SOLUTION: A woman has a total of $12,000 to invest. She invests part of the money in an account that pays 6% per year and the rest in an account that pays 8% per year. If the interest earned

Question 1152349: A woman has a total of $12,000 to invest. She invests part of the money in an account that pays 6% per year and the rest in an account that pays 8% per year. If the interest earned in the first year is $820, how much did she invest in each account?
She invested $ at 6% and $ at 8%.
Found 2 solutions by ankor@dixie-net.com, greenestamps:
Answer by ankor@dixie-net.com(22740) (Show Source): You can put this solution on YOUR website!
A woman has a total of $12,000 to invest.
She invests part of the money in an account that pays 6% per year and the rest in an account that pays 8% per year.
If the interest earned in the first year is $820, how much did she invest in each account?
:
let a = amt invested at 8%
the total is 12000, therefore
(12000-a) = amt invested at 6%
:
.08a + .06(12000-a) = 820
.08a + 720 - .06a = 820
.08a - .06a = 820 - 720
.02a = 100
a = 100/.02
a = $5000 invested at 8%
then
12000 - 5000 = $7000 invested at 6%
:
:
:
Check the amt of interest of each
.08(5000) = 400
.06(7000) = 420
----------------
total int: 820

She invested $7000 at 6% and $5000 at 8%.
Answer by greenestamps(13198) (Show Source): You can put this solution on YOUR website!

If an algebraic solution is required, then the solution provided by the other tutor is a good presentation of the usual method for solving the problem.

This kind of problem -- as well as a great deal of similar types of problems -- is very common on competitive middle school or high school math contests. Here is a quick way to solve this kind of problem with a few quick calculations -- much faster than with the equation used in the formal algebraic method.

(1) Find out how the actual interest of $820 compares to the interest that would have been earned if the whole amount were invested at each of the two rates:
$12,000 at 6% --> $720 interest
actual -- $820 interest
$12,000 at 8% --> $960 interest

From $720 to $960 is a difference of $240; from $720 to $820 is a difference of $100. So the actual interest is "100/240 = 5/12 of the way" from $720 to $960.

Since the actual interest if 5/12 of the way from the amount it would have been if all the money had been invested at 6% to the amount it would have been if it had all been invested at 8%, that means 5/12 of the total was invested at 8%.

ANSWER: 5/12 of the $12,000, or $5000, at 8%; the other $7000 at 6%.

CHECK: .08(5000)+.06(7000) = 400+420 =820

All the words of explanation might make this seem like a lengthy process; but it is not. Being good with mental arithmetic, I solve this kind of problem typically in 7-8 seconds once I have read the given information.

Without all the words, here are the needed calculations:
.06(12000) = 720; .08(12000) = 960
960-720 = 240; 820-720 = 100; 100/240 = 5/12
(5/12)*12000 = 5000

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