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Algebra Difficulty 2.8 Junior Find the answer Canada

Of the $100$ people who are
watching a movie, $47$ bought
popcorn, more than $40$ bought a
drink, $x$ bought both popcorn and a
drink, and $2x$ bought neither. If
$x>0$ , what is the largest
possible value of $x$ ?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Let $y$ be the number of
people who bought a drink. It is given that $y>40$ .

The number of people who bought popcorn but did not buy a drink is $47-x$ .

The number of people who bought a drink but did not buy popcorn is $y-x$ .

The number of people who bought both is given to be $x$ , and the number of people who bought
neither is given to be $2x$ .

Every person falls into exactly one of the cases above, so we have that
$100 = (47-x) + (y-x) + x + 2x$
which can be solved for $y$ to get
$y = 53-x$ . It is given that $y>40$ , so we must have $53-x > 40$ , which can be rearranged to
get $x < 53-40$ or $x<13$ . Since $x$ is an integer, we must have $x\leq 12$ .

Note that if $x=12$ , the purchases
satisfy the conditions:

Purchases
Number of people

popcorn and drink
$12$

popcorn and no drink
$35$

drink and no popcorn
$29$

nothing
$24$

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Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.