Question

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.

\[
104 = 2^3 \times 13 \\
360 = 2^3 \times 3^2 \times 5
\]
The GCD is \( 2^3 = 8 \).

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

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