A bank pays interest to its depositors compounded yearly. If a deposit becomes Rs. 54,000/- at the end of 3 rd year and Rs. 64,800/- at the end of 6 th year, what is the principal invested in the deposit?
45,000
Principal Invested Calculation with Compound Interest
This problem involves calculating the initial principal amount deposited in a bank account where interest is compounded yearly. We are given the amount accumulated after 3 years and after 6 years.
Let the principal amount be \( P \). Let the annual interest rate be \( r \). The formula for compound interest is:
\( A = P(1 + r)^n \)
where \( A \) is the amount after \( n \) years.
Using Given Information to Formulate Equations
We are given the following information:
- Amount at the end of the 3rd year, \( A_3 = 54000 \)
- Amount at the end of the 6th year, \( A_6 = 64800 \)
Using the compound interest formula, we can write these as:
- Equation 1: \( 54000 = P(1 + r)^3 \)
- Equation 2: \( 64800 = P(1 + r)^6 \)
Finding the Compound Factor over the Period
Notice that the period between the 3rd year and the 6th year is \( 6 - 3 = 3 \) years. The amount at the end of the 6th year is the amount at the end of the 3rd year compounded for another 3 years.
We can also see this by dividing Equation 2 by Equation 1:
\( \frac{A_6}{A_3} = \frac{P(1 + r)^6}{P(1 + r)^3} \)
\( \frac{64800}{54000} = (1 + r)^{6-3} \)
\( \frac{64800}{54000} = (1 + r)^3 \)
Calculating the Growth Factor
Now, let's calculate the value of \( (1 + r)^3 \):
\( (1 + r)^3 = \frac{64800}{54000} \)
We can simplify the fraction by dividing both the numerator and denominator by 100:
\( (1 + r)^3 = \frac{648}{540} \)
Both 648 and 540 are divisible by common factors. Let's divide by 108 (since \( 6 \times 108 = 648 \) and \( 5 \times 108 = 540 \)):
\( (1 + r)^3 = \frac{648 \div 108}{540 \div 108} = \frac{6}{5} \)
So, \( (1 + r)^3 = 1.2 \).
Determining the Principal Amount
We can substitute this value of \( (1 + r)^3 \) back into Equation 1:
\( 54000 = P(1 + r)^3 \)
\( 54000 = P \times 1.2 \)
To find the principal \( P \), we rearrange the equation:
\( P = \frac{54000}{1.2} \)
To divide by 1.2, we can write 1.2 as \( \frac{12}{10} \) or multiply the numerator and denominator by 10:
\( P = \frac{54000 \times 10}{1.2 \times 10} = \frac{540000}{12} \)
Now, perform the division:
\( P = 45000 \)
Therefore, the principal amount invested in the deposit was Rs. 45,000.
Important Questions from Miscellaneous
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2. Distance between the longitudes is maximum on the Equator.
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