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

Problem

In the diagram, $QRS$ is a straight line. What is the measure of $\angle RPS$ ?

$\text{ (A) }\ 27^{\circ} \qquad\text{ (B) }\ 47^{\circ} \qquad\text{ (C) }\ 48^{\circ} \qquad\text{ (D) }\ 65^{\circ} \qquad\text{ (E) }\ 67^{\circ}$

Solution 1

We can find the total amount of time that he spent on each part of the day, and then add everything up.

$\frac{1}{3} \times 24 = 8$ hours spent sleeping

$\frac{1}{4} \times 24 = 6$ hours spent sleeping

$\frac{1}{8} \times 24 = 3$ hours spent sleeping

We can see that he spent $8 + 6 + 3 = 14 + 3 = 17$ 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:

$24 - 17 = 7$ 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:

$\frac{1}{3} + \frac{1}{4} + \frac{1}{8} = \frac{1 \times 8}{3 \times 8} + \frac{1 \times 6}{4 \times 6} + \frac{1 \times 3}{8 \times 3} = \frac{8}{24} = \frac{6}{24} + \frac{3}{24}$

$=\frac{8 + 6 + 3}{24} = \frac{14 + 3}{24} = \frac{17}{24}$

Subtracting this from the whole amount, we see:

$\frac{24}{24} - \frac{17}{24} = \frac{7}{24}$ of the day is remaining

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

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

Problem

Solution 1

Solution 2