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# PRMO 2016 Problem No 5 | Set Theory Problem

Try this beautiful Set theory Problem based on Set theory from PRMO - 2016.

## Set Theory Problem - PRMO 2016 Problem 5

In a school there are 500 students. Two-thirds of the students who do not wear glasses, do not bring lunch. Three-quarters of the students who do not bring lunch do not wear glasses. Altogether, 60 students whowear glasses bring lunch. How many students do not wear glasses and do not bring lunch?

### Key Concepts

Set theory

Intersection and Union of Sets

Solving equations of several variables

## Suggested Book | Source | Answer

Challenges and Thrills of Pre-college Mathematics

PRMO 2016 Problem 5

240

## Try with Hints

Divide the children into four parts :

m: Those who don't wear glasses, and don't bring lunch.

n: Those who wear glasses but don't bring lunch.

p: Those who bring lunch but don't wear glasses.

q : Those who do bring lunch and wear glasses.

Try to find information and form equations

There can be following equations formed:

$m + n + p + q =500.$

$m = \frac{2}{3}(m + p)$

$m=\frac{3}{4}(m+n)$

$q=60$

Try to solve by finding the values of the different variables

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