Musicat wrote:Shawn invested one half of his savings in a bond that paid simple interest for 2 years and received $550 as interest. He invested the remaining in a bond that paid compound interest, interest being compounded annually, for the same 2 years at the same rate of interest and received $605 as interest. What was the value of his total savings before investing in these two bonds?
A. $5500
B. $11000
C. $22000
D. $2750
E. $44000
OA: D
Hmm... I probably did this an unorthodox way but here goes nothing.
First, recognize that simple interest accrues at a constant rate over any time interval. The only relevant sum is the initial principal. So, the $550 in simple interest accrued over two years means he earns $275 in simple interest each and every year.
Now, this also means on the compounded interest bond, he also earned $275 in interest in Year 1 (because it has not yet compounded and the principal and interest rate are identical to the other bond). It follows then that he makes $330 in interest in Year 2 on that bond (605 - 275 = 330).
From this we can make the following equations:
275 = P*r, where P = A/2 (A being the value we are trying to solve for, i.e. the initial savings)
330 = (P+Pr)*r (from the compounded interest bond - he now has P+Pr at the start of year 2 and earns interest at rate r, which totals $330)
We already know Pr = 275, so substituting that into our second equation & then simplifying:
330 = (P + 275)r
330 = Pr + 275r
Substituting Pr = 275 again...
330 = 275 + 275r
55 = 275r
r = 0.2
This means the interest rate was 20%.
We can use this value of r to solve for P using our first equation, and then double P to find A.
275 = P/5
P = 1375
So A = 2P = 2750 and the answer is
D.
