using the provided information:
- After 3 years, the amount is Rs. 13,380: $ P(1 + r)^3 = 13380 \quad \quad (Equation \, 1) $
- After 6 years, the amount is Rs. 20,070: $ P(1 + r)^6 = 20070 \quad \quad (Equation \, 2) $
Step 2: Determine the Growth Factor
To find the value of $ (1 + r)^3 $, which represents the growth factor over a 3-year period, we can divide Equation 2 by Equation 1:
$ \frac{P(1 + r)^6}{P(1 + r)^3} = \frac{20070}{13380} $Simplify the left side using exponent rules ($ \frac{a^m}{a^n} = a^{m-n} $):
$ (1 + r)^{6-3} = \frac{20070}{13380} $ $ (1 + r)^3 = \frac{20070}{13380} $Now, simplify the fraction $ \frac{20070}{13380} $. We can divide both the numerator and the denominator by 10:
$ (1 + r)^3 = \frac{2007}{1338} $Both 2007 and 1338 are divisible by 3 (since the sum of their digits is divisible by 3). $2+0+0+7=9$ and $1+3+3+8=15$.
$ (1 + r)^3 = \frac{2007 \div 3}{1338 \div 3} = \frac{669}{446} $Let's test if this fraction simplifies further or represents a known ratio. We notice that $ 13380 \times 1.5 = 20070 $. Thus, the ratio $ \frac{20070}{13380} $ is equal to 1.5, or $ \frac{3}{2} $.
So, we have:
$ (1 + r)^3 = \frac{3}{2} $Step 3: Calculate the Principal Sum (P)
Now we can use Equation 1, $ P(1 + r)^3 = 13380 $, and substitute the value of $ (1 + r)^3 $ we just found:
$ P \times \left(\frac{3}{2}\right) = 13380 $To find P, multiply both sides by $ \frac{2}{3} $:
$ P = 13380 \times \frac{2}{3} $ $ P = \frac{13380 \times 2}{3} $ $ P = \frac{26760}{3} $ $ P = 8920 $Conclusion
The original sum of money (the principal) is Rs. 8,920.
Important Questions from Compound Interest
A person borrowed Rs. 10000 on compound interest at the rate of 40 percent per annum. If the interest is compounded half yearly, then what will be the amount to be paid after 1.5 years?
In how many years will a sum of Rs.1875 amount to Rs.2187 at 8 percent p.a. compound interest?
A sum of money has increased by 45% in 9 years at simple interest. What will be the compound interest of Rs. 12,000 after 3 years at the same rate?
At a certain rate of compound interest a certain sum amounts to Rs. 64800 in 4 years and Rs. 93312 in 6 years. What is the compound interest earned in fifth year?
Karuna has an option to invest in the following three schemes. If she wants to invest Rs P, then which scheme will be most beneficial for her?
Scheme that comes with lower interest rates and terms;
I. 28% p.a. interest, compounded every quarter, for one year.
II. 28% p.a. interest, compounded every half year, for one year.
III. 28% p.a. interest, for one year, compounded every year.
