Simple and Compound Interest Problem

• Jan 14th 2011, 12:41 AM
dumluck
Simple and Compound Interest Problem
Hi All,

Q: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?

1. $5500 2.$ 11000
3. $22000 4.$ 2750
5. $44000 Answer Explanation... 1. Interest for the first year of the simple compound bond is 275/2 -$275.
2. So we need to determine the rate of interest based on this so...

605 - 550 = 55. That's the difference between the interest earned on the simple vs compound interest bonds.

55/275 * 100/1 = 11/55 * 100/1 = 20% Interest

3. 275 represents 20% interest of a number
275/20 * 100/1 = 55/4 * 100/1 = $1375. 4. This represents half the money so 1375*2 =$2750. (D).

My questions is: Why are we using 55. I.E. The difference between the two interest to determine the interest in 2. What does this 55 represent (besides the difference between the two?)

• Jan 14th 2011, 08:20 AM
Soroban
Hello, dumluck!

I'm not impressed with their explanation.

Quote:

Q: 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, 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?

. . $\displaystyle 1.\;\$5500 \qquad 2.\;\$11000 \qquad 3.\;\$22000 \qquad 4.\;\$2750 \qquad 5.\;\$44000$Let$\displaystyle \,r$be the annual interest rate for both accounts. Let$\displaystyle \,P$be the amount invested in each account. He invested$\displaystyle \,P$dollars at$\displaystyle \,r$percent simple interest for 2 years. . . This earned: .$\displaystyle P(r)(2) \:=\:2Pr$dollars in interest. So we have: .$\displaystyle 2Pr \:=\:550 \quad\Rightarrow\quad P \:=\:\dfrac{275}{r}$.[1] He invested$\displaystyle \,P$dollars at$\displaystyle \,r$percent compounded annualy for 2 years. . . This grew to: .$\displaystyle P(1+r)^2 $dollars in two years. His interest is: .$\displaystyle P(1+r)^2 - P \;=\;P\left[(1+r)^2 - 1\right] $dollars. . . So we have: .$\displaystyle P(2r + r^2) \:=\:605$.[2] Substitute [1] into [2]: .$\displaystyle \dfrac{275}{r}(2r + r^2) \:=\:605$. . . . . .$\displaystyle 2 + r \:=\:\dfrac{605}{275}\:=\:2.2 \quad\Rightarrow\quad r \:=\:0.2$Substitute into [1]: .$\displaystyle P \:=\:\dfrac{275}{0.2} \:=\:1375$Hence, he invested$\displaystyle \$1375$ in each account.

Therefore, he invested a total of $\displaystyle \$2750.\$

• Jan 14th 2011, 08:41 AM
Wilmer
Quote:

Originally Posted by dumluck
My questions is: Why are we using 55. I.E. The difference between the two interest to determine the interest in 2. What does this 55 represent (besides the difference between the two?)

Code:

Simple interest: 0            1375.00 1  275.00  1650.00 2  275.00  1925.00 Compound interest: 0            1375.00 1  275.00  1650.00 2  330.00  1980.00
The 55 is the interest on the 275 (275 * .20 = 55) paid at end of 1st year
(in the compound interest case, of course); makes 2nd year interest = 330