SOLUTION: How can $45,000 be invested part at 10% annual simple interest and the remainder of 5% annual simple interest so that the interest earned by the two accounts will be equal the amou

Question 1019123: How can $45,000 be invested part at 10% annual simple interest and the remainder of 5% annual simple interest so that the interest earned by the two accounts will be equal the amount to invest 10% is?

Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
it can't.

if the first part is invested at 10% and the second part is invested at 5%, the composite interest rate will be less than 10% unless the part that is invested at 5% is equal to 0.

you want a composite interest rate of 10%.
you invest x at 10%.
you invest y at 5%.

x + y = 45,000

.10*x + .05*y = .10 * (x + y)

simplify to get .10*x + .05*y = .10*x + .10*y

subtract .10*x and .10*y from both sides of the equation to get:

.10*x + .05*y - .10*x - .10*y = 0

combine like terms to get -.05*y = 0

solve fory to get y = 0.

since x + y = 45,000, and y = 0, then x must be equal to 45,000.

you must invest 0 at 5% and 45,000 at 10% to get a total of 45,000 at 10%.

so, the problem you posed is wrong because, in order for it to work, one of the parts must be greater than 10% and one of the parts must be less than 10%.

for example:

assume the first part is invested at 20% and the second part at 5% and you want the total to be invested at 10%.

you get:

x + y = 45,000
.20 * x + .05 * y = .10 * (x + y)

simplify the second equation to get:

.20 * x + .05 * y = .10 * x + .10 * y

subtract .10 * x and .10 * y from both sides of the equation to get:

.20 * x + .05 * y - .10 * x - .10 * y = 0

combine like terms to get .10 * x - .05 * y = 0

from x + y = 45,000, solve for y to get y = 45,000 - x

replace y with 45,000 - x in the equationn of .10 * x - .05 * y = 0 to get:

.10 * x - .05 * (45,000 - x) = 0

simplify to get .10 * x - 2250 + .05 * x = 0

combine like terms to get .15 * x - 2250 = 0

add 2250 to both sides to get .15 * x = 2250

divide both sides by .15 to get x = 2250/.15 = 15,000

since x + y = 45,000, then y = 45000 - 15000 = 30,000

you have 15,000 invested at 20% and 30,000 invested at 5%.

your total interest will be 3000 + 1500 = 4500.

4500 / 45,000 = 10%.

it works, but only if x is invested at greater than 10% and y is invested at less than 10%.

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