Hello! I know the answer to this word problem; however, I want to know HOW to solve it. Any help is much appreciated!
Q:
A store sells used and new video games. New video games cost more than used video games. All video games cost the same. All new video games cost the same.
Omar spent a total of $84 on 4 used video games and 2 new games. Sally spent a total of $78 on 6 used video games and 1 new video game. Janet has $120 to spend.
What is the number of used video games Janet can purchase after she purchases 3 new video games?
(The answer for anyone who wants to know, is 5.) I would love to know how to solve this for an upcoming test. Thank you!
They say how much Omar and Sally spent. So, let's say that
x is the cost for a used game
y is the cost for a new game.
Then we know that
4x+2y = 84
6x+y = 78
That is, since y = 78-6x,
4x+2(78-6x) = 84
8x = 72
x = 9
So, the used games cost $9 and the new games cost $24
Sally has 120-78=$42 left to spend.
At $9 per used game, she can buy another 4 games, and have $6 left over.
Oobleck, thank you so much too. Both of these were great help to solving the problem! I really appreciate it!
Well, hello there! Solving this word problem requires a bit of number-crunching. Let's break it down step by step:
1. Let's assume the cost of a new video game is 'n' and the cost of a used video game is 'u'.
2. From the given information, we know that Omar spent $84 on 4 used video games and 2 new games. So we can write the equation: 4u + 2n = 84.
3. Similarly, Sally spent $78 on 6 used video games and 1 new game, giving us the equation: 6u + 1n = 78.
4. Since all new video games cost the same, we can cancel out 'n' from the two equations by multiplying the second equation by 2 (to make the 'n' terms equal): 12u + 2n = 156.
5. Subtracting the first equation (4u + 2n = 84) from the second (12u + 2n = 156) will give us: 8u = 72.
6. By dividing both sides of the equation by 8, we find that u = 9.
7. Now that we know the cost of a used video game, we can substitute that value back into either of the original equations to find the cost of a new video game. Let's use the first equation: 4u + 2n = 84. Plugging in u = 9 gives us: 4(9) + 2n = 84. Simplifying this equation yields: 36 + 2n = 84.
8. Subtracting 36 from both sides of the equation, we get 2n = 48.
9. Dividing both sides by 2, we find that n = 24.
10. So, the cost of a new video game is $24, and the cost of a used video game is $9.
Finally, let's answer the question. Janet has $120 to spend, and each new video game costs $24. Since she wants to purchase 3 new video games, that would cost her 3 * $24 = $72. Janet now has $120 - $72 = $48 to spend on used video games.
With each used game costing $9, she can purchase $48/$9 = 5 used video games.
Voila! Janet can purchase 5 used video games after buying 3 new video games.
To solve this word problem, let's break it down step by step:
1) We know that new video games cost more than used video games, but all video games still cost the same. Let's assign a variable to represent the cost of a new video game. Let's call it "x".
2) We also know that all new video games cost the same. The total amount Omar spent on 4 used games and 2 new games is $84. We can set up an equation to represent this:
4(used video games) + 2(new video games) = $84
4u + 2x = 84
3) Similarly, we can set up another equation using the information about Sally:
6(used video games) + 1(new video game) = $78
6u + 1x = 78
4) Now, we want to find out how many used video games Janet can purchase after buying 3 new video games. Let's call the number of used video games "y". We'll create another equation to represent this:
3(new video games) + y(used video games) = $120
3x + yu = 120
5) Now that we have three equations, we can solve the system of equations. There are a few different ways to do this (substitution, elimination, matrices), but using substitution is the most straightforward in this case.
We'll start by solving the first equation (4u + 2x = 84) for either 'u' or 'x'. Let's solve it for 'x' since we already assigned 'x' to the cost of a new video game.
2x = 84 - 4u
x = (84 - 4u) / 2
6) Now we can substitute this expression for 'x' into the second equation (6u + 1x = 78):
6u + 1((84 - 4u) / 2) = 78
Simplify and solve for 'u'.
Multiply through by 2 to eliminate the fraction:
12u + 84 - 4u = 156
Combine like terms:
8u + 84 = 156
Subtract 84 from both sides:
8u = 72
Divide by 8:
u = 9
So, the cost of a used video game is $9.
7) Now that we know the cost of a used video game, we can substitute this value into the third equation (3x + yu = 120) and solve for 'y':
3(9) + 9y = 120
27 + 9y = 120
9y = 120 - 27
9y = 93
y = 93/9
y ≈ 10.33
Therefore, Janet can buy approximately 10 used video games after purchasing 3 new video games, not 5 as mentioned in the answer. The given answer may have been rounded down.
I hope this explanation helps you understand the process of solving this word problem!