Bunuel wrote:
If X is invested in a bank at a rate of simple interest of y% p.a. for two years, then the interest earned is 800. if X is invested at y% p.a., for two years when the interest is compounded annually, the interest is 820. What is the value of X?
A. 8000
B. 6000
C. 5000
D. 4000
E. 3000
Solution:
Using the simple interest formula (Principal x rate x time = Interest) we have:
X * y/100 * 2 = 800
Xy/50 = 800
y = 40,000/X
Using the compound interest formula [P(I + r/n)^rt - P = I, where P = principal, r = annual interest rate, n = the number of compounding periods per year, t = time in years, and I = the total interest earned, we have:
X(1 + y/100)^2 - X = 820
X(1 + 2y/100 + y^2/10,000 - 1) = 820
Since y = 40,000/X, we have:
X(2(40,000/X)/100 + (40,000/X)^2/10,000) = 820
X(800/X + 160,000/X^2) = 820
800 + 160,000/X = 820
160,000/X = 20
X = 8,000
Alternate Solution:The question mentions no units, so we will use dollars for clarity.
Since the interest is simple, exactly half of the interest of $800 is the annual interest; in other words, the principal of X acquires an interest of 800/2 = $400 in one year.
When the interest is compounded annually, we see that 820 - 800 = $20 more interest is earned. This is because in the second term, the principal acquiring interest is X + 400 dollars instead of X dollars. Thus, the extra interest of $20 is y percent of $400. Let’s use this to calculate y:
400 * y/100 = 20
y = 5
Now that we know y = 5, let’s use the fact that X dollars generate $400 in interest in one year at 5 percent:
X * 5/100 = 400
X = $8,000
Answer: A