GMAT Math: Interest
GMAT includes two types of interest related problems:
1. Simple Interest
2. Compound Interest
Simple interest is the interest duration of the investment, the amount invested, and the interest rate.
Simple interest is calculated using the formula I = (n* P * r)
n = number of years for which the amount was invested
p = principle amount
r = rate of interest
Example: What is the interest earned in two years by a $4000 investment at 5% simple interest?
Interest = I = n* P * r
I = 2* $4000* 5%
= 2 * 4000 * 0.05
$400 at 5% earns $400 simple interest in 2 years.
Compound interest is different from simple interest beginning with the second earnings period, when interest accrued during the first period is added to the original principal balance. Given the same principal amount and rate of interest, earnings will be the same during the first period for simple and compound interest investments.
From the second period onward, however, the principal increases as each period's earnings are added to it, and each new interest calculation uses an ever-larger figure for P. This makes compound interest increasingly greater than simple interest; as the number of accounting periods increases, compound interest will amount to much more than simple interest would have been.
Compound interest is calculated using the formula
I = A – P
A is the total amount accumulated after n years, including both principal and interest.
Example: What is the compound interest earned by a two-year investment of $4000 at 5% compounded annually?
A = 4410
I = A – P
I = 4410 – 4400
I = 410
Interest earned through annual compounding is $10 more than simple interest because for the second year, interest is also accrued on the interest earned in the first year.
For more frequent compounding, prorate the annual interest rate.
Semiannual compounding: divide the rate by 2
Quarterly compounding: divide the rate by 4
Monthly compounding: Divide the rate by 12
GMAT Problem: James invested half his yearly savings in a bond that paid simple interest for 2 years, on which he has received $700 interest. He invested the remaining savings in a bond that paid the same rate compounded annually for the same 2 years, for which he has received $770. What was the total value of the savings he invested?
A. $ 5500
B. $ 1400
C. $ 2000
D. $ 3500
E. $ 3800
For half his savings, James earned an extra $70 (770 – 700) because the second year's interest was calculated on the original investment and interest accrued the first year. The other half his savings, invested in a simple-interest bond, paid $350 each year ($700/2).
To calculate the interest rate, divide the extra income by the amount of interest earned in the first year:
(70 / $350) * 100 = (one / 5) * 100 = 0.2 * 100 = 20%
James earned 20% in simple interest on half his savings.
If $350 is 20% of the investment in one of the bonds, then he invested (350 / 20)*100 = 17.5 * 100 = $1750 in it.
Since he invested equal amounts in the two bonds, double that amount for his total investment, which was all his savings: $1750 * 2 = $3500.
Hence the answer is $3500. Option (D).
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