<p>At the Biltmore Hotel,, 86 guests are in rooms with king size beds and 57 guests are in rooms with bath tubs. If 83 of the guests are in rooms with a king size bed OR a bathtub, how many guests are in rooms with both features?
a)60
b)56
c)36
d)30
e) 27
I thought the answer was 60 but its 30. I dont understand why. </p>
<p>In the xy coordinate plane , the point (2,r) is a distance of 13 from the point (14,2. Which of the following could be r?
A)15
b)6
c)5
d)-1
e)-3</p>
<p>For the first question: 86 + 57 = 143 - 83 = 60/2 = 30
You need to divide 60 by 2 to get 30, because 60 is counting each person twice. If 30 people are in a room with a king size bed, and 30 people are in a room with a bathtub, and all of these people are in rooms with both features, those 30 people must be the same 30 people.</p>
<p>For the second question:</p>
<p>Use the distance formula (Pythagorean theorem)</p>
<p>13 = sqrt((14-2)^2 + (2-r)^2) = sqrt(144 + (2-r)^2)</p>
<p>169 = 144 + (2-r)^2</p>
<p>25 = (2-r)^2</p>
<p>2-r = 5 or 2-r = -5, so r = -3 or 7. -3 is one of the answer choices, E.</p>
<p>You should also be alert for pythagorean triples in this kind of question. Notice that the x value went from 2 to 14, a run of 12…and the distance, which is also the hypotenuse, was 13. That is NOT a coincidence! The other leg of the triangle will be 5 – 5,12,13 is a well-known pythagorean triple. So now you know that the other point is either 5 up or 5 down from (14,2). That gives 7 or -3 as the possible values.</p>
<p>Some people say that with calculators, you don’t need to know your pythag triples any more. But good “number sense” often leads to easier, less algebraic solutions.</p>