GMAT 700+ Math Question 16

The given question is GMAT 700+ Math problem solving question combining concepts in number properties and linear equations. The question is a GMAT hard math word problem.

Question 16: 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.

1. 14
2. 11
3. 2
4. 8
5. 5

Video Explanation

Explanatory Answer | GMAT Word Problem

Step 1: Assign Variables | Frame Equations

Let ‘x’ and ‘y’ be the number of tickets sold at $11 and $7 respectively.
Then, total number of tickets sold = x + y

Collection by selling 'x' $11 tickets = 11x
Collection by selling 'y' $7 tickets = 7y
Total collection by selling (x + y) tickets = 11x + 7y

A fourth of all those who bought a ticket also spent $4 each on refreshments at the movie hall.
i.e., \\frac{1}{4})(x + y) spent $4 on refreshments.
∴ Collections from sale of refreshments = \\frac{1}{4})(x + y) × 4 = (x + y)

Total collection from tickets and refreshments = $124
i.e., 11x + 7y + (x + y) = 124
Or 12x + 8y = 124
Divide the equation by 4: 3x + 2y = 31

Step 2: Solve 3x + 2y = 31 for x and y using these 2 facts

Fact 1: x and y are positive integers.
Fact 2: \\frac{1}{4}) (x + y) is an integer because a fourth of total tickets sold should be an integer. i.e., (x + y) should be a multiple of 4.

Let us list down the different possibilities that satisfy the equation and fact 1

Of all the different positive integer values of x and y that satisfy the equation, the only combination (x, y) = (7, 5) is divisible by 4

So, the movie hall sold five $7 tickets.

Choice E is the correct answer.