In a class there are 18 boys who are over 160 cm tall. If these boys constitute three-fourths of the boys and the total number of boys is two-thirds of the number of students in the class, then what is the number of girls in the class?

### Concept & Strategy This is a sequential fraction problem. Work backward from the known absolute quantity (18 boys over 160cm) through the given fractions to find the total number of boys, then the total class size, and finally the number of girls by subtraction. ### Step-by-Step Solution * **Given:** Boys > 160cm = 18. These 18 boys = $\frac{3}{4}$ of total boys. Total boys = $\frac{2}{3}$ of total students. * **Calculation:** Find the total number of boys (let it be $B$): $$ \frac{3}{4}B = 18 $$ $$ B = 18 \times \frac{4}{3} = 24 \text{ boys} $$ * Find the total number of students in the class (let it be $S$): $$ \frac{2}{3}S = B $$ $$ \frac{2}{3}S = 24 $$ $$ S = 24 \times \frac{3}{2} = 36 \text{ students} $$ * Calculate the number of girls: $$ \text{Girls} = \text{Total Students} - \text{Total Boys} $$ $$ \text{Girls} = 36 - 24 = 12 $$ ### Exam Strategy & Shortcut Visualize the fractions as a chain: $18 = \frac{3}{4} \times \text{Boys} \implies \text{Boys} = 24$. Boys represent $\frac{2}{3}$ of the class. If $\frac{2}{3}$ of the class are boys, then the remaining $\frac{1}{3}$ of the class must be girls. Since 2 parts (boys) = 24, 1 part (girls) = $24 \div 2 = 12$. This skips calculating the total class size entirely! ### Common Pitfall A common error is to calculate the total number of students (36) and accidentally select that as the final answer, failing to perform the final subtraction step to find the number of girls. Always re-read the specific final question being asked. ### Final Answer Therefore, the correct answer is **12**.