Problems on Sets

Lessons

Senior 3 · Mathematics

Problems on Sets

4 lessons

Lesson 2 of 4

Solving problems with two sets

24 min

A question with a range for an answer

A coffee cooperative in Huye has 50 members. 32 of them also grow maize and 27 also grow beans. How many grow both crops? From these facts alone, nobody can give one number. But you can be sure that the answer is at least 9 and at most 27, and by the end of this lesson you will know why.

In S1 you solved two-set problems in which every number but one was given. At S3 the information arrives in other forms: as a ratio between regions ("twice as many"), as percentages, or with a region left open. You will need the Venn diagram, the formula and, often, an equation.

One formula, five uses

You already know the rule $n(A \cup B) = n(A) + n(B) - n(A \cap B)$ , and that every element of $U$ is either in $A \cup B$ or in neither set. Rearranged, these two facts answer almost every two-set question.

You want	Region(s)	Formula
in both	2	$n(A \cap B) = n(A) + n(B) - n(A \cup B)$
in $A$ only	1	$n(A \cap B') = n(A) - n(A \cap B)$
in exactly one	1 and 3	$n(A \,\Delta\, B) = n(A) + n(B) - 2\,n(A \cap B)$
in neither	4	$n(A' \cap B') = n(U) - n(A \cup B)$
not in $A$	3 and 4	$n (A^{'}) = n (U) - n (A)$

Why the "exactly one" formula has a 2: region 1 holds $n(A) - n(A \cap B)$ elements and region 3 holds $n(B) - n(A \cap B)$ . Adding them subtracts the overlap once for each set.

Definition

Consistent data: counts for a Venn diagram are consistent (possible) when every region receives a whole number that is zero or more, and the regions add up to $n (U)$ .

Notation: none; you test it by filling in every region.

Example: "Of 50 learners, 35 walk, 30 come by bus, 10 do both, and every learner does at least one" is inconsistent, because $35 + 30 - 10 = 55$ , which is more than 50.

Worked example 1: a ratio between two regions

Every one of the 42 learners in a class studies French ( $F$ ), Kiswahili ( $K$ ) or both. Twice as many study French only as study Kiswahili only, and 6 study both languages. How many study (a) French (b) Kiswahili?

Solution Let $x$ be the number who study Kiswahili only. Then $2 x$ study French only. Nobody is outside both circles.

$n (U) = 42$

The regions add up to the class: $2 x + 6 + x + 0 = 42$ , so $3 x = 36$ and $x = 12$ .

(a) $n (F) = 2 x + 6 = 24 + 6 = 30$ . (b) $n (K) = x + 6 = 12 + 6 = 18$ .

Check: $n(F \cup K) = 30 + 18 - 6 = 42$ . ✓

Worked example 2: percentages

In a survey of 400 households in one sector (illustrative figures), 70% owned a mobile phone ( $P$ ), 45% owned a radio ( $R$ ) and 10% owned neither. (a) What percentage owned both? (b) How many households owned a radio but no phone?

Solution Work in percentages; the whole survey is 100%.

(a) $n(P \cup R) = 100\% - 10\% = 90\%$ , so $n(P \cap R) = 70\% + 45\% - 90\% = 25\%$ .

(b) Radio only $= 45\% - 25\% = 20\%$ , and 20% of 400 is $0.20 \times 400 = 80$ households.

Check: $45\% + 25\% + 20\% + 10\% = 100\%$ . ✓

Worked example 3: two facts about the same unknown

A tour company took 60 visitors round Rwanda. 36 visited Volcanoes National Park ( $V$ ) and 6 visited neither Volcanoes nor Akagera National Park ( $A$ ). The number who visited both parks was one third of the number who visited Akagera. How many visited Akagera?

Solution Let $x$ be the number who visited both. Then $n (A) = 3 x$ , so Akagera only $= 3 x - x = 2 x$ , and Volcanoes only $= 36 - x$ .

$n (U) = 60$

Add the four regions: $(36 - x) + x + 2 x + 6 = 60$ , so $42 + 2 x = 60$ , $x = 9$ , and $n(A) = 3 \times 9 = 27$ .

Check: the regions are $27, 9, 18, 6$ , which add up to 60 ✓, and 9 is one third of 27 ✓.

The greatest and the least possible overlap

Return to the cooperative: $n (U) = 50$ , $n (M) = 32$ (maize) and $n (B) = 27$ (beans).

Greatest overlap. The overlap is part of $B$ , so it cannot be bigger than 27. It equals 27 when every bean grower also grows maize, that is $B \subseteq M$ . Then 5 grow maize only and $50 - 32 = 18$ grow neither.
Least overlap. The union cannot be bigger than the whole cooperative: $n(M \cup B) \le 50$ . So $n(M \cap B) = 32 + 27 - n(M \cup B) \ge 59 - 50 = 9$ . It equals 9 when every member grows at least one of the crops.

The two extreme cases. Every other arrangement has an overlap between 9 and 27.

Important

Rule (bounds for the overlap). For two sets in a universal set $U$ ,

n(A) + n(B) - n(U) \;\le\; n(A \cap B) \;\le\; \text{the smaller of } n(A) \text{ and } n(B).

If $n (A) + n (B) - n (U)$ is negative, the least possible overlap is 0.

Why: the right-hand bound holds because $A \cap B$ is a subset of both $A$ and $B$ . The left-hand bound holds because $n(A \cup B) \le n(U)$ , and $n(A \cap B) = n(A) + n(B) - n(A \cup B)$ .

Worked example 4: judging a claim

In an S3 class, 80% passed Mathematics and 65% passed English. (a) Find the least and greatest possible percentages that passed both. (b) Keza says that 10% of the class failed both subjects. Is that possible?

Solution

(a) Least: $80\% + 65\% - 100\% = 45\%$ . Greatest: the smaller of 80% and 65%, which is 65%.

(b) If 10% failed both, then $n(M \cup E) = 90\%$ and the overlap is $80\% + 65\% - 90\% = 55\%$ . Since $45\% \le 55\% \le 65\%$ , the regions are $25 %$ , $55 %$ , $10 %$ , $10 %$ , all zero or more and adding up to 100%. Keza's figure is possible.

Warning

Percentages that add up to more than 100% are not a mistake. In Worked example 2, $70\% + 45\% = 115\%$ . The extra is there because the 25% who own both were counted in both groups. Only the four regions must add up to 100%.

Tip

Put the unknown where the question is silent. Let $x$ be the region that nobody gives you, usually the overlap. Write every other region in terms of $x$ , then add all four regions and set the total equal to $n (U)$ .

Steve Crow: Solving Word Problems With Venn Diagrams Two Sets (7 min). A short refresher on turning a two-set word problem into a Venn diagram. Then try the ratio, percentage and greatest-and-least problems in this lesson, which go further than the video.

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Practice

Draw a Venn diagram for every question. The last three are harder.

$n (U) = 80$ , $n (A) = 45$ , $n (B) = 38$ and $n(A \cap B) = 15$ . Find (a) $n(A \cup B)$ (b) the number in exactly one set (c) $n(A' \cap B')$ .
Each of 36 learners likes tea, milk or both. 20 like tea and 22 like milk. How many like both, and how many like exactly one?
Of 120 traders at a market, three times as many sell only fruit as sell only vegetables. 20 sell both and 12 sell neither. How many sell fruit?
In a survey, 55% of the people read a newspaper, 62% listen to the radio and 8% do neither. What percentage do both?
In a class of 45, the number who play volleyball only equals the number who play basketball only. 9 play both games and 6 play neither. How many play volleyball?
$n (U) = 40$ , $n (A) = 25$ and $n (B) = 21$ . Find the least and greatest possible values of (a) $n(A \cap B)$ (b) $n(A' \cap B')$ .
$n(A) = 2\,n(B)$ , $n(A \cap B) = 5$ and $n(A \cup B) = 40$ . Find $n (A)$ and $n (B)$ .
A report says: "Every one of our 50 learners walks to school or comes by bus. 35 walk, 30 come by bus and 10 do both." Show that the report contains a mistake.

Show answersHide answers

(a) $45 + 38 - 15 = 68$ (b) $45 + 38 - 2 \times 15 = 53$ (c) $80 - 68 = 12$
Both: $20 + 22 - 36 = 6$ . Exactly one: $36 - 6 = 30$ .
Vegetables only $x$ , fruit only $3 x$ : $3 x + 20 + x + 12 = 120$ , so $x = 22$ . Fruit: $3 \times 22 + 20 = 86$ .
Union $= 92 %$ ; both $= 55\% + 62\% - 92\% = 25\%$ .
$x + 9 + x + 6 = 45$ , so $x = 15$ . Volleyball: $15 + 9 = 24$ .
(a) Least $25 + 21 - 40 = 6$ , greatest 21. (b) Neither $= 40 - (46 - n(A \cap B)) = n(A \cap B) - 6$ , so from 0 up to 15.
With $n (B) = b$ : $2 b + b - 5 = 40$ , so $b = 15$ . $n (B) = 15$ , $n (A) = 30$ .
$n(W \cup B) = 35 + 30 - 10 = 55$ , but there are only 50 learners. The data are inconsistent.

Sources and credits

1.4: Set Operations with Two Sets, from Contemporary Mathematics by Donna Kirk and contributors, OpenStax, provided by LibreTexts, licensed under CC BY 4.0 (licence read on the page, 30 September 2026). Adapted by Ganzaa.
The table of rearranged formulas, the rule for the greatest and least overlap and the consistency test. Written by Ganzaa.
Video: Solving Word Problems With Venn Diagrams Two Sets, Steve Crow, embedded from YouTube.

The Rwandan worked examples, the diagrams, the practice items and the quiz questions were written by Ganzaa.

1A question with a range for an answer

2One formula, five uses

3The greatest and the least possible overlap

A question with a range for an answer

One formula, five uses

The greatest and the least possible overlap