SOLUTION: Ms. Green invested $1000 at 4% and $6000 at 6%. How much must she invest at 10% to make her total annual income 7.5% of her total investment?

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Question 73129This question is from textbook
: Ms. Green invested $1000 at 4% and $6000 at 6%. How much must she invest at 10% to make her total annual income 7.5% of her total investment? This question is from textbook

Answer by bucky(2189) About Me  (Show Source):
You can put this solution on YOUR website!

Let's work our way through this problem one step at a time:
.
$1000 invested at 4% will return 1000%2A%28.04%29+=+40 dollars per year.
.
$6000 invested at 6% will return 6000%2A%28.06%29+=+360 dollars per year.
.
Totaled together Ms Green already gets 360+%2B+40 = $400 dollars per year on two investments.
.
So far, so good.
.
Now she is going to invest an unknown sum of money (call it S) at 10% so her return on
this money is going to be S%2A%28.10%29 or in more standard form the number comes before the
letter. So her return is going to be %280.10%29%2AS
.
Let's write the total of all the interest she will be getting as:
.
400+%2B+%280.10%29%2AS.
.
What is the total amount she will have invested? We already know about her two investments
of $1000 and $6000. To that she will add the amount S. So her total investment will be:
.
1000%2B6000%2BS which reduces to S+%2B+7000
.
She wants to make 7.5% or 0.075 on this entire investment. So we can write her desired return
as:
.
%280.075%29%2A%28S%2B7000%29
.
And we can set that equal to the return she will be getting from her individual investments to
get:
.
%280.075%29%2A%28S%2B7000%29+=+400+%2B+%280.10%29%2AS
.
We can solve this equation to find S using normal algebraic operations.
.
First do the distributed multiplication on the left side:
.
%280.075%29%2AS+%2B+%280.075%29%2A7000+=+400+%2B+%280.10%29%2AS
.
Which further reduces to:
.
%280.075%29%2AS+%2B+525+=+400+%2B+%280.10%29%2AS
.
Next, to eliminate the 525 on the left side, subtract 525 from both sides to get:
.
%280.075%29%2AS+=+-125+%2B+%280.10%29%2AS
.
Then to eliminate the S term on the right side, subtract (0.10)*S from both sides. This
results in:
.
%28-0.025%29%2AS+=+-125
.
Finally, divide both sides by (-0.025) to solve for S. When you do you finally reach the answer:
.
S+=+%28-125%29%2F%28-0.025%29+=+5000
.
So Ms Green needs to invest $5000 at 10% to end up with a return rate of 7.5% on the total
amount she has invested.
.
Hope this helps you to do two things. First to analyze the problem so you can determine
what to do. And second, to do the mathematical work to find the answer.