Joel and Sharon had $400 altogether. Joel gave away 3/7 of his money to Sharon. Sharon then gave 3/8 of her money to Joel.

Question

After that, they had the same amount of money.

 

Question: How much money did Joel have at first?

Answer 1

We started with Joel and Sharon had $400 total

Let J = starting money for Joel

Let S = starting money for Sharon

 

J + S = 400

 

Joel gave away (3/7)J to S and Sharon gave 3/8(S) to J and they had equal amounts at that time ($200 each)

 

J - (3/7)J + (3/8)S = S - (3/8)S + (3/7)J

(4/7)J + (3/8)S = (5/8)S + (3/7)J

Putting this all to the left side of the = sign we have:

(4/7)J - (3/7)J + (3/8)S - (5/8)S = 0

(1/7)J - (2/8)S = 0

(1/7)J - (1/4)S = 0

To simplify this by eliminating fractions multiply through by 28

4J - 7S = 0

 

We now have two equations we can easily work with

 

J + S = 400    eqn 1

4J - 7S = 0     eqn 2

 

Solve for one variable in terms of the other in eqn 1... Let's try solving for J

 

J = 400-S

Substitute that into eqn 2 and solve for S

 

4(400-S) - 7S = 0

1600 - 4S -7S = 0

-11S = -1600

S = (-1600)/(-11)

S = 145.45

 

Sharon started with $145.45

 

Joel started with 400 - 145.45 = $254.55 

 

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Another way to set up the equations is using the fact that they both had $200 after the exchange.

From that we have:

 

(4/7)J + (3/8)S = 200  eqn 1

(3/7)J + (5/8)S = 200  eqn 2

 

Let's eliminate one variable by adding the equations.  Multiply eqn 1 by -3 and eqn 2 by 4.

 

(-12/7)J - (9/8)S = -600     ...  (eqn 1)(-3)

(12/7)J + (20/8)S = 800   ...   (eqn 2)(4)

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11/8 S = 200 

 

S = 200(8/11)

 

S = $145.45

 

J = 400 - 145.45 = $254.55 

 

Answer 2

For this question it is important to work backwards. At the end, Joel and Sharon have $200 each. Therefore, Sharon had 200 after giving away 3/8ths of her share or retaining 5/8ths of her share. So 200=(5/8)S, where S is the amount of money Sharon had before giving away 3/8ths of her share. We can conclude that S=320. Because we know the total amount of money that Joel and Sharon have, we know that Joel had $80 after giving away 3/7ths of his initial share, or keeping 4/7ths of his share. So 80=(4/7)J, where J is the amount of money Joel initial had to begin with. We multiply by 4/7ths because this is the fraction he kept in order for him to have $80 after it all. Therefore J=$140. So Joel initially had $140 and Sharon had $260. In order to have reached the correct answer (and show all the necessary work) you must know how to isolate the desired variable by multiplying both sides of the equation by the reciprocal of the fraction.