Help with Algebra problem needed-deals with two variable probs:

[Catherine]

My ds and I are working through the long section of word problems in Foerster's at the end of chapter 7. He is really conceptually stuck on the problems that have one variable defined in terms of another. Second part of problem number 33 (p. 333) for example-he did it in his head (sigh-still struggling with this) and came up with the father's contribution as being $80 (mother's contribution) plus $400 ($480) times 30%=$144. Mother's contribution was $120 (father's contribution) plus $400 times 20%=$104.

 

The worked problem is:

x=$ mother pays

y=$ father pays

x=0.2(400+y)

y+0.3(400+x)

x=80+0.2y

substituting:

y=120+0.3(80+0.2y)

y=$153.19

x=$110.64

 

Why the different answers? I think its because the mother is not now contributing $80, but a larger sum based on the father's contribution. But I do understand why he's wondering how we can go into a problem without knowing either party's contribution, and define the contribution in terms of the other's contribution, and come up with a firm answer? Can anyone (thank you if you've gotten this far!!) help me to clarify this? He thinks -if I keep going, it will be infinity.

[regentrude]

It would help if you could post the actual problem.

[Catherine]

Calvin earns $400 working at the supermarket to help pay his share of a vacation trip. To supplement his earnings, Calvin's mother will give him an additional 20% of what he earns and his father will give him an additional 30% of what he earns.

a. How much will he get from each parent?

b. Calvin is good at business deals and asks if his mother will give him 20% of the total he gets from working and from his father. He also asks if his father will give 30% of the total he gets from working and from his mother. His parents agree, provided Calvin can figure out the right amounts. How much will Calvin get from each parent under these conditions?

[Teachin'Mine]

nm - just re-read the problem - my answer is definitely wrong.

Edited October 14, 2010 by Teachin'Mine

[regentrude]

came up with the father's contribution as being $80 (mother's contribution) plus $400 ($480) times 30%=$144. Mother's contribution was $120 (father's contribution) plus $400 times 20%=$104.

 

This is incorrect because the mother's contribution is not just 20% of 400, but more, and how much more is based on the father's contribution.

In variables,

the mother's contribution is 20%*(400+Dad's contribution)

and Dad's contribution is not yet known.

 

The father's contribution, OTOH, is based on the mother's contribution and not simply 30% of 400

but 30%*(400+Mom's contribution)

and Mom's contribution is not yet known.

This will be very hard to solve in his head; he will need to write out the system of equations the way you typed them and solve the coupled equations.

(your answers above are correct)

[Melissa in NC]

In general, you can solve any system of equations as long as the number of variables equal the number of equations.

 

If you have x, y, and z, you will need three independent equations to solve for all three variables.

 

When I was studying in college, I would solve equations with 10 variable and 10 equations. Don't panic, there are special techniques to solve the problems.

 

In this case:

Eq 1 x=0.2(400+y)

Eq 2 y=0.3(400+x)

 

Thus 2 variables x,y and two equations. The problem has one answer.