Same girls, smaller class — Visual lesson
Same part, a changing whole
A percentage compares a part with a whole. When only boys leave, the number of girls stays the same but the total decreases, so the girls’ percentage increases.
Solve the classroom question
Initially, 40% of a class are girls. Only 10 boys leave. Girls are now 50% of the remaining class. No other students join or leave. What was the original total?
A) 50 · B) 40 · C) 60 · D) 100
Try it before revealing the steps below.
Picture the solved class
Each equal-sized token represents one student. These counts check the answer; they are not extra information given in the question.
Follow the reasoning
- Initially girls = 40% and boys = 60% of the original total.
- Finally girls are 50%, so remaining boys and girls are equal.
- The 10 boys who left account for the initial excess of boys over girls.
- Initial excess = (60% − 40%) of original total = 20% of original total.
- 20% of original total = 10, so original total = 10 × 100/20 = 50.
- Check: initially 20 girls + 30 boys = 50; finally 20 girls + 20 boys = 40, and 20/40 = 50%.
Algebra: keep the girls fixed
Let original total be x. Girls = 0.4x; remaining total = x − 10. Thus 0.4x = 0.5(x − 10) = 0.5x − 5; 0.1x = 5; x = 50.
Answer: 50 students originally. The final total is 40.
Avoid these mistakes
- 40% and 50% use different totals. A rise of 10 percentage points is not the loss of 10 students.
- Do not write 0.4x = 0.5x − 10. Half of (x − 10) is 0.5x − 5.
- The final total is 40; the requested original total is 50.
- The 20-percentage-point boys–girls gap shortcut works here because the final groups are equal (50% girls). For other final percentages use the invariant equation.
Five original practice questions
Choose an option to check your answer and read the explanation.
- Initially 30% of a class are girls. Only 15 boys leave, and girls become 40% of the remaining class. No other change occurs. Find the original total.
Show answer
Correct answer: C
Unchanged girls: 30/100 × x = 40/100 × (x − 15). Hence (40 − 30)x = 600, so x = 60. Check: 18 girls and 42 boys initially; 27 boys remain, total 45. Girls = 18/45 = 40%.
Hint: Write the number of girls before and after as the same quantity.
- Initially 40% of a class are girls. Only 12 boys leave, and girls become 50% of the remaining class. No other change occurs. Find the original total.
Show answer
Correct answer: B
Unchanged girls: 40/100 × x = 50/100 × (x − 12). Hence (50 − 40)x = 600, so x = 60. Check: 24 girls and 36 boys initially; 24 boys remain, total 48. Girls = 24/48 = 50%.
Hint: Write the number of girls before and after as the same quantity.
- Initially 25% of a class are girls. Only 18 boys leave, and girls become 40% of the remaining class. No other change occurs. Find the original total.
Show answer
Correct answer: C
Unchanged girls: 25/100 × x = 40/100 × (x − 18). Hence (40 − 25)x = 720, so x = 48. Check: 12 girls and 36 boys initially; 18 boys remain, total 30. Girls = 12/30 = 40%.
Hint: Write the number of girls before and after as the same quantity.
- Initially 40% of a class are girls. Only 20 boys leave, and girls become 60% of the remaining class. No other change occurs. Find the original total.
Show answer
Correct answer: C
Unchanged girls: 40/100 × x = 60/100 × (x − 20). Hence (60 − 40)x = 1200, so x = 60. Check: 24 girls and 36 boys initially; 16 boys remain, total 40. Girls = 24/40 = 60%.
Hint: Write the number of girls before and after as the same quantity.
- Initially 20% of a class are girls. Only 16 boys leave, and girls become 25% of the remaining class. No other change occurs. Find the original total.
Show answer
Correct answer: B
Unchanged girls: 20/100 × x = 25/100 × (x − 16). Hence (25 − 20)x = 400, so x = 80. Check: 16 girls and 64 boys initially; 48 boys remain, total 64. Girls = 16/64 = 25%.
Hint: Write the number of girls before and after as the same quantity.
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