Problem Solving

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

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.

Let's say he saved 2x dollars.

The x at simple interest gives us x * p/100 * 2 = 550.

The x at compound interest gives us x * (1 + p/100) * (1 + p/100) = x + 605.

The first equation gives us p/100 = 275/x.

Replacing p/100 with 275/x in the second equation:

x * (1 + 275/x) * (1 + 275/x) = x + 605

x * (1 + 550/x + 275²/x²) = x + 605

x + 550 + 275²/x = x + 605

275²/x = 55

275² = 55x

275 * 5 = x

Since the savings is 2x, we just double this, for 275 * 5 * 2, or D.