Question
The figure is a circle divided into three equal parts: one semi-circle at the top and two quarter-circles at the bottom. The semi-circle is colored red, the bottom left quarter-circle is colored blue, and the bottom right quarter-circle is colored orange.
What percentage of the figure is blue and yellow?
What percentage of the figure is blue and yellow?
Answers
GPT-4o mini
To find the percentage of the figure that is blue and yellow, we first need to analyze the composition of the circle.
1. **Total Area of the Circle**:
Let's denote the radius of the circle as \( r \).
The area \( A \) of the circle is given by:
\[
A = \pi r^2
\]
2. **Area of the Semi-Circle (Red)**:
The area of the semi-circle is half of the area of the whole circle:
\[
A_{\text{semi-circle}} = \frac{1}{2} \pi r^2
\]
3. **Area of Each Quarter-Circle**:
The area of a quarter-circle is a quarter of the area of the whole circle:
\[
A_{\text{quarter-circle}} = \frac{1}{4} \pi r^2
\]
Since there are two quarter-circles, the total area of both quarter-circles (blue and orange) is:
\[
A_{\text{two quarter-circles}} = 2 \times \frac{1}{4} \pi r^2 = \frac{1}{2} \pi r^2
\]
4. **Total Area of the Figure**:
The entire figure, consisting of one semi-circle and two quarter-circles, has a total area of:
\[
A_{\text{total}} = A_{\text{semi-circle}} + A_{\text{two quarter-circles}} = \frac{1}{2} \pi r^2 + \frac{1}{2} \pi r^2 = \pi r^2
\]
5. **Area of Blue and Orange Sections**:
Since the blue section is one quarter-circle and the orange section is another quarter-circle, their combined area (the two bottom areas) is:
\[
A_{\text{blue}} + A_{\text{orange}} = \frac{1}{4} \pi r^2 + \frac{1}{4} \pi r^2 = \frac{1}{2} \pi r^2
\]
6. **Percentage of Blue and Orange**:
To find the percentage of the figure represented by the blue and orange sections:
\[
\text{Percentage} = \left(\frac{A_{\text{blue}} + A_{\text{orange}}}{A_{\text{total}}}\right) \times 100 = \left(\frac{\frac{1}{2} \pi r^2}{\pi r^2}\right) \times 100 = \left(\frac{1}{2}\right) \times 100 = 50\%
\]
Thus, the percentage of the figure that is blue and orange is **50%**. Note that there is no yellow mentioned in your description; thus, I assume you might have meant orange instead.
1. **Total Area of the Circle**:
Let's denote the radius of the circle as \( r \).
The area \( A \) of the circle is given by:
\[
A = \pi r^2
\]
2. **Area of the Semi-Circle (Red)**:
The area of the semi-circle is half of the area of the whole circle:
\[
A_{\text{semi-circle}} = \frac{1}{2} \pi r^2
\]
3. **Area of Each Quarter-Circle**:
The area of a quarter-circle is a quarter of the area of the whole circle:
\[
A_{\text{quarter-circle}} = \frac{1}{4} \pi r^2
\]
Since there are two quarter-circles, the total area of both quarter-circles (blue and orange) is:
\[
A_{\text{two quarter-circles}} = 2 \times \frac{1}{4} \pi r^2 = \frac{1}{2} \pi r^2
\]
4. **Total Area of the Figure**:
The entire figure, consisting of one semi-circle and two quarter-circles, has a total area of:
\[
A_{\text{total}} = A_{\text{semi-circle}} + A_{\text{two quarter-circles}} = \frac{1}{2} \pi r^2 + \frac{1}{2} \pi r^2 = \pi r^2
\]
5. **Area of Blue and Orange Sections**:
Since the blue section is one quarter-circle and the orange section is another quarter-circle, their combined area (the two bottom areas) is:
\[
A_{\text{blue}} + A_{\text{orange}} = \frac{1}{4} \pi r^2 + \frac{1}{4} \pi r^2 = \frac{1}{2} \pi r^2
\]
6. **Percentage of Blue and Orange**:
To find the percentage of the figure represented by the blue and orange sections:
\[
\text{Percentage} = \left(\frac{A_{\text{blue}} + A_{\text{orange}}}{A_{\text{total}}}\right) \times 100 = \left(\frac{\frac{1}{2} \pi r^2}{\pi r^2}\right) \times 100 = \left(\frac{1}{2}\right) \times 100 = 50\%
\]
Thus, the percentage of the figure that is blue and orange is **50%**. Note that there is no yellow mentioned in your description; thus, I assume you might have meant orange instead.