To determine the number of red pencils in the box, let's follow these steps systematically:
1. Understand the given probability:
- The probability of picking a red pencil from the box is given as [tex]\(\frac{3}{11}\)[/tex].
2. Identify the total probability for any event:
- The total probability for picking any color pencil from the box (whether red or blue) is 1 (since there are no other colors).
3. Determine the probability of picking a blue pencil:
- Since the only other color is blue, the probability of picking a blue pencil is:
[tex]\[
1 - \frac{3}{11} = \frac{11}{11} - \frac{3}{11} = \frac{8}{11}
\][/tex]
4. Given the number of blue pencils:
- There are 16 blue pencils in the box.
5. Set up the proportion for the blue pencils:
- We know that the probability of picking a blue pencil can also be expressed in terms of the number of blue pencils divided by the total number of pencils. Let's denote the total number of pencils in the box as [tex]\(N\)[/tex].
The probability of picking a blue pencil is:
[tex]\[
\frac{\text{Number of blue pencils}}{N} = \frac{16}{N} = \frac{8}{11}
\][/tex]
6. Solve for the total number of pencils [tex]\(N\)[/tex]:
[tex]\[
\frac{16}{N} = \frac{8}{11}
\][/tex]
Cross-multiplying to solve for [tex]\(N\)[/tex]:
[tex]\[
16 \times 11 = 8 \times N
\][/tex]
[tex]\[
176 = 8N
\][/tex]
[tex]\[
N = \frac{176}{8} = 22
\][/tex]
Therefore, there are a total of 22 pencils in the box.
7. Determine the number of red pencils:
- Now that we know the total number of pencils, we can find the number of red pencils by noting that:
[tex]\[
\frac{\text{Number of red pencils}}{N} = \frac{3}{11}
\][/tex]
Let's denote the number of red pencils by [tex]\(R\)[/tex]:
[tex]\[
\frac{R}{22} = \frac{3}{11}
\][/tex]
Cross-multiplying to solve for [tex]\(R\)[/tex]:
[tex]\[
R \times 11 = 3 \times 22
\][/tex]
[tex]\[
11R = 66
\][/tex]
[tex]\[
R = \frac{66}{11} = 6
\][/tex]
Thus, the number of red pencils in the box is [tex]\(6\)[/tex].
