<p>At the Biltmore Hotel , 86 guests are in rooms with king size beds
and 57 guests are in rooms with bathtubs. If 83 of the guests are
in rooms with only a bathtub OR a king size bed, how many guests
are in rooms with both features?</p>
<p>Bump…</p>
<p>You have been given three unknowns and three numbers. So this calls for solving a system of three equations with three variables.</p>
<p>Let x = number of guests in rooms with only a king-sized bed
Let y = number of guests in rooms with only a bathtub
Let z = number of guests in rooms with both a king-sized bed and a bathtub</p>
<p>Translating the given information into math equations, we have
x + z = 86
y + z = 57
x + y = 83</p>
<p>Since all equations have variables with a coefficient of one, use elimination or substitution to solve for x, y, and z.
x = 56 , y = 27, and z = 30. Therefore, 30 guests are in rooms with both features.</p>
<p>Thank you, knowthestuff!</p>
<p>Another way to think about this, if you like Venn diagrams: Picture a circle that represents the 86 guests in rooms with king-sized beds, and another circle that represents the 57 in rooms with bathtubs. The circles overlap. The region of their overlap contains the guests who have both–but they are counted twice, once among the 86 with king-sized beds and once among the 57 with bathtubs. (Shouldn’t this question be updated to jacuzzis or something?)</p>
<p>Add the 86 inside the king-sized-bed circle and the 57 inside the bathtub-circle: 143.
Subtract the ones that we know have only one or the other: 143 - 83 = 60.
But, people in this group have been counted twice, once in the king-sized-bed group and once in the bathtub group, so divide by 2: 60/2 = 30 with both features in their rooms, same as above.</p>
<p>Some people will prefer the algebraic approach to this one–just depends how you like to think about it.</p>