Tough word problem!

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cbteach64

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Oct 27, 2020
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This problem is driving my class nuts. I am trying to find an easy way to explain it to them. Here goes...Jade, Matt, and Zane collect stamps. Jade and Matt have 220 stamps more than Zane. Zane and Matt have 140 more than Jade. Jade and Zane have 500 stamps. How many stamps does Matt have?
 
Have you let J be the number of stamps Jade has, M be the number of stamps Matt has and Z be the number of stamps Zane has.
Then form 3 equations. Come back when you have done that.
 
Let's break it into parts:
  • Jade and Matt have 220 stamps more than Zane.
  • Zane and Matt have 140 more than Jade.
  • Jade and Zane have 500 stamps.
  • How many stamps does Matt have?
Now let's turn those into equations:
  • [MATH]J + M = 220 + Z[/MATH]
  • [MATH]Z + M = 140 + J[/MATH]
  • [MATH]J + Z = 500[/MATH]
  • [MATH]M =[/MATH] [MATH]?[/MATH]
This system of equations can be solved by substituting expressions for variables in such a way that produces new equations with fewer variables... Consider the following:
  • If [MATH]J + M = 220 + Z[/MATH]
    • then [MATH]J = 220 + Z - M[/MATH]
  • If [MATH]Z + M = 140 + J[/MATH]
    • then [MATH]Z + M = 140 + (200 + Z - M)[/MATH]
This modifies the second equation, which began with three variables, into an equation containing only two variables.

A little algebra can then produce an equation in the form of "[MATH]Z =[/MATH] [MATH]...[/MATH]" that contains only [MATH]M[/MATH] as a variable. This can in turn be used to substitute both [MATH]J[/MATH] and [MATH]Z[/MATH] in one of the equations with expressions that contain only [MATH]M[/MATH] as a variable. Once we're there, we can solve for [MATH]M[/MATH].
 
Mr. Bland said:
Let's break it into parts:
  • Jade and Matt have 220 stamps more than Zane.
  • Zane and Matt have 140 more than Jade.
  • Jade and Zane have 500 stamps.
  • How many stamps does Matt have?
Now let's turn those into equations:
  • [MATH]J + M = 200 + Z[/MATH]
  • [MATH]Z + M = 110 + J[/MATH]
  • [MATH]J + Z = 500[/MATH]
  • [MATH]M =[/MATH] [MATH]?[/MATH]
Click to expand...

  • [MATH]J + M = {\color{red}220} + Z[/MATH]
  • [MATH]Z + M = {\color{red}140} + J[/MATH]
  • [MATH]J + Z = 500[/MATH]
  • [MATH]M =[/MATH] [MATH]?[/MATH]
 
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