Set Theory & Venn Diagrams

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Survey & Caselet Word Problems

⏱ 4 min read🧩 4 question types🎯 12 practice Q

The idea in one minute

Surveys, languages and subscriptions are set questions in disguise. Convert each sentence into a region count or a percentage, then read off the answer. Checking that every region stays non-negative catches bad data.

Turn sentences into regions

A survey caselet is a set question wearing words. Underline each group name and each joining word.

"like tea" — a single count.
"like tea and coffee" — the overlap.
"like none of these" — the neither block.

Write the four-region sum and fill what is given. The one unknown region falls out of the total.

Worked: a survey of 250 students has 130 taking Mathematics and 120 taking Biology, while 40 take neither.

At least one subject: $250 - 40 = 210$ .
Both: $130 + 120 - 210 = 40$ .
Check: only Maths $90$ $+$ only Biology $80$ $+$ both $40$ $+$ neither $40$ $= 250$ .

Percentage surveys

Work on a base of 100 when everything is in percent. The arithmetic is the same; only the units change.

Worked: 65% read A, 48% read B, 20% read neither.

At least one $= 80\%$ .
Both $= 65 + 48 - 80 = 33\%$ .
Exactly one $= 80 - 33 = 47\%$ .

When a percent must become people, multiply it by the survey total.

Tip: Never mix counts and percents in one line. Convert everything to one scale first.

Surveys with three groups

Languages and three-product surveys use the three-set union formula. Keep a small table: three singles, three pairs, the triple and none.

Worked: 55% tea, 45% coffee, 35% milk; pairs 18%, 15%, 12%; triple 6%.

Union $= 135 - 45 + 6 = 96\%$ , so none $= 4\%$ .

Note: With "every person likes at least one", the union must equal 100%. That pins a hidden triple.

Checking the data

Real surveys sometimes carry impossible numbers. Regions catch them.

The union cannot exceed the total.
Each "only" region must stay non-negative.

Worked: 200 people; singles 120, 100, 90; pairs 60, 50, 45; triple 30.

Union $= 310 - 155 + 30 = 185$ , so none $= 15$ . A report claiming "no one said none" would be wrong.

Watch: If a region turns negative, do not force an answer — the data is inconsistent.

CAT-style caselets

A caselet gives four or five facts and asks two or three sub-questions. The hidden unknown is usually the triple intersection.

Write the union formula with $t$ for the triple.
Use "everyone covered" or the total to fix the union.
Solve for $t$ , then fill all seven regions.
Answer each sub-question by reading the regions.

Worked: 60% tea, 50% coffee, 45% milk; pairs 25%, 20%, 15%; everyone drinks at least one.

Union $= 155 - 60 + t = 95 + t$ .
The union must be $100\%$ , so $t = 5\%$ .
Every sub-question is now a read-off: only tea $= 25\%$ , both tea and coffee only $= 20\%$ .

Tip: Percentages multiply cleanly on a base of 100. Keep the base fixed through the whole caselet.

Question types you will see

Each type: how to recognise it, the method step by step, and one question to try.

Type 1very common3 practice Q

Survey with a 'neither' block

How to spot it:

A count-based survey of two groups with 'n like neither' or 'n like none'.

n(A \cap B) = n(A) + n(B) - (n(U) - \text{neither})

Method

Union $=$ total $-$ neither.
Apply the union formula for the missing region.
Check the four regions add up.

Why it works:

The neither block is the complement of the union, so removing it exposes the union.

Try this

In a survey of 500 people, 300 like mangoes, 260 like apples and 80 like neither. How many like both?

Show solution

At least one: $500 - 80 = 420$ .
Both: $560 - 420 = 140$ .

Answer

140

Type 2very common5 practice Q

Percentage-based caselet

How to spot it:

All figures are percentages of people, families or students; two or three groups.

(A \cup B \cup C)\% = a + b + c - p_{AB} - p_{BC} - p_{CA} + t

Method

Work on a base of 100.
Find the union percent.
Subtract from 100 for 'none', or peel regions as asked.

Why it works:

Percentages are counts on a base of 100, so every set formula carries over unchanged.

Try this

In a survey, 55% like tea, 45% like coffee and 35% like milk. 18% like tea and coffee, 15% like coffee and milk, 12% like tea and milk, and 6% like all three. What percentage like none?

Show solution

Union: $135 - 45 + 6 = 96\%$ .
None: $100 - 96 = 4\%$ .

Answer

Type 3common2 practice Q

Exactly-one split in a survey

How to spot it:

The survey asks how many use 'only one' product, language or service.

n(A) + n(B) - 2\,n(A \cap B)

Method

Find the overlap first.
Subtract twice the overlap from the plain sum.
Or take the union minus the overlap.

Why it works:

Exactly one keeps the wings only, so the shared middle is removed completely.

Try this

In a group of 250 people, 130 play cricket, 110 play hockey and 40 play both. How many play exactly one game?

Show solution

Union: $130 + 110 - 40 = 200$ .
Exactly one: $200 - 40 = 160$ .

Answer

160

Type 4common2 practice Q

Reverse-solve a hidden region in a caselet

How to spot it:

The caselet gives the union (or 'everyone covered') and hides one region — often the triple or a wing.

Method

Name the hidden region $t$ .
Write the union equation with $t$ .
Solve, then fill the remaining regions.
Read every sub-question off the regions.

Why it works:

The union equation is linear in the hidden region, so one extra fact pins it down.

Try this

Every member of a club of 120 members knows at least one of Hindi or English. 85 know Hindi and 70 know English. How many know only Hindi?

Show solution

Both: $85 + 70 - 120 = 35$ .
Only Hindi: $85 - 35 = 50$ .

Answer

Formula sheet

Union of two groups (percent)

(A \cup B)\% = a\% + b\% - (A \cap B)\%

Same formula, percent units.

Neither (percent)

100 - (a + b - ab)

$ab$ is the both percent here.

Union of three groups (percent)

(A \cup B \cup C)\% = a + b + c - p_{AB} - p_{BC} - p_{CA} + t

Inclusion–exclusion in percent units.

Exactly one (percent)

a + b - 2ab

Two groups; percent units.

Shortcuts that save time

⚡ Percent questions: work on 100

Assume 100 people, place every percent in its region, and read the answer as a percent.

Example

In a survey, 65% read newspaper A, 48% read B and 20% read neither. What percentage read both?

Show solution

At least one: $80\%$ .
Both: $65 + 48 - 80 = 33\%$ .

Answer

33%

⚡ Neither is the leftover of the union

Union first, then subtract from 100. The neither percent is never inside the circles.

Example

60% of families have a car, 55% have a bike and 35% have both. What percentage have neither?

Show solution

Union: $60 + 55 - 35 = 80\%$ .
Neither: $100 - 80 = 20\%$ .

Answer

20%

⚡ Bound the overlap before computing

A quick max-min check on the overlap catches impossible survey data and wrong options.

Example

In a group of 100 people, 70 like tea and 80 like coffee. Between which limits does the number who like both lie?

Show solution

Minimum: $70 + 80 - 100 = 50$ .
Maximum: $\min(70, 80) = 70$ .
Both must lie between 50 and 70.

Answer

between 50 and 70

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Mixing counts and percentages in one equation — convert everything to one scale first.

Mistake 02

Reading '65% read A' as 65 people — multiply the percent by the survey total.

Mistake 03

Dropping the neither block when checking the total — regions plus neither must equal the total.

Mistake 04

Accepting impossible survey data — a negative region means the numbers cannot all be true.

Mistake 05

Forgetting the triple sits inside every pair sum when solving a percentage caselet.

Mistake 06

Answering in percent when the question asks for a number of people (or the reverse).

Quick revision

Read this the night before the exam.

Percent formulas mirror count formulas on a base of 100.
Neither percent $= 100 -$ union percent.
Exactly one: $a + b - 2ab$ (percent units).
Three-group union: singles $-$ pairs $+$ triple, all in percent.
Union fixed at 100% when everyone is covered.
A negative region flags impossible data.

Practice: 12 questions

Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.

Topic test · 12 questions

Suggested time 9 min · wrong answers go to your mistake notebook automatically.

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