Some money problems involve one quantity divided into two investments paying different interest rates. Such questions are phrased ‘‘How much was invested at ___%?’’ or ‘‘How much was invested at each rate?’’
Examples
A woman had
$10,000 to invest. She deposited her money into two accounts-one paying 6%
interest and the other 7 ½ %. interest. If at the end of the year the total
interest earned was $682.50, how much was originally deposited in each account?
You could either let x represent the amount
deposited at 6% or at
7 ½ %. Here,
we will let x represent the amount deposited into the 6% account. Because the
two amounts must sum to 10,000, 10,000 – x is the amount deposited at 7 ½ %.
The amount of interest earned at 6% is 0.06x, and the amount of interest earned
at 7 ½ %. is 0.075(10,000 – x). The total amount of interest is $682.50, so 0.06x
+ 0.075(10,000 – x) = 682.50.
The woman
deposited $4500 in the 6% account and 10,000 – x = 10.000 – 4500 = $5500 in the
7 ½ % account.
Practice
1. A
businessman invested $50,000 into two funds which yielded profits of 16 ½ % and
18%. If the total profit was $8520, how much was invested in each fund?
2. A
college student deposited $3500 into two savings accounts, one with an annual
yield of 4 ¾ % and the other with an annual yield of 5 ¼ %. If he earned
$171.75 total interest the first year, how much was deposited in each account?
3. A
banker plans to lend $48,000 at a simple interest rate of 16% and the remainder
at 19%. How should she allot the loans in order to obtain a return of 18 ½ %?
Solutions
1. Let
x represent the amount invested at 16 ½ %. Then 50,000 – x represents the
amount invested at 18%. The profit from the 16 ½ % account is 0.165x, and the
profit from the 18% investment is 0.18(50.000 – x). The sum of the profits is
$8520.
The
amount invested at 16 ½ % is $32,000, and the amount invested at 18% is 50,000 –
x = 50,000 – 32,000 = $18,000.
2. Let
x represent the amount deposited at 4 ¾ %. Then the amount deposited at 5 ¼ %
is 3500 – x. The interest earned at 4 ¾ % is 0.0475x; the interest earned at 5 ¼
% is 0.0525(3500 – x). The sum of these two quantities is 171.75.
$2400
was deposited in the 4 ¾ % account, and 3500 – x = 3500 – 2400 = $1100 was
deposited in the 5 ¼ % account.
3. Let
x represent the amount to be loaned at 16%, so 48,000 – x represents the amount
to be loaned at 19%. The total amount of return should be 18 ½ % of 48,000
which is 0.185(48,000) = 8880.
$8000
should be loaned at 16%, and 48,000 – x = 48,000 – 8000 = $40,000 should be
loaned at 19%.
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