there are 23 boys and 35 girls. after q boys and (q+4) girls leave, the probability of selecting a boy at random is 2/5. find q

We are given the following problem:

- Initially, there are 23 boys and 35 girls.
- After \( q \) boys and \( q + 4 \) girls leave, the probability of selecting a boy at random is \( \frac{2}{5} \).
- We need to find the value of \( q \).

### Step 1: Set up the total number of boys and girls after the change

Initially, the total number of boys is 23, and the total number of girls is 35.

- After \( q \) boys leave, the number of boys remaining is \( 23 - q \).
- After \( q + 4 \) girls leave, the number of girls remaining is \( 35 - (q + 4) = 35 - q - 4 = 31 - q \).

### Step 2: Express the total number of people after the change

The total number of people remaining is:
\[
(23 - q) + (31 - q) = 54 - 2q
\]

### Step 3: Set up the probability equation

We are told that the probability of selecting a boy at random is \( \frac{2}{5} \). This means that the probability of selecting a boy can be written as:
\[
\frac{\text{Number of boys}}{\text{Total number of people}} = \frac{23 - q}{54 - 2q}
\]
We are given that this probability is \( \frac{2}{5} \). So, we have the equation:
\[
\frac{23 - q}{54 - 2q} = \frac{2}{5}
\]

### Step 4: Solve the equation

Now, cross-multiply to eliminate the fractions:
\[
5(23 - q) = 2(54 - 2q)
\]
Expanding both sides:
\[
115 - 5q = 108 - 4q
\]

### Step 5: Simplify and solve for \( q \)

Now, solve for \( q \):
\[
115 - 108 = 5q - 4q
\]
\[
7 = q
\]

### Final Answer:
The value of \( q \) is 7.

Let me know if you need more details or have any questions!

Here are 5 related questions:
1. What happens if \( q \) is larger than 23?
2. How does the total population change after \( q \) boys and \( q + 4 \) girls leave?
3. What if the probability of selecting a boy is \( \frac{3}{5} \) instead of \( \frac{2}{5} \)?
4. How would the problem change if the number of girls leaving was different from \( q + 4 \)?
5. Can we use this method to solve for unknowns in other probability scenarios?

**Tip:** Always carefully write down expressions for the remaining quantities before setting up the probability equation!

Explore how to solve a probability problem involving boys and girls leaving a group. Learn step-by-step how to find the value of q using probability equations and adjustments to the group composition. Suitable for high school students studying probability and equations.

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