KBAT Problems
KBAT Problem: Classifying with Sets
A KBAT problem where a survey result must be split with a Venn diagram, the challenge is turning words into regions before any counting.
Draw the regions
Students take Maths, Science or both. Draw two overlapping circles inside the universal set and fill the overlap first, working outward.
Placing the "both" number in the middle before the singles is the step most students miss.
Count from the diagram
Once each region has a number, any question, how many take only Maths, how many take neither, is read straight off the diagram. A full answer shows the completed Venn diagram, not just the final count.
Understand the problem
In a class of 40 students, 25 study Additional Mathematics, 18 study Physics, and 7 study both. How many study only Add Maths, how many only Physics, and how many neither?
What it really asks: turn the words into Venn-diagram regions correctly, because the totals 25 and 18 already include the 7 who do both, that overlap is the trap.
Plan and solve
- Draw two overlapping circles inside a rectangle for the 40 students. Put the overlap first: both = 7.
- Only Add Maths = 25 − 7 = 18, because the 25 includes the 7 who also do Physics.
- Only Physics = 18 − 7 = 11, by the same reasoning.
- Students taking at least one = 18 + 7 + 11 = 36, so neither = 40 − 36 = 4.
Check and a variant
Check that every region adds to the total: 18 + 11 + 7 + 4 = 40, which matches the class size. A variant the examiner could add: instead of giving the overlap, tell you 4 study neither and ask how many study both.
Working backwards, at least one = 40 − 4 = 36, and n(A∩B) = 25 + 18 − 36 = 7, the same diagram read in reverse, which is a classic harder twist.
Frequently asked questions
Why can't I just add 25 and 18 to get the students taking a subject?
Because the 7 who take both are counted inside both the 25 and the 18, so adding gives 43 and double-counts them. The correct total is 25 + 18 − 7 = 36.
Subtracting the overlap once is the whole point of the formula, and forgetting it is the most common set-question error.
Do I have to draw the Venn diagram or can I just use the formula?
Draw it. The diagram earns method marks, keeps the overlap visible so you do not double-count, and lets you read off any region the question asks for.
The formula alone can answer one part, but the labelled diagram answers every part and shows the marker your reasoning at a glance.
What does 'neither' mean and where does it sit on the diagram?
Neither means students outside both circles, they take neither subject. On the diagram they sit inside the rectangle but outside the two circles.
Find them by subtracting everyone in the circles from the universal total, here 40 − 36 = 4. Forgetting this outside region is a common way to lose an easy mark.