Question 535556
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
{{{1.50*x}}} or {{{1.5*x}}}
(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
{{{1.25*y}}}
The total amount is
{{{1.5x+1.25y=105.5}}}
If decimals bother you (or your teacher), we can multiply everything by some number to get an equivalent equation without decimals:
{{{4*(1.5x+1.25y)=4*105.5}}} ---> {{{6x+5y=422}}}
The total number of items sold is
{{{x+y=78}}}
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):
{{{x+y=78}}} ---> {{{y=78-x}}}
We substitute that expression in the other equation:
{{{6x+5*(78-x)=422}}} ---> {{{6x+5*78-5*x)=422}}} ---> {{{6x+390-5x)=422}}} ---> {{{x+390=422}}}--->{{{x=422-390}}}--->{{{x=32}}}
Then we substitute that value for x in {{{y=78-x}}}
{{{y=78-32}}}--->{{{y=46}}}
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.