Problem
Both A and B is part of A or B
In one class, 2/5 of the students like soccer, 8/15 like baseball, and 4/15 like neither sport. I need the fraction of the class that likes both soccer and baseball.
Fractions
Your answer
How to solve
Strategy Draw a Venn Diagram — The problem is about 'both', 'neither', and 'one or the other' — exactly what a Venn diagram organizes. I split the work into two small steps: first find the fraction in the soccer-or-baseball circle (the complement of 'neither'), then see how much the two circles overlap, which is the 'both' part.
1STEP 1
Find the 'soccer or baseball' part
The class is 1, and 4/15 like neither, so 11/15 like soccer, baseball or both.
1 - 4/15 = 15/15 - 4/15 = 11/15
The two circles plus the 'neither' region must fill the whole class, so the circles together cover everyone who is not in 'neither'.
5.NF.A.1Identify Subproblems2STEP 2
Add the two circles with a common denominator
Adding the two circles double-counts the middle: 6/15 + 8/15 = 14/15.
2/5 + 8/15 = 6/15 + 8/15 = 14/15
Soccer-lovers and baseball-lovers share an overlap, so summing both totals double-counts that shared middle slice of the Venn diagram.
5.NF.A.1Draw A Venn Diagram3STEP 3
The double-counted part is 'both'
The excess, 14/15 - 11/15 = 1/5, is exactly the part that was counted twice.
14/15 - 11/15 = 3/15 = 1/5
However much the sum of the circles exceeds their actual union is the slice that lives in both circles at once.
5.NF.A.2Draw A Venn DiagramThe extra 3/15 is exactly the share of students who like both sports.
Why?
Adding the soccer share to the baseball share counts anyone who likes both once inside each share, so those students get counted twice.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
Why?
The true share who like soccer or baseball counts each of those students once, so the gap between the sum and the truth is one whole extra count of the both group.
Why?
The class splits into soccer only, baseball only, both, and neither, with no student missed and none in two places.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Answer
Check the regions add to the whole: only-soccer 6/15 - 3/15 = 3/15, only-baseball 8/15 - 3/15 = 5/15, both 3/15, neither 4/15. Sum = 3/15 + 5/15 + 3/15 + 4/15 = 15/15 = 1. The 'both' fraction 3/15 = 1/5 is smaller than either single sport, which makes sense for an overlap.
Takeaway
When you add two overlapping groups, the part counted twice is exactly the 'both' group.
- Find the 'soccer or baseball' part
- Add the two circles with a common denominator
- The double-counted part is 'both'
Where next?
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