Two-Way Tables and Venn Diagram Reasoning
A two-way table organises data by two categories at once, such as year group and favourite subject, in rows and columns with row and column totals. A Venn diagram shows how groups (sets) overlap using labelled circles inside a rectangle. 11+ papers test reading, completing and reasoning about missing values in both, often combined with basic probability.
Before you start
Make sure you're comfortable with these topics first:
Method
- In a two-way table, check that every row sums to its row total and every column sums to its column total; use this to find any missing value by subtraction.
- Read the question carefully to identify exactly which row, column or single cell is being asked about.
- In a Venn diagram, the overlapping region shows items in both sets; the region outside both circles but inside the rectangle shows items in neither.
- To find a missing value in a Venn diagram, use the total number of items and any given regions, subtracting known regions from the total step by step.
- For 'only' questions, such as 'how many like tennis only', make sure the overlap region has already been subtracted so it is not double-counted.
- Double-check a completed table or diagram by confirming its rows, columns and overall total all sum correctly.
Worked example
A two-way table shows how 50 pupils travel to school. There are 24 boys and 26 girls in total. 18 boys walk to school and 6 boys cycle. 14 girls walk to school. Work out how many girls cycle to school.
- Note the total number of girls is 26.
- Note the number of girls who walk is 14.
- Since each girl either walks or cycles, subtract: 26 - 14 = 12.
- Girls who cycle to school = 12. Check the boys: 18 + 6 = 24, which matches the given total of 24 boys.
Practice questions
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Q1A two-way table shows 40 pupils. 22 are in Year 5 and the rest are in Year 6. How many are in Year 6?Show answer
Answer: 18
Q2In a two-way table, 15 pupils like tea only, 9 like coffee only, and 6 like both. How many pupils like at least one of tea or coffee?Show answer
Answer: 30 (15 + 9 + 6)
Q3A Venn diagram shows 40 pupils in total. 25 play football, 18 play tennis, and 10 play both. How many play neither sport?Show answer
Answer: 7 (football only 15, tennis only 8, at least one = 15 + 8 + 10 = 33, neither = 40 - 33 = 7)
Q4A two-way table shows that 12 pupils own a cat and a dog, 20 own a cat only, and 15 own a dog only. How many pupils own at least one of a cat or a dog?Show answer
Answer: 47 (12 + 20 + 15)
Q5In a Venn diagram of 30 pupils, 8 like both apples and bananas. If 17 like apples in total, how many like apples only?Show answer
Answer: 9 (17 - 8)
Q6A two-way table for 60 pupils shows 35 study French and 32 study Spanish, with 10 studying both. How many pupils study neither language?Show answer
Answer: 3 (French only 25, Spanish only 22, at least one = 25 + 22 + 10 = 57, neither = 60 - 57 = 3)
Q7True or False: in a Venn diagram, the region where two circles overlap should be counted twice when finding the total number in either set.Show answer
Answer: False
Exam-style questions
Written in the style of a 11+ exam paper, with a full mark scheme.
A two-way table shows the recreation choices of 45 pupils. 28 pupils are boys and the rest are girls. Of the boys, 16 chose swimming. Of the girls, 9 chose swimming. (a) How many girls are there? (b) How many pupils in total chose swimming?
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A Venn diagram shows the pets owned by 50 pupils, split into 'owns a dog' and 'owns a cat'. 21 pupils own a dog, 19 own a cat, and 7 own both. (a) How many pupils own a dog only? (b) How many pupils own neither a dog nor a cat?
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Free printable worksheet
Want more practice on paper? Download the two-way tables and venn diagram reasoning worksheet pack - 10 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 26 of 11+ Maths Workbook 3, the whole course as one free printable PDF.
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