In a school, the total number of students (boys plus girls) is 150. If the number of boys is x, the number of girls becomes x% of the total number of students. How many boys are there in the school?

Introduction / Context:
This question combines percentages and simple algebra to relate the numbers of boys and girls in a school. It tests your ability to translate a verbal relationship into an equation, where the number of girls is given as a percentage of the total number of students. Problems like this appear frequently in aptitude tests under percentage and ratio topics and help reinforce the interpretation of expressions such as "x% of a total".

Given Data / Assumptions:

• The total number of students in the school is 150.
• The number of boys is x.
• The number of girls is therefore 150 - x.
• The number of girls is equal to x% of the total number of students.
• The total number of students is the sum of boys and girls.

Concept / Approach:
The main concept is that x% of the total number of students can be written as (x/100) * 150. Since the number of girls is given to be exactly this value, we can equate 150 - x to (x/100) * 150. That equation contains only one unknown, x, which represents the number of boys. Solving this linear equation gives the required value of x. Understanding how to convert "x% of 150" and correctly set up the equality is the key step here.

Step-by-Step Solution:
Step 1: Let the number of boys be x.Step 2: Since the total number of students is 150, the number of girls is 150 - x.Step 3: The question states that the number of girls equals x% of the total students.Step 4: x% of 150 is (x/100) * 150 = 1.5x.Step 5: Set up the equation: 150 - x = 1.5x.Step 6: Rearrange to get 150 = 1.5x + x = 2.5x.Step 7: Solve for x: x = 150 / 2.5 = 60.Step 8: Therefore, there are 60 boys in the school.

Verification / Alternative check:
With x = 60, the number of boys is 60 and the number of girls is 150 - 60 = 90. According to the condition, girls should be x% of the total, i.e., 60% of 150. Now calculate 60% of 150: (60/100) * 150 = 0.6 * 150 = 90. This matches the actual number of girls, confirming that the equation and solution are correct. Thus, 60 boys satisfy the relationship given in the problem.

Why Other Options Are Wrong:
If x were 70, the girls would be 80, but 70% of 150 is 105, not 80. If x were 80, the girls would be 70, but 80% of 150 is 120, which does not match. For x = 90, girls would be 60, whereas 90% of 150 is 135. Finally, x = 75 would mean girls are 75, but 75% of 150 is 112.5, again not equal. Therefore, only x = 60 satisfies the required condition that girls equal x% of the total students.

Common Pitfalls:
A common mistake is to misinterpret x% of the total as simply x, or to write 150 * x instead of (x/100) * 150. Some learners also mistakenly set x equal to 150 - x rather than relating 150 - x to a percentage of 150. Another error is to treat x% as x/10, which is incorrect. To avoid such mistakes, always remember that y% of a quantity T is (y/100) * T and set up the equation carefully before solving.

Final Answer:
The number of boys in the school is 60.