How much did Tony pay for each game?

Question

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?

 

Answer 1

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

Answer 2

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.

 

Answer 3

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

Answer 4

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