Question 1208467
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A movie hall sold tickets to one of its shows in two denominations, $11 and $7. 
A fourth of all those who bought a ticket also spent $4 each on refreshments at the movie hall. 
If the total collections from tickets and refreshments for the show was $124, 
how many $7 tickets were sold? 
Note: The number of $11 tickets sold is different from the number of $7 tickets sold.
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<pre>
Let x be the number of movie tickets sold at $11.
Let y be the number of movie tickets sold at  $7.


We know that x+y, the total number of visitors, is a multiple of 4.
Also, we know that the total revenue was $124 and that x =/= y.


So, we write this equation for the total revenue

    11x + 7y + {{{4*((x+y)/4)}}} = 124,   (1)


and we look for a solution to it in non-negative integer numbers such that and x=/= y and x+y is a multiple of 4.


We simplify equation (1)

    12x + 8y = 124.


We re-write (2) in an equivalent form

    x = {{{(124 -8y)/12}}} = {{{(31-2y)/3}}}.


31 - 2y  is divisible by 3 for y = 2, 5, 8, 11, 14, giving for x these values, respectively

                                   9, 7, 5,  3,  1.


Thus we see that these pairs (x,y) = (9,2), (7,5), (5,8), (3,11), (1,14) are the potential solutions.

But then we check for the sum x+y to be a multiple of 4,  and we see that the only a pair,

satisfying this condition, is (x,y) = (7,5).


<U>ANSWER</U>.  7 tickets at $11  and  5 tickets at $7 dollars.


<U>CHECK</U>.   7*11 + 5*7 + {{{((7+5)/4)*4}}} = 77 + 35 + 12 = 124 dollars as the total revenue, including refreshments.  ! correct !
</pre>

Solved.


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Notice that the condition x=/= y is excessive: it is not used in the solution.