Beachwear Overlap Count
There are 190 people on the beach. 110 are wearing sunglasses, 70 are wearing bathing suits, and 95 are wearing a hat. Everyone is wearing at least one of these items. 30 are wearing both bathing suits and sunglasses. 25 are wearing both bathing suits and a hat. 40 are wearing both sunglasses and a hat. How many people are wearing all three items?
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Solution
Let $ x $ be number of people wearing all three items. Since 30 people are wearing bathing suits and sunglasses, we know that $ 30 - x $ are wearing just bathing suits and sunglasses. Similarly, $ 25 - x $ are wearing just bathing suits and hats, while $ 40 - x $ are wearing just sunglasses and a hat.
To find the number of people wearing just sunglasses, we subtract the people who are wearing sunglasses with other items from the total number of people wearing sunglasses, which is $ 110 - (30 - x) - (40 - x) - x = 40 + x $. Similarly, the number of people wearing just hats is $ 30 + x $, while the number of people wearing just bathing suits is $ 15 + x $.
Since the total number of people on the beach is 190, and everyone is wearing one of the items, we have: \begin{align*}
190 &= (15 + x) + (40 + x) + (30 + x) \\
&\qquad+ (25 - x ) + (30 - x) + (40 - x) + x\\
&= 180 + x.\end{align*}
We can then solve for $ x $, so the number of people on the beach wearing all three items is $ x = \boxed{10} $.
Freeflows