Track / Stage 1 / 215 of 240 #215 of 2444

Problem 215

Combinatorics Difficulty 1.6 Multiple choice CEMC Pascal · Canada · 2023

In a survey, 100 students were asked if they like lentils and
were also asked if they like chickpeas. A total of 68 students like
lentils. A total of 53 like chickpeas. A total of 6 like neither lentils
nor chickpeas. How many of the 100 students like both lentils and
chickpeas?

Pick one

Official solution

Suppose that $x$ students like
both lentils and chickpeas.

Since 68 students like lentils, these 68 students either like chickpeas
or they do not.

Since $x$ students like lentils and
chickpeas, then $x$ of the 68
students that like lentils also like chickpeas and so $68-x$ students like lentils but do not
like chickpeas.

Since 53 students like chickpeas, then $53-x$ students like chickpeas but do not
like lentils.

We know that there are 100 students in total and that 6 like neither
lentils nor chickpeas.

We use a Venn diagram to summarize this information:

[[IMAGE0]]

Since there are 100 students in total, then $(68-x) + x + (53-x) + 6 = 100$ which gives
$127 - x = 100$ and so $x = 27$ .

Therefore, there are 27 students that like both lentils and
chickpeas.

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