Amount of Money Each Had

[nycmathguy]

Michael had 80% more money than Alvin. Vanesa had 40% less than Alvin. Michael and Alvin gave Vanessa some money in the ratio of 3:2, respectively. In the end, Vanessa had twice as much money as before. Given that Michael had $144 more than Vanessa in the end, how much money did each of them have at first?

This one is tricky.

Michael = x + 0.80.

Vanessa = (x + 0.80) - 0.40

I then see the following sentence in the paragraph:

"Michael and Alvin gave Vanessa some money in the ratio of 3:2, respectively."

This threw me in for a loop. I don't know how to input this information into the equation I am trying to make.

The problem continues with the following words:

"In the end, Vanessa had twice as much money as before."

By now, I am totally lost.

Here comes the question:

"Given that Michael had $144 more than Vanessa in the end, how much money did each of them have at first?"

Let x = amount of money each had at the start.

I need help setting up the right equation. I decided to break it down piece by piece to demonstrate my inability to form the needed equation from the given words in the problem.

You say?

 

[MathLover1]

M = amount of money that Michael has to start with.
A = amount of money that Alvin has to start with.
V = amount of money that Vanessa has to start with.
X = amount of money that Michael gives to Vanessa.
Y = amount of money that Alvin gives to Vanessa.

M = 1.8 * A (Michael has 80% more money than Alvin).
V = .6 * A (Vanessa has 40% less money than Alvin).

X/Y = 3/2
from this, you can derive X = 3/2 * Y and Y = 2/3 * X.

you are given that Michael and Alvin give Vanessa some money in the ratio of 3/2, and that, after they give her the money, she has twice as much as she had originally.
the equation for that becomes:
2V = V + X + Y is the equation that models this.
from this equation, you can derive V = X + Y.

you are given that, after Michael has given Vanessa some money, that Michael has 144 dollars more than Vanessa has.

M - X = 2V + 144.

since 2V is equal to V + X + Y, then you get:

M - X = V + X + Y + 144.

since V = .6 * A, you get:

M - X = .6 * A + X + Y + 144.

since M is equal to 1.8 * A, you get:

1.8 * A - X = .6 * A + X + Y + 144.

add X to both sides of this equation and subtract .6 * A from both sides of this equation to get:

1.2 * A = 2 * X + Y + 144.

since Y is equal to 2/3 * X, this equation becomes:

1.2 * A = 2 * X + 2/3 * X + 144.

combine like terms to get:

1.2 * A = 8/3 * X + 144.

you know that V = X + Y and that Y = 2/3 * X.
from this you can derive that V = 5/3 * X.
from this you can derive that X = 3/5 * V.
you know that V = .6 * A.
.6 is equivalent to 3/5.
therefore you can derive that X = 3/5 * V which becomes X = 3/5 * 3/5 * A which results in X = 9/25 * A.

your equation of 1.2 * A = 8/3 * X + 144 becomes:

1.2 * A = 8/3 * 9/25 * A + 144.

combine like terms to get:

1.2 * A = 72/75 * A + 144.


subtract 72/75 * A from both sides of this equation to get:

1.2 * A - 72/75 * A = 144

1.2 is equal to 90/75, therefore you get:

90/75 * A - 72/75 * A = 144.

combine like terms to get:

18/75 * A = 144.

solve for A to get:

A = 75 / 18 * 144 = 600.

M = 1.8 * A = 1080.
V = .6 * A = 360.

your solution should be:

M = 1080 (what Michael had to start with)
V = 360 (what Vanessa had to start with)
A = 600 (what Alvin had to start with)

you also know that V = X + Y which led you to V = 5/3 * X.
from this, you derived that X = 3/5 * V
this makes X = 3/5 * 360 = 216.

you know that Y = 2/3 * X.
this makes Y = 144.

you now have:

M = 1080
V = 360
A = 600
X = 216
Y = 144.

you already have your solution.

that is:

M = 1080
V = 360
A = 600

the calculation of X and Y is to confirm the solution is correct.

you are given that Michael has 144 more than Vanessa after Vanessa was given the money by Michael and Alvin.

she had double what he had originally, then she had 720.

360 + 216 + 144 = 720 which is 2 * 360, so that part is confirmed.

Michael had 1080 to start.
after he gave Vanessa 216, he had 864 left.

864 - 720 = 144, so the part that he had 144 more than Vanessa after all the exchanges of money were complete is also confirmed.
the solution is confirmed to be good

 

[nycmathguy]

M = amount of money that Michael has to start with.
A = amount of money that Alvin has to start with.
V = amount of money that Vanessa has to start with.
X = amount of money that Michael gives to Vanessa.
Y = amount of money that Alvin gives to Vanessa.

M = 1.8 * A (Michael has 80% more money than Alvin).
V = .6 * A (Vanessa has 40% less money than Alvin).

X/Y = 3/2
from this, you can derive X = 3/2 * Y and Y = 2/3 * X.

you are given that Michael and Alvin give Vanessa some money in the ratio of 3/2, and that, after they give her the money, she has twice as much as she had originally.
the equation for that becomes:
2V = V + X + Y is the equation that models this.
from this equation, you can derive V = X + Y.

you are given that, after Michael has given Vanessa some money, that Michael has 144 dollars more than Vanessa has.

M - X = 2V + 144.

since 2V is equal to V + X + Y, then you get:

M - X = V + X + Y + 144.

since V = .6 * A, you get:

M - X = .6 * A + X + Y + 144.

since M is equal to 1.8 * A, you get:

1.8 * A - X = .6 * A + X + Y + 144.

add X to both sides of this equation and subtract .6 * A from both sides of this equation to get:

1.2 * A = 2 * X + Y + 144.

since Y is equal to 2/3 * X, this equation becomes:

1.2 * A = 2 * X + 2/3 * X + 144.

combine like terms to get:

1.2 * A = 8/3 * X + 144.

you know that V = X + Y and that Y = 2/3 * X.
from this you can derive that V = 5/3 * X.
from this you can derive that X = 3/5 * V.
you know that V = .6 * A.
.6 is equivalent to 3/5.
therefore you can derive that X = 3/5 * V which becomes X = 3/5 * 3/5 * A which results in X = 9/25 * A.

your equation of 1.2 * A = 8/3 * X + 144 becomes:

1.2 * A = 8/3 * 9/25 * A + 144.

combine like terms to get:

1.2 * A = 72/75 * A + 144.


subtract 72/75 * A from both sides of this equation to get:

1.2 * A - 72/75 * A = 144

1.2 is equal to 90/75, therefore you get:

90/75 * A - 72/75 * A = 144.

combine like terms to get:

18/75 * A = 144.

solve for A to get:

A = 75 / 18 * 144 = 600.

M = 1.8 * A = 1080.
V = .6 * A = 360.

your solution should be:

M = 1080 (what Michael had to start with)
V = 360 (what Vanessa had to start with)
A = 600 (what Alvin had to start with)

you also know that V = X + Y which led you to V = 5/3 * X.
from this, you derived that X = 3/5 * V
this makes X = 3/5 * 360 = 216.

you know that Y = 2/3 * X.
this makes Y = 144.

you now have:

M = 1080
V = 360
A = 600
X = 216
Y = 144.

you already have your solution.

that is:

M = 1080
V = 360
A = 600

the calculation of X and Y is to confirm the solution is correct.

you are given that Michael has 144 more than Vanessa after Vanessa was given the money by Michael and Alvin.

she had double what he had originally, then she had 720.

360 + 216 + 144 = 720 which is 2 * 360, so that part is confirmed.

Michael had 1080 to start.
after he gave Vanessa 216, he had 864 left.

864 - 720 = 144, so the part that he had 144 more than Vanessa after all the exchanges of money were complete is also confirmed.
the solution is confirmed to be good

Wow! Now that I see how much work you put into finding the answer, I don't feel so bad. Honestly, I just don't understand what leads you through a problem like this one. The HOW is not clear.

Marvelous work at your end. Allow me to be honest again. I just don't have enough training in mathematics to even come close to the unbelievable amount of work that went into putting all this together leading to the answer.

This is a GRE exam question. In my opinion, it is totally senseless to have a question that requires so much calculation on a times test to only go half way through. A job well-done!

What you did here can't be learned. Either I get it or not. I play a fairly decent fingerstyle guitar but nothing close to Ana Vidovic or Christopher Parkening, or the late Andres Segovia, etc. Practice is over-rated. I can practice answering math word problems until the end of time and NEVER be as talented as you are in any way, shape or form. This is the truth. This is reality. This is a humbling reality.