Word problems with Venn diagrams

Sur­veys often report how many peo­ple like one thing, how many like another, and how many like both. Sets and Venn dia­grams turn such reports into exact counts: how many like nei­ther, how many like only one, how many like all three. The skill is in trans­lat­ing the words of the prob­lem into the right set, so this les­son starts with a dic­tio­nary of phrases and then solves prob­lems with two and three sets.

From words to sets#

"Each per­son likes at least one" means nobody is out­side both cir­cles, so $n(A \cup B)$ is the whole group.

Two sets#

Exam­ple 1#

In a group of 70 peo­ple, 37 like cof­fee, 52 like tea, and each per­son likes at least one of the two drinks. How many like both cof­fee and tea?

Let $C$ and $T$ be the sets of peo­ple who like cof­fee and tea. Then $n(C) = 37$, $n(T) = 52$ and $n(C \cup T) = 70$. By the for­mula,

$$n(C \cup T) = n(C) + n(T) - n(C \cap T) \implies 70 = 37 + 52 - n(C \cap T)$$

So $n(C \cap T) = 89 - 70 = 19$. Nine­teen peo­ple like both.

Going fur­ther: only cof­fee $= 37 - 19 = 18$, and only tea $= 52 - 19 = 33$. Check: $18 + 19 + 33 = 70$.

Exam­ple 2#

In a class of 35 stu­dents, 24 like to play cricket and 16 like to play foot­ball. Each stu­dent likes to play at least one of the two games. How many like to play both?

$n(C \cup F) = 35$, so $35 = 24 + 16 - n(C \cap F)$, giv­ing $n(C \cap F) = 40 - 35 = 5$.

Exam­ple 3#

In a class of 60 stu­dents, 25 learn music, 30 learn dance and 10 learn both. How many learn nei­ther?

$n(M \cup D) = 25 + 30 - 10 = 45$. The num­ber who learn nei­ther is $n\left((M \cup D)'\right) = 60 - 45 = 15$.

Exam­ple 4#

A sur­vey shows that 63% of the peo­ple in a city like tea and 76% like cof­fee. If $x\%$ like both tea and cof­fee, find the pos­si­ble val­ues of $x$.

Think of the city as 100 peo­ple, so $n(U) = 100$, $n(T) = 63$, $n(C) = 76$ and $n(T \cap C) = x$.

Upper limit. $T \cap C$ is a sub­set of $T$ and of $C$, so $x \le \min\{63, 76\} = 63$.

Lower limit. $n(T \cup C)$ can­not be more than the whole pop­u­la­tion: $63 + 76 - x \le 100$, so $139 - x \le 100$ and $x \ge 39$.

So $39 \le x \le 63$. At least 39% and at most 63% of the peo­ple like both drinks.

Three sets#

With three sets, fill in the Venn dia­gram from the inside out: first the cen­tre (all three), then the "two only" regions, then the "only one" regions. Each region is found by sub­tract­ing what is already known.

Exam­ple 5#

Out of 800 boys in a school, 224 played cricket, 240 played hockey and 336 played bas­ket­ball. Of these, 64 played both bas­ket­ball and hockey, 80 played cricket and bas­ket­ball, 40 played cricket and hockey, and 24 played all three games. How many boys did not play any game?

Let $C$, $H$, $B$ be the three sets.

By the for­mula:

$$n(C \cup H \cup B) = 224 + 240 + 336 - 64 - 80 - 40 + 24 = 640$$

So the num­ber who played no game is $800 - 640 = 160$.

Region by region, which also shows how many played only one game:

The seven regions add up to $24 + 56 + 40 + 16 + 128 + 160 + 216 = 640$, as before, and $800 - 640 = 160$ boys played no game.

Exam­ple 6#

In a sur­vey of 100 peo­ple about three news­pa­pers $A$, $B$ and $C$: 40 read $A$, 35 read $B$, 30 read $C$, 12 read $A$ and $B$, 10 read $B$ and $C$, 8 read $A$ and $C$, and 5 read all three. Find how many read at least one paper, none, exactly one, and exactly two.

Here $n(A) + n(B) + n(C) = 105$ and the pair­wise inter­sec­tions add up to 30.

At least one: $105 - 30 + 5 = 80$. None: $100 - 80 = 20$.

Exactly two: $30 - 3 \times 5 = 15$. Exactly one: $105 - 2 \times 30 + 3 \times 5 = 60$.

Check: $60 + 15 + 5 = 80$.

Prac­tice#

1. In a group of 50 peo­ple, 30 speak Tel­ugu and 35 speak Eng­lish, and every­one speaks at least one of the two. How many speak both?
2. In a group of 65 peo­ple, each likes cricket or ten­nis or both. If 40 like cricket and 10 like both, how many like ten­nis? How many like ten­nis only?
3. In a class of 120 stu­dents, 70 study Math­e­mat­ics, 60 study Sci­ence and 30 study both. How many study nei­ther?
4. In a town, 55% of the peo­ple read an Eng­lish paper and 70% read a Tel­ugu paper. If $x\%$ read both, find the range of pos­si­ble val­ues of $x$.
5. Of 200 peo­ple, 120 read paper $P$, 90 read paper $Q$, and 50 read nei­ther. How many read both?
6. Of 100 stu­dents, 40 play foot­ball, 45 cricket and 35 hockey; 15 play foot­ball and cricket, 12 cricket and hockey, 10 foot­ball and hockey, and 5 play all three. How many play at least one game? How many play none?
7. For ques­tion 6, how many play exactly one game, and how many play exactly two?
8. For ques­tion 6, how many play cricket only?
9. In a colony of 5000 fam­i­lies, 50% watch chan­nel P, 30% watch Q and 20% watch R; 10% watch P and Q, 6% Q and R, 8% P and R, and 4% watch all three. How many fam­i­lies watch P only? How many watch none of the three?

Answers#