An officer has R$. $10,00,000$. He planned to deposit the amount in Bank A but his senior officer suggests him to deposit it in Bank B.
| Bank-A | Bank-B |
|---|---|
| Compound interest Annually $-12 \% p.a. | Compound Interest Semiannually $-10 \% p.a. |
(a) Which formula can be used for finding compound interest semiannually?
(b) If the officer chooses the Bank B for 3 years, find the interest he gets.
(c) If he had not followed his senior officer's suggestion, what would be his profit or loss?
Answer (2)
The formula for semiannual compound interest is A = P ( 1 + 2 r ) 2 t . If the officer deposits in Bank B for 3 years, he earns approximately R 340095.64. I f h e ha d c h ose n B ank A , h e w o u l d ha v e a p ro f i t o f R 64832.36 instead.
;
The formula for compound interest semiannually is: A = P ( 1 + 2 r ) 2 t .
The interest earned in Bank B after 3 years is: R$ 340095.64.
The interest earned in Bank A after 3 years is: R$ 404928.
If the officer had not followed his senior officer's suggestion, his profit would be: R$ 64832.36 .
Explanation
Problem Analysis We are given a problem involving compound interest calculations for two banks, Bank A and Bank B. Bank A offers compound interest annually at a rate of 12%, while Bank B offers compound interest semiannually at a rate of 10%. We need to determine the formula for semiannual compounding, calculate the interest earned in Bank B over 3 years, and find the profit or loss if the officer had chosen Bank A instead.
Formula for Semiannual Compounding (a) The formula for compound interest compounded n times per year is given by: A = P ( 1 + n r ) n t where:
A is the final amount
P is the principal amount
r is the annual interest rate
n is the number of times interest is compounded per year
t is the number of years For semiannual compounding, n = 2 .
Interest Earned in Bank B (b) For Bank B, we have:
P = R$ 10 , 00 , 000
r = 10% = 0.10
n = 2 (semiannually)
t = 3 years Using the formula, we calculate the final amount A :
A = 1000000 ( 1 + 2 0.10 ) 2 × 3 A = 1000000 ( 1 + 0.05 ) 6 A = 1000000 ( 1.05 ) 6 A = 1000000 × 1.340095640625 A = R$ 1340095.64 The interest earned is the final amount minus the principal: I n t eres t = A − P = 1340095.64 − 1000000 = R$ 340095.64
Profit/Loss Calculation (c) For Bank A, we have:
P = R$ 10 , 00 , 000
r = 12% = 0.12
n = 1 (annually)
t = 3 years Using the compound interest formula, we calculate the final amount A :
A = 1000000 ( 1 + 0.12 ) 3 A = 1000000 ( 1.12 ) 3 A = 1000000 × 1.404928 A = R$ 1404928 The interest earned in Bank A is: I n t eres t A = A − P = 1404928 − 1000000 = R$ 404928 Now, we compare the interest earned in Bank A and Bank B: P ro f i t / L oss = I n t eres t A − I n t eres t B = 404928 − 340095.64 = R$ 64832.36 Since the result is positive, there would be a profit of R$ 64832.36 if the officer had chosen Bank A.
Final Answer The formula for semiannual compound interest is A = P ( 1 + 2 r ) 2 t . The interest earned in Bank B is R$ 340095.64. If the officer had chosen Bank A, there would be a profit of R$ 64832.36.
Examples
Understanding compound interest is crucial in personal finance. For instance, when planning for retirement, knowing how different investment options compound over time helps in making informed decisions. Comparing annual versus semiannual compounding can significantly impact long-term savings. This problem illustrates how choosing the right investment strategy can lead to substantial gains over a few years, highlighting the importance of financial literacy.
