In a college union there are 48 students. The ratio of the number of boys to the number of girls is 5 : 3. How many girls must be added so that the new ratio of boys to girls becomes 6 : 5?

Introduction / Context:
This question involves changing ratios with a fixed original total and an unknown increment. We are told the initial ratio of boys to girls in a college union and the total number of students. The task is to determine how many girls should be added to achieve a new specified ratio. This type of problem is common in aptitude tests and requires careful handling of ratios and linear equations.

Given Data / Assumptions:
• Total number of students initially = 48. • Ratio of boys to girls initially = 5 : 3. • Some number of girls (say x) are added; number of boys remains the same. • After adding x girls, the new ratio of boys to girls becomes 6 : 5.

Concept / Approach:
The method is to convert ratios into actual numbers using a common multiplier. Initially, if the ratio of boys to girls is 5 : 3 and the total is 48, we can find the actual numbers of boys and girls. Then, we add an unknown number x to the girls and set up an equation reflecting the new ratio 6 : 5. Solving that equation gives the number of girls to be added. This is a straightforward combination of ratio interpretation and simple algebra.

Step-by-Step Solution:
Step 1: Let the common multiplier for the initial ratio be k. Then boys = 5k and girls = 3k. Step 2: Total students = 5k + 3k = 8k. We are given that this equals 48. Step 3: So 8k = 48 gives k = 6. Step 4: Therefore, initial number of boys = 5 * 6 = 30, and initial number of girls = 3 * 6 = 18. Step 5: Let x be the number of girls to be added. Then the new number of girls is 18 + x. The number of boys remains 30. Step 6: The new ratio of boys to girls is given as 6 : 5. So 30 : (18 + x) = 6 : 5. Step 7: Write this as a fraction equation: 30 / (18 + x) = 6 / 5. Step 8: Cross multiply: 30 * 5 = 6 * (18 + x). Step 9: Compute: 150 = 108 + 6x. Step 10: Subtract 108 from both sides: 150 − 108 = 6x, so 42 = 6x. Step 11: Divide by 6: x = 7.

Verification / Alternative Check:
After adding 7 girls, the new number of girls is 18 + 7 = 25. The number of boys is still 30. The new ratio of boys to girls is 30 : 25. Simplify this by dividing both numbers by 5, giving 6 : 5, which matches the required new ratio. Therefore, the calculation is correct.

Why Other Options Are Wrong:
• 6 would give new girls = 24 and ratio 30 : 24 = 5 : 4, not 6 : 5. • 12 would give new girls = 30 and ratio 30 : 30 = 1 : 1. • 17 would give new girls = 35 and ratio 30 : 35 = 6 : 7, none of which match 6 : 5.

Common Pitfalls:
Some learners mistakenly change both the number of boys and girls, even though the question clearly says girls are added, not total students redistributed. Another common mistake is to treat the ratio itself as if it were actual numbers and add directly to the ratio terms. Always convert ratios to actual counts before applying changes, and then re-form the ratio as needed.

Final Answer:
The number of girls that must be added is 7.