Let's go through the problem step by step to understand how many more boys should join the queue to maintain the ratio of 3:4 after 8 more girls join.
1. Initial Ratio:
The initial ratio of boys to girls is given as 3:4.
Let's say there are 3x boys and 4x girls, where x is a multiplier for the ratio. For simplicity, we can assume x = 1 to find the initial quantities, but any value of x would work as long as the ratio remains 3:4.
- Number of boys = 3x
- Number of girls = 4x
2. Initial Assumption:
For simplicity, let’s initially assume:
- Number of boys = 3
- Number of girls = 4
3. Girls Joining the Queue:
8 more girls join the queue, so the new number of girls:
[tex]\( \text{New number of girls} = 4 + 8 = 12 \)[/tex]
4. Maintaining the Ratio:
To maintain the ratio of boys to girls as 3:4 with the new number of girls (12), we need to find out the new number of boys that will keep the ratio at 3:4.
The ratio 3:4 means for every 4 girls, there should be 3 boys.
Therefore, we set up the proportion to find out how many boys are needed:
[tex]\[
\frac{\text{Number of boys}}{\text{Number of girls}} = \frac{3}{4}
\][/tex]
Given the new number of girls is 12:
[tex]\[
\frac{\text{Number of boys}}{12} = \frac{3}{4}
\][/tex]
Solving for the number of boys:
[tex]\[
\text{Number of boys} = 12 \times \frac{3}{4} = 12 \times 0.75 = 9
\][/tex]
5. How Many More Boys Needed:
Initially, we had 3 boys. Now we need 9 boys to maintain the ratio. Therefore, the number of additional boys needed is:
[tex]\[
\text{Additional boys needed} = 9 - 3 = 6
\][/tex]
Summary:
To maintain the ratio of boys to girls at 3:4 after 8 more girls join the queue, 6 more boys should join the queue.
