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How much did Tony pay for each game?

These are two math problems I need help with:
1. Tony paid the same price for a number of computer games totaling $138.25. Willy bought the same number of games for $117.25. Willy paid $3 less for each game than Tony did. How much did Tony pay for each game?
 
2. Damon gave Shaquan a 20-yard head-start in a race. Damon can run 3 yards a second while Shaquan can run 2.5 yards a second. Damon crosses the finish line 2 seconds ahead of Shaquan. How many yards did Damon run?
 
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4 Answers

1.  Let x be the number of games for Tony and Willy, let y be the price per game for Tony and y-3 the price per game for Willy.
 
Tony:
 
xy=138.25
 
Willy:
 
x(y-3)= 117.25
 
xy-3x=117.25
 
138.25-3y= 117.25
 
-3x=-21
 
x=7
 
 
Substitute into Tony's  equation:
 
7y=138.25
 
y=19.75
 
Tony paid $19.75 for each game.
 
 
2.  Use the formula d=rt
 
Since this is a race to the finish line both runners run the same distance. Shaquan adds 2 seconds to his time because it takes him two seconds longer to run the same distance as Damon. Damon subtracts 8 seconds from his time because Shaquan had a 20 yard head start.
 
20yards/2.5yards/second=8 seconds
 
since d=d we can set rt=rt:
 
2.5(t+2)=3(t-8)
 
2.5t+5=3t-24
 
.5t=29
 
t=58s
 
d= 3(58-8) = 150 yards
1.  Willy saved 21 dollars buying the same number of games.
 
138.25 - 117.25 = 21.00
 
Since the money he saved was $3 per game, 
 
Money saved
--------------                   =   number of games each bought.
Money saved per game
 
 
 21/3 = 7.
 
They bought 7 games each.
 
 
Tony:
 
$138.25/7 games = $19.75/game.
 
 
 
 
2.
 
Forgot about the head start!!!!  :)
 
 
Distance = rate * time
 
They run the same distance D.
 
Let t be the time that it took Damon to run the distance.
 
Damon
    D = 3yd/s * (t) 
 
Shaquan took 2 seconds longer than Damon, so his time was t+2, and his distance was 20 yards shorter.
 
    D - 20 = 2.5yd/s * (t+2)
 
Solve for D:
 
D = 2.5(t+2) + 20
 
Set them equal to each other:
 
3(t) = 2.5(t+2) + 20
 
3t = 2.5t + 5 + 20
 
3t - 2.5t = 25
 
.5t = 25
 
t = 25/.5 = 25*2 = 50s
 
 
That's the time for Damon.
 
The question wants the distance.
 
D = r*t
D = 3*50
 
D = 150 yards.
 
 
Double check:
 
If D = 150, then Shaquan only had to run 150-20=130 yards, because he had a headstart.
 
 
If Damon ran the race in 50 seconds, and Shaquan took 2 seconds longer, then it took Shaquan 50+2 = 52 seconds.
 
Does that check out?
 
D for Shaquan = (2.5 yds/s)(52s) = 130yds.  Check.
 
1. Lets think about what this problem is telling us.
  • Tony bought a number of video games (n) at a fixed price (p)
  • The total of these games was $138.25
  • We can write this as an equation: 
  • n*p = $138.25

  • Willy bought the same number of games as Tony (n)
  • Willy paid $3 less per game (p-3)
  • Willy's total was $117.25
  • This can be written as: n*(p-3) = $117.25
 
The problem asks us to find the price Tony paid per game (p)
  • We have two equations and two variables in each.
  • If we re-arrange the equations, we can get (n) on the left side of each
    • n = 138.25 / p
    • n = 117.25 / (p-3)
 
Now we have two different expressions for the value n, but we know that n = n
  • So we can set the expressions for n equal to each other
    • 138.25 / p = 117.25 / (p-3)
  • At this point, we can solve the new equation for p
    • 138.25 * (p-3) = 117.25 * p
    • 138.25p - 414.75 = 117.25p
    • 138.25p - 117.25p = 414.75
    • 21p = 414.75
    • p = 414.75 / 21 = 19.75
 
Tony paid $19.75 for each game
 
An alternate method:
  • We can take the difference of the total prices ($138.25 - $117.25 = $21)
  • Divide it by the price difference per game ( $21 / $3 = 7 )
    • This is because the total difference divided by the per item difference will give us the count of items
  • Therefore, each person bought 7 games.
  • Now we can divide the total price paid by Tony by the number of games he bought (based on the equation from the first method):
  • $138.25 / 7 games = $19.75 per game
 
You can use either of these methods to solve the second problem

Comments

You are correct, but I want to use algebra, choose a variable and convert English to algebraic statements.
Parviz, the first method I demonstrated converted English to algebraic statements. The first 8 steps explain the conversion. The variables I used are n and p. I demonstrated choosing the variable p, as the question asks about the price Tony paid.
nP = 138.25
n(p-3) = 117.25
 
      138.25 = 117.25
         p          p-3
 
     P( 117.25) = p( 138.25)- 3( 138.25)
 
      P( 138.25- 117.25) = 3( 138.25)
 
       P = 3( 138.25 = 19.75
                  21
          20/yd
     S--------- D --------
       2.5y/s       3.0Y/s
      
    20 + 2.5(t+2) = 3 t
       20 +5 = 3t-2.5t
 
        25 = 0.5t         t = 50 second
 
         50* 3yd/s = 150yd Damon Yards