The central angle of sector R is 31 degrees and the central angle of sector Q is 225 degrees. what is the probability that the spinner does not land on Q or R? simplify your answer and write it as a proper fraction

Answer 1

To determine the probability that the spinner does not land on sectors Q or R, we first need to find the angle measures of these sectors, and then the total angle for the entire circle.

1. Total angle in a circle: $$360\text{ degrees}$$

2. Central angle of sector R: $$31\text{ degrees}$$

3. Central angle of sector Q: $$225\text{ degrees}$$

4. Total angle of sectors Q and R: $$31+225=256\text{ degrees}$$

5. Angle not covered by sectors Q and R: $$360-256=104\text{ degrees}$$

6. Probability that the spinner does not land on Q or R: The probability $P$ is the angle not landing on these sectors divided by the total angle in a circle: $$P=\frac{104}{360}$$

7. Now, simplify this fraction: Find the greatest common divisor (GCD) of 104 and 360.

   $$104=2^3\times 13360=2^3\times 3^2\times 5$$ The GCD is $2^3=8$.

   Now, divide both the numerator and denominator by their GCD: $$\frac{104÷8}{360÷8}=\frac{13}{45}$$

Thus, the probability that the spinner does not land on sectors Q or R is: $$\boxed{{\displaystyle \frac{13}{45}}}$$