Question 1208467
<font color=black size=3>
x = number of $11 tickets sold
y = number of $7 tickets sold
x+y = total number of tickets
0.25(x+y) = number of people who bought a refreshment


A = 11x = amount spent on the $11 tickets
B = 7y = amount spent on the $7 tickets
C = 4*0.25(x+y) = x+y = amount spent on refreshments


A+B+C = total amount spent
A+B+C = 124
11x+7y+x+y = 124
12x+8y = 124
4(3x+2y) = 124
3x+2y = 124/4
3x+2y = 31
Using trial-and-error, a graph, or the Extended Euclidean Algorithm, you will find these nonnegative integer solutions<pre>
(x, y) = (1, 14)
(x, y) = (3, 11)
(x, y) = (5, 8)
(x, y) = (7, 5)
(x, y) = (9, 2)</pre>Each time x goes up by 2, y decreases by 3.


Plug each of those coordinates into 0.25(x+y) to see which results in an integer.
<pre>
(x,y) = (1, 14) ----> 0.25(x+y) = 0.25(1+14) = 0.25(15) = 3.75
(x,y) = (3, 11) ----> 0.25(x+y) = 0.25(3+11) = 0.25(14) = 3.5
(x,y) = (5, 8) -----> 0.25(x+y) = 0.25(5+8) = 0.25(13) = 3.25
(x,y) = (7, <font color=red>5</font>) -----> 0.25(x+y) = 0.25(7+<font color=red>5</font>) = 0.25(12) = <font color=blue size=4><b>3</b></font>
(x,y) = (9, 2) -----> 0.25(x+y) = 0.25(9+2) = 0.25(11) = 2.75
</pre>Of those results only <font color=blue>3</font> is an integer. Everything else is a decimal value.
The result <font color=blue>3</font> corresponds to (x,y) = (7,<font color=red>5</font>)
Another way to find this is to note that x+y = 7+<font color=red>5</font> = 12 is a multiple of 4. Every other sum is not a multiple of 4 (eg: x+y=1+14 = 15).



Question: how many $7 tickets were sold?
Answer: <font color=red>5</font>
</font>