Money Distribution Help

Question

A sum of money is to be divided between Amanda, Bill, and Carol. Amanda receives $1 plus 1/3 of what is left. Bill then receives $6 plus 1/3 of what remains. Carol receives the rest, which amounts to $40. How much money did Bill receive?

Answer 1

We need variable names for the original sum and the amount each person receives:

S, A, B and C

 

Pick the statements apart to find pieces that you can write expressions/equations for.

"Amanda receives $1 plus ...:

A = 1 + ...

1/3 of what is left. "

What's left is (S-1), so 

A = 1 + (S-1)/3

 

"Bill then receives $6 plus 1/3 of what remains."

What remains is now S-A-6.

B = 6 + (S-A-6)/3

 

"Carol receives the rest, which amounts to $40."

C = 40 = A - B

 

Finally, we know that the total (S) is the sum of the other three.

S = A + B + C

 

Now you have some equations, so it's a math problem instead of a word problem.

I'm assuming that you're supposed to use matrix algebra to solve.

Rearrange each of the equations to have all of the variables on the left and numbers on the right.  That will give the coefficients you need to populate the matrix.

 

A = 1 + (S-1)/3 = 1 + (1/3)S - 1/3

A - (1/3)S = 2/3

 

B = 6 + (S-A-6)/3 = 6 + (1/3)S - (1/3)A -2

(1/3)A + B - (1/3)S = 4

 

C = 40 = A - B

- A + B + C = 40

 

S = A + B + C

- A - B - C + S = 0

 

Use those four equations to solve.