Expert: Sherry Wallin - 9/24/2009

Question
Hello:

(A) 5% X $2000 = $100

(B) 2% X $1000 = $20

This is a total return or average of $120/$3000 = 4%

If I want to have an average or total return of 3%, what amounts must be in the A account and what dollar amount must be in the B account?

And also what percentage must the A and B accounts be changed to so that the average or total return is 3% but keep the dollar amounts in each account, A $2000 and B $1000, the same?

Try to explain in basic arithmetic if possible.

I thank you for your reply.  

Answer
Kenneth~
   You want to figure out how much money needs to be put in account A at 5% return and how much in account B at a 2% return so that you get 3% of $3000.  3% of $3000 = $90 total return. So let the amount in account A be x and the amount in account B be y so that x + y = 3000 and .05x + .02y = 90.
If you multiply the last equation by 100 you can get rid of the decimals and the result is
5x+2y = 9000. Now take the 1st equation and multiply it by -5 getting -5x + -5y = -15,000. Now add these two new equations together and you will notice that the x's go away leaving you with -3y=-6000
solve for y and you get y = 2000. Put this value of y back into one of the original equations:
x + y = 3000 -> x + 2000 = 3000 -> x = 1000. This means put $2000 in account A and $1000 in account B to realize an average return of 3% of $3000.

Note this is using algebra, this is one of the reasons we have algebra because these kind of problems are easily solved.

To figure the second half of your problem (just using basic arithmetic). You can play around with the numbers. The rate of return is $90 at 3% of $3000. So you want x% 2000 + y% 1000 = 90. One solution is to take 1% of 2000 = 20 and 7% of 1000 = 70 and since 20 + 70 = 90 you have satisfied the 3% return using the original amounts in account A and in account B. Note there are others and you could find them playing around or you could write the equation of the line and find another point on the line. For instance if you take 2% of 2000 you get 40 and then you need to figure out how many % of 1000 makes up the difference between the 40 and 90 which is 50. So take 5% of 1000 and get 50 and 50 + 40 = 90, another solution.
x% 2000 + y% 1000 = 90. translates into 2ox + 10y = 90, solve for y getting y =-2x + 9. Now choose an x, say 3, and solve for y -2(3)+9 = -6+9 = 3 so 3% of 2000 + 3% of 1000 should equal 90.
60 + 30 = 90 so that is another solution.

Math Prof