SOLUTION: At a recent concert, souvenirs were available for the following prices: Sweatshirts for $65.00 Hats for $23.00 Key Rings for $4.00 During a recent period, 33 items were sold fo

Question 1164906: At a recent concert, souvenirs were available for the following prices:
Sweatshirts for $65.00
Hats for $23.00
Key Rings for $4.00
During a recent period, 33 items were sold for $1,149.00. If the number of sweatshirts sold was twice the number of key rings sold, find the number of each item sold.
Let X represent the number of sweatshirts sold.
Let Y represent the number of hats sold.
Let Z represent the number of key rings sold

Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52926)   (Show Source): You can put this solution on YOUR website!
.

I prefer to solve this problem using ONLY ONE unknown R which is the number of key rings. Then the number of sweatshirts is 2R, according to the condition, and the number of hats is the rest (33 - R - 2R) = (33-3R) items. Then I write the money equation (also called "the revenue equation") 4R + 65*(2R) + 23*(33-3R) = 1149 dollars. Next, I simplify this equation step by step 4R + 130R + 23*33 - 69R = 1149 65R = 1149 - 23*33 65R = 390 R = 390/65 = 6. ANSWER. 6 key rings; 2*6 = 12 Sweatshirts; and the rest (33-6 - 12) = 15 hats.

Solved.

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Using single unknown and single equation is the MAJOR LESSON for you to learn from my solution.

There is no need to explain that obvious fact that it reduces the volume of calculations,
reduces your efforts and diminish the chances to make errors.

If you want to see many other similar solved problems, look into the lesson
    - Advanced word problems to solve using a single linear equation
in this site.

Your major task for looking into this lesson is to learn on HOW TO recognize such problems (solved in one unknown)
from the first glance.

Answer by greenestamps(13216)   (Show Source): You can put this solution on YOUR website!

As instructed...

Let X represent the number of sweatshirts sold.
Let Y represent the number of hats sold.
Let Z represent the number of key rings sold.

Given:

(1) x+y+z = 33
(2) 65x+23y+4z = 1149
(3) x = 2z

Substitute (3) into (1) and (2) and simplify:

2z+y+z = 33
(4) y+3z = 33

65(2z)+23y+4z = 1149
(5) 23y+134z = 1149

Multiply (4) by 23 and compare to (5):

23y+69z = 759
23y+134z = 1149
65y = 390

You can finish from there:
solve the last equation for y
substitute into (4) to solve for z
substitute into (3) to solve for x

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