Solving the School Ratio Problem with Multiple Step Reasoning

Let's tackle a common school problem involving ratios, specifically dealing with the number of boys to girls in a group. The initial problem we have is that in a group of pupils, the ratio of boys to girls is 4:1. If 24 more boys arrive and the ratio becomes 7:1, how many girls are there? We will solve this step-by-step.

Initial Ratios and Solution

First, let's understand the initial setup. The problem states that the ratio of boys to girls is 4:1. Let the number of boys be 4x and the number of girls be x. The total number of pupils is then 4x x 5x. Given that the total number of pupils is 432, we can set up the equation:

5x 432

Solving for x:

x 432 / 5 86.4

Since the number of pupils must be a whole number, we can verify this with the correct arithmetic. Instead, let's directly solve the equation:

5x 432

x 432 / 5 86.4

This indicates a calculation error. Let's correct it:

4x x 432

5x 432

x 432 / 9 48

Therefore, the initial number of boys (4x) is:

4 * 48 192

The initial number of girls (x) is:

48

When 24 more boys arrive, the number of boys becomes 192 24 216. According to the new ratio of 7:1, we set up the equation:

frac21{216} / 48 7 / 6

Let's cross multiply to solve for the new number of girls (G):

1344 * 6 1512 * 216 / 7

7G 1512 - 1344 168

G 168 / 7 24

Thus, the number of girls remains 48, and the increase in the number of girls is:

24

This concludes the solution to the school ratio problem, demonstrating a step-by-step approach to solving such problems.

Additional Worked Solutions

Here are alternative solutions to the same problem for different perspectives:

Let the number of boys initially be 5x and the number of girls be 4x. Initially, we have 5x 4x 432. Solving for x:

x 432 / 9 48

Therefore, the initial number of boys is 5 * 48 240, and the number of girls is 4 * 48 192. When 24 more boys arrive, the number of boys becomes 240 24 264. Let the new number of girls be G. The new ratio is 264 / G 7 / 6. Solving for G:

1584 1848 / G

G 1848 / 1584 24

The increase in the number of girls is 24.

Using the initial numbers, we know the number of boys is 5/9 * 432 240 and the number of girls is 432 - 240 192. If 24 more boys arrive, the number of boys becomes 240 24 264. Let the new number of girls be G. The new ratio is 264 / G 7 / 6. Solving for G:

264 * 6 1848 / G

G 1848 / 1584 24

The increase in the number of girls is 24.

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Conclusion

This problem demonstrates the importance of understanding ratios and how changes in one part of the ratio can affect the other. By breaking down each step and solving systematically, we can ensure a clear and accurate solution to the problem.