If $20\%$ of $A = 30\%$ of $B = \frac{1}{6}$ of $C$, then $A : B : C$ is

### Concept & Equating Multiple Variables To find the ratio $A : B : C$ from an equation where multiple fractional or percentage parts are equated, convert everything to fractions, set the entire equation equal to a constant $k$, and express each variable in terms of $k$. $$ xA = yB = zC = k $$ ### Step-by-Step Solution * **Given:** $20\%$ of $A = 30\%$ of $B = \frac{1}{6}$ of $C$. * **Convert to fractions:** Change the percentages to their simplest fractional forms. $\frac{20}{100}A = \frac{30}{100}B = \frac{1}{6}C$ $\frac{1}{5}A = \frac{3}{10}B = \frac{1}{6}C$ * **Equate to a constant:** Let $\frac{1}{5}A = \frac{3}{10}B = \frac{1}{6}C = k$. * **Solve for each variable:** $A = 5k$ $B = \frac{10k}{3}$ $C = 6k$ * **Form the ratio:** Substitute the $k$ values into $A : B : C$. $5k : \frac{10k}{3} : 6k$ * **Simplify the ratio:** Divide out the $k$, then multiply the entire sequence by $3$ to eliminate the denominator. $(5 \times 3) : \left(\frac{10}{3} \times 3\right) : (6 \times 3)$ $15 : 10 : 18$ ### Exam Strategy & Shortcut Once you have the coefficients as fractions ($1/5$, $3/10$, $1/6$), the ratio of the variables is simply the ratio of their reciprocals. $A : B : C = 5 : \frac{10}{3} : 6$. Multiply by $3$ to clear the fraction instantly: $15 : 10 : 18$. ### Common Pitfall A frequent error is forgetting to multiply *every* term in the ratio by the least common multiple (LCM) when removing the fraction, leaving terms unbalanced. ### Final Answer Therefore, the correct answer is **$15 : 10 : 18$**.