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medium importance~1-2 Q in Tier 111 formulas⚡ 8 shortcuts4 subtopics

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Two-circle counting

⏱ 4 min read🧩 5 question types🎯 15 practice Q

The idea in one minute

Two circles make four parts: only A, only B, both, and neither. Every two-circle question is a sum or a difference of these four parts. Fill them in, then read off the answer.

The four parts

Two groups A and B split everyone into four parts.

Part	Meaning
Only A	In A, not in B
Both	In A and in B
Only B	In B, not in A
Neither	In neither group

Only A, both and only B together make the union. The union is "at least one". Neither is what is left of the total.

The main formula

When you add A and B, the "both" part is counted twice. Subtract it once to fix that.

n(A \cup B) = n(A) + n(B) - n(A \cap B)

Rule: Union = A + B − both. Neither = total − union.

Example: 150 families. 90 have a car, 70 have a two-wheeler, 40 have both. Union = 90 + 70 − 40 = 120. Neither = 150 − 120 = 30.

Finding the only parts

"Only A" means A without the overlap. So only A = A − both.

In the same example, only car = 90 − 40 = 50. Only two-wheeler = 70 − 40 = 30. Check: 50 + 40 + 30 + 30 = 150.

Tip: After you fill the four parts, add them. The sum must equal the total. This catches every slip.

Working backwards

Sometimes the overlap is the unknown. Rearrange the main formula.

\text{both} = A + B - \text{union}

Example: 45 students. 28 play cricket, 22 play football, and each plays at least one. Union = 45. Both = 28 + 22 − 45 = 5.

Percentages

Percent questions work the same way, with a total of 100.

Example: 60% like tea, 45% like coffee, and 15% like neither. Union = 100 − 15 = 85. Both = 60 + 45 − 85 = 20%.

Watch: If the question gives a percent and a headcount, change everything to one form first.

Biggest and smallest overlap

Sometimes the overlap is not given. You then find the smallest and the largest possible value.

Largest overlap = the smaller of A and B. One group sits fully inside the other.
Smallest overlap = A + B − total, or 0 if that is negative. The groups spread out as much as possible.

Example: 100 people. 70 speak Hindi and 60 speak English. Smallest both = 70 + 60 − 100 = 30. Largest both = 60.

Note: A negative answer for the smallest overlap means the groups can be apart. Then the smallest overlap is 0.

Reading the wording

Exam words hide the four parts. Turn each phrase into a part before you calculate.

Phrase	Part
"at least one", "either or both"	Union
"both", "and"	Overlap
"only A", "A but not B"	A − both
"exactly one", "only one of the two"	Only A + only B
"neither", "none"	Total − union

Example: 80 people. 50 read English news, 40 read Hindi news, 20 read both. Exactly one language = 50 + 40 − 2 × 20 = 50. Neither = 80 − (50 + 40 − 20) = 10.

Question types you will see

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

Type 1very common2 practice Q

Union and neither

How to spot it:

Two group totals, their overlap and the size of the whole group. It asks for at least one, neither or only one.

n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

Method

Union = A + B − both.
Neither = total − union.
Only A = A − both.

Why it works:

The overlap is counted twice in A + B, so it is removed once.

Try this

Of 150 families, 90 have a car, 70 have a two-wheeler and 40 have both. How many have neither?

Show solution

Union = 90 + 70 − 40 = 120.
Neither = 150 − 120 = 30.

Answer

Type 2common2 practice Q

Find the overlap

How to spot it:

The total and both group sizes are given. Everyone is in at least one group. It asks for the overlap.

both = A + B − union

Method

Write the union. If everyone is in a group, union = total.
Add the two group sizes.
Subtract the union.

Why it works:

The extra people in A + B over the union are exactly the overlap.

Try this

In a class of 45, 28 play cricket and 22 play football. Every student plays at least one. How many play both?

Show solution

Union = 45.
A + B = 28 + 22 = 50.
Both = 50 − 45 = 5.

Answer

Type 3common2 practice Q

Percentages in two sets

How to spot it:

The data are in percent. It asks for the percent in both or in neither.

Method

Take the whole group as 100%.
Union = 100 − neither.
Both = A + B − union.

Why it works:

Percent is a count out of 100, so the same formula works.

Try this

In a town, 60% like tea, 45% like coffee and 15% like neither. What percent like both?

Show solution

Union = 100 − 15 = 85%.
Both = 60 + 45 − 85 = 20%.

Answer

20%

Type 4occasional2 practice Q

Biggest and smallest overlap

How to spot it:

The overlap is not given. It asks for the 'least' or 'greatest' possible number in both.

Smallest = A + B − total; largest = smaller set

Method

Smallest overlap: A + B − total. Use 0 if negative.
Largest overlap: the smaller of A and B.
Any overlap between these two is possible.

Why it works:

The groups overlap least when spread apart, and most when one sits inside the other.

Try this

100 people: 70 speak Hindi and 60 speak English. Find the least and the greatest number who speak both.

Show solution

Least = 70 + 60 − 100 = 30.
Greatest = smaller group = 60.

Answer

Least 30, greatest 60

Type 5occasional

Exactly one group

How to spot it:

The words 'only one of the two' or 'exactly one'. Only-A and only-B are added.

Exactly one = A + B − 2 × both

Method

Subtract the overlap from A to get only A.
Subtract the overlap from B to get only B.
Add the two.

Why it works:

The overlap is taken out of both groups, so it is subtracted twice.

Try this

40 people like A, 30 like B and 12 like both. How many like exactly one of the two?

Show solution

Only A = 40 − 12 = 28.
Only B = 30 − 12 = 18.
Total = 28 + 18 = 46.

Answer

Formula sheet

Two-set union

|A \cup B| = |A| + |B| - |A \cap B|

The overlap is counted twice on the right, so subtract it once.

Only A

|A \setminus B| = |A| - |A \cap B|

Only A is A minus the overlap.

Neither

\text{neither} = \text{total} - |A \cup B|

Everyone outside both circles.

Exactly one

|A| + |B| - 2|A \cap B|

Only A plus only B.

Shortcuts that save time

⚡ Add the four parts to check

After you fill only A, both, only B and neither, add them. If the sum is not the total, one part is wrong.

Example

150 families: 90 have a car, 70 have a two-wheeler, 40 have both. Check the parts.

Show solution

Only car 50, both 40, only two-wheeler 30, neither 30.
50 + 40 + 30 + 30 = 150.

Answer

The parts add to 150, so all are right.

⚡ Overlap limits

The overlap can never be more than the smaller group. It can never be less than A + B − total.

Example

100 people: 70 speak Hindi, 60 speak English. Find the smallest and largest overlap.

Show solution

Smallest = 70 + 60 − 100 = 30.
Largest = the smaller group = 60.

Answer

30 and 60

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Adding 90 + 70 and calling 160 the union.

The overlap is counted twice. Union = 90 + 70 − 40 = 120.

Mistake 02

Treating 'both' as 'only both'.

'Both' is the overlap. 'Only A' removes it from A.

Mistake 03

Forgetting to subtract the union from the total to get neither.

Neither = total − union.

Mistake 04

Giving a negative number as the smallest overlap.

If A + B − total is negative, the smallest overlap is 0.

Mistake 05

Mixing percent and headcount in one sum.

Change all numbers to percent, or all to counts, first.

Quick revision

Read this the night before the exam.

Union = A + B − both.
Neither = total − union.
Only A = A − both.
Both = A + B − union.
Exactly one = A + B − 2 × both.
Largest overlap = the smaller set. Smallest overlap = A + B − total (not below 0).
Add the four parts at the end. They must equal the total.

Practice: 15 questions

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

Topic test · 10 questions

Suggested time 6 min · wrong answers go to your mistake notebook automatically.

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