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?
Apparently the answer is 25 but I cannot get to it. I tried using Venn diagrams and still couldn't get to it.
On
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 at least 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 $x = \boxed{10}$.
The answer is not $25$...
By PIE, we have that $190=110+70+95-(30+25+40)+x$
Simplifying, we get that $x=10$.
Hence, $10$ people are wearing all three.