Neeraj lent Rs. 65536 for 2 years at 12
% per annum, compounded annually. How much 2 more could he earn if the interest were compounded half-yearly?
Initial value, P = Rs.65536
Interest rate, R = (25/2)% per annum
Time, n = 2 years
∵ Compounded annually.
∴ Amount (A) = P [1 + R/100]n [Where, P = Present value
R = Annual interest rate
n = Time in years]
∴ A = 65536 [1 +(25/2) /100]2
⇒ A = 65536 [1 + 1/8]2
⇒ A = 65536 [9/8]2
⇒ A = 65536 × 9/8 × 9/8
⇒ A = 65536 × 81/64
⇒ A = 1024 × 81
⇒ A = 82944
∴ Amount = Rs.82944
∴ Compound interest = Rs.(82944 – 65536) [∵CI = A – P]
= Rs.17408
Now,
∵ Compounded half-yearly.
∴ Amount (A) = P [1 + (R/2)/100]2n [Where, P = Present value
R = Annual interest rate
n = Time in years]
∴ A = 65536 [1 + (25/4)/100]4 [R = (25/2)% and n = 2 years]
⇒ A = 65536 [1 + 1/16]4
⇒ A = 65536 [17/16]4
⇒ A = 65536 × 17/16 × 17/16 × 17/16 × 17/16
⇒ A = 65536 × 83521/65536
⇒ A = 1 × 83521
⇒ A = 83521
∴ Amount = Rs.83521
∴ Compound interest = Rs.(83521 – 65536) [∵CI = A – P]
= Rs.17985
Now,
Difference between interests compound half-yearly and yearly,
= Rs.(17985 – 17408)
= Rs.577