Question 535556: The concession stand is selling hot dogs and hamburgers during a game. At halftime, they sold a total of 78 hot dogs and hamburgers and brought in $105.50. How many of each item did they sell if hamburgers sold for $1.50 and hot dogs sold for $1.25?
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! IF YOU ARE STUDYING ALGEBRA,
you are used to working with variables, and know that the most popular names for those variables are x and y.
Let x be the number of burgers.
Let y be the number of hot dogs.
The amount, in dollars that hamburgers brought in is
 or 
(Zeros at the end, after a decimal point have no meaning in algebra, and are ignored. Calculators ignore them too).
The amount, in dollars that hot dogs brought in is
The total amount is
If decimals bother you (or your teacher), we can multiply everything by some number to get an equivalent equation without decimals:
 ---> 
The total number of items sold is
Now we have a system of 2 linear equations with 2 variables to solve.
You probably know that there are two very popular methods to do it: substitution and elimination.
Substitution:
We solve for y (you could choose to solve for x):
---> 
We substitute that expression in the other equation:
 --->  --->  --->  ---> --->
Then we substitute that value for x in
 --->
IF YOU NEVER STUDIED ALGEBRA your strategies would be guess and check or you could calculate the effect of talking each person wanting a hot dog into ordering a hamburger instead. (For only an extra $0.25, they get more protein and less fat). If everyone ordered hot dogs, all 78 items sold would be hot dogs and the total would be $97.50. To get $105.50 you need another $8.00, meaning 32 of those orders to switch from hot dog to burger, leaving 32 burgers and 78-32=46 hot dogs.
Guess and check would start by guessing a number for hot dogs of hamburgers and checking the number for the other item and the total.
You could guess 40 hot dogs ($50=40*$1.25), so that would mean 38 (78-40=38) hamburgers ($57=38*$1.50) That would be a total of $107, which is too much. Less of the expensive item (the burger) was sold. So you try 50 hot dogs ($62.50=50*$1.25), which leaves 28 (78-50=28) burgers ($42=$1.50*28), for a total of $104.50, which is too little, but you are a bit closer. You need a guess between 40 and 50 hot dogs, a bit closer to 50 than to 40. Guessing 46 hot dogs would get you the answer. Other wise, you would need one or two more guesses.
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