Anil deposited Rs 2,00,000 in a Bank at the rate of $10 \%$ p.a. compound.
(a) Write the formula for finding quarterly compound interest.
(b) If the bank provides the semi annual compound interest instead of the annual compound interest at the same rate of interest, how much more interest would be received by Anil? Find it.
Answer (2)
Calculate the quarterly compound interest using the formula A = P ( 1 + 400 R ) 4 T and C I = A − P , which results in approximately Rs. 43,680.58.
Calculate the annual compound interest using the formula A 1 = P ( 1 + 100 R ) T and C I 1 = A 1 − P , which results in Rs. 42,000.
Calculate the semi-annual compound interest using the formula A 2 = P ( 1 + 200 R ) 2 T and C I 2 = A 2 − P , which results in Rs. 43,101.25.
Determine the difference between semi-annual and annual compound interest: D i ff ere n ce = C I 2 − C I 1 = R s .1101.25 . Therefore, Anil would receive Rs. 1101.25 more with semi-annual compounding.
Q u a r t er l y I n t eres t : R s .43680.58 , D i ff ere n ce : R s .1101.25
Explanation
Understanding the Problem First, let's break down the problem. We are given a principal amount of Rs. 2,00,000, an annual interest rate of 10%, and a time period of 2 years. We need to calculate the compound interest when compounded quarterly and find the difference in interest earned when compounded semi-annually instead of annually.
Calculating Quarterly Compound Interest (a) To find the compound interest when compounded quarterly, we use the formula: A = P ( 1 + 400 R ) 4 T where:
A is the amount after T years
P is the principal amount (Rs. 2,00,000)
R is the annual interest rate (10%)
T is the time period in years (2)
Plugging in the values: A = 200000 ( 1 + 400 10 ) 4 × 2 A = 200000 ( 1 + 0.025 ) 8 A = 200000 ( 1.025 ) 8 A = 200000 × 1.21840289646
A = 243680.579292
The compound interest (CI) is calculated as: C I = A − P C I = 243680.579292 − 200000 C I = 43680.579292 So, the quarterly compound interest is approximately Rs. 43,680.58.
Calculating the Difference (b) To find the difference in compound interest when compounded semi-annually instead of annually, we first calculate the compound interest when compounded annually using the formula: A 1 = P ( 1 + 100 R ) T A 1 = 200000 ( 1 + 100 10 ) 2 A 1 = 200000 ( 1.1 ) 2 A 1 = 200000 × 1.21 A 1 = 242000
The compound interest ( C I 1 ) is: C I 1 = A 1 − P C I 1 = 242000 − 200000 C I 1 = 42000
Next, we calculate the compound interest when compounded semi-annually using the formula: A 2 = P ( 1 + 200 R ) 2 T A 2 = 200000 ( 1 + 200 10 ) 2 × 2 A 2 = 200000 ( 1 + 0.05 ) 4 A 2 = 200000 ( 1.05 ) 4 A 2 = 200000 × 1.21550625 A 2 = 243101.25
The compound interest ( C I 2 ) is: C I 2 = A 2 − P C I 2 = 243101.25 − 200000 C I 2 = 43101.25
The difference between the semi-annual and annual compound interest is: D i ff ere n ce = C I 2 − C I 1 D i ff ere n ce = 43101.25 − 42000 D i ff ere n ce = 1101.25 So, Anil would receive Rs. 1101.25 more if the interest were compounded semi-annually instead of annually.
Final Answer (c) The quarterly compound interest is approximately Rs. 43,680.58, and Anil would receive Rs. 1101.25 more if the interest were compounded semi-annually instead of annually.
Examples
Understanding compound interest is crucial in personal finance. For instance, when planning for retirement, knowing how different compounding frequencies affect your investment returns can significantly impact your savings. Suppose you invest in a fund with an initial deposit and regular contributions. Calculating the future value with different compounding periods (annually, semi-annually, quarterly) helps you project potential retirement income and make informed decisions about your investment strategy. This ensures you maximize your returns and achieve your financial goals effectively.
The quarterly compound interest for Anil is approximately Rs. 43,680.58. Additionally, if his interest were compounded semi-annually, he would earn Rs. 1101.25 more compared to annual compounding.
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