View source for User:Anabel.disher/Sandbox/CEMC Pascal 2009 Problem 12

==Problem==
In the diagram, <math>QRS</math> is a straight line. What is the measure of <math>\angle RPS</math>?

<math> \text{ (A) }\ 27^{\circ} \qquad\text{ (B) }\ 47^{\circ} \qquad\text{ (C) }\ 48^{\circ} \qquad\text{ (D) }\ 65^{\circ} \qquad\text{ (E) }\ 67^{\circ}</math>
==Solution 1==
We can find the total amount of time that he spent on each part of the day, and then add everything up.

<math>\frac{1}{3} \times 24 = 8</math> hours spent sleeping

<math>\frac{1}{4} \times 24 = 6</math> hours spent sleeping

<math>\frac{1}{8} \times 24 = 3</math> hours spent sleeping

We can see that he spent <math>8 + 6 + 3 = 14 + 3 = 17</math> hours in total.

Finally, we can subtract this from the total number of hours in the day to see how many hours he has remaining:

<math>24 - 17 = 7</math> hours

~anabel.disher
==Solution 2==
We can combine all of the fractions to see how much time was spent of the day, and then subtract that amount from the whole:

<math>\frac{1}{3} + \frac{1}{4} + \frac{1}{8} = \frac{1 \times 8}{3 \times 8} + \frac{1 \times 6}{4 \times 6} + \frac{1 \times 3}{8 \times 3} = \frac{8}{24} = \frac{6}{24} + \frac{3}{24}</math>

<math>=\frac{8 + 6 + 3}{24} = \frac{14 + 3}{24} = \frac{17}{24}</math>

Subtracting this from the whole amount, we see:

<math>\frac{24}{24} - \frac{17}{24} = \frac{7}{24}</math> of the day is remaining

Multiplying this by the number of hours in a day, we get <math>\frac{7}{24} \times 24 = 7</math>