SOLUTION: After watching a major musical production at the theater, the patrons can purchase souvenirs. If a family purchases 6 t-shirts, the video, and a stuffed animal, their total is $135

Question 1136979: After watching a major musical production at the theater, the patrons can purchase souvenirs. If a family purchases 6 t-shirts, the video, and a stuffed animal, their total is $135.
A couple buys 2 t-shirts, the video, and 5 stuffed animals for their nieces and spends $175. Another couple buys 2 t-shirts, the video, and a stuffed animal, and their total is $75. What is the cost of each item?
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

Let's solve the problem first using logical reasoning. Then we can solve it using algebra, and we will see that the solution using formal algebra does EXACTLY the same thing as the informal solution.

The difference between the purchases by the first and third groups is 4 t-shirts (they both buy the video and one stuffed animal; one group buys 6 t-shirts while the other buys only 2). The difference in the total cost between those two purchases is $60; that means each t-shirt costs $60/4 = $15.

Knowing the cost of each t-shirt, we can determine the costs of the other items purchased by the three groups:
1st group: 1 video and 1 stuffed animal for ($135-$90) = $45
2nd group: 1 video and 5 stuffed animals for ($175-$30) = $145
3rd group: 1 video and 1 stuffed animal for ($75-$30) = $45 (same as the 1st group; so we won't need this)

Comparing these parts of the purchases by the first and second groups shows that the extra 4 stuffed animals cost an extra $100; that means the cost of each stuffed animal is $100/4 = $25.

Now, knowing the cost of each t-shirt and each stuffed animal, we can determine the cost of the video using any one of the group's purchases. For example, the third group bought 2 t-shirts for $30 and 1 stuffed animal for $25, for a total of $55; the cost of the video is then $75-$55 = $20.

ANSWER: t-shirt $15; video $20; stuffed animal $25

Now let's do the formal algebra....

(1) 6t + v + a = 135
(2) 2t + v +5a = 175
(3) 2t + v + a = 75

The coefficients of both v and a are the same in equations (1) and (3), so subtracting one equation from the other lets us solve for a:
(4)

The cost of each t-shirt is $15.

Substitute t-15 in each of the original equations:

(5) 90 + v + a = 135 --> v + a = 45
(6) 30 + v +5a = 175 --> v +5a = 145
(7) 30 + v + a = 75 --> v + a = 45 (same as (5), so we won't need this)

The coefficient of v is the same in both (5) and (6), so subtracting one equation from the other eliminates v, allowing us to find a:

The cost of each stuffed animal is $25.

Substituting that value for a in (7) then gives us the value of v:

The cost of the video is $20.

Again note that the formal algebra does exactly the same thing as a solution using logical reasoning. The algebra simply uses mathematical notations instead of words.
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