39. A bag contains 3 red pens, 7 blue pens, and some black pens. If a pen is picked randomly from the bag, the probability of picking a red pen is [tex]\frac{1}{6}[/tex]. What is the number of black pens?

A. 3
B. 6
C. 8
D. 10



Answer :

To answer the question, we need to determine the number of black pens in the bag.

Given:
- Number of red pens = 3
- Number of blue pens = 7
- Probability of picking a red pen = [tex]\(\frac{1}{6}\)[/tex]
- Let the number of black pens be [tex]\(x\)[/tex]

The total number of pens in the bag is the sum of red, blue, and black pens. Therefore, the total number of pens is:
[tex]\[ \text{Total number of pens} = 3 + 7 + x = 10 + x \][/tex]

The probability of picking a red pen is given by the ratio of the number of red pens to the total number of pens. Therefore, we have the following equation for the probability:
[tex]\[ \frac{3}{10 + x} = \frac{1}{6} \][/tex]

Next, we solve this equation to find the value of [tex]\(x\)[/tex]:
1. Cross-multiply to solve for [tex]\(x\)[/tex]:
[tex]\[ 3 \cdot 6 = (10 + x) \cdot 1 \][/tex]
[tex]\[ 18 = 10 + x \][/tex]

2. Isolate [tex]\(x\)[/tex] by subtracting 10 from both sides of the equation:
[tex]\[ 18 - 10 = x \][/tex]
[tex]\[ x = 8 \][/tex]

So, the number of black pens is [tex]\(8\)[/tex]. Thus, the correct answer is:
[tex]\[ \boxed{8} \][/tex]