A fraction is such that if the double of the numerator and the triple of the denominator is changed by $+10$ percent and $-30$ percent respectively, then we get 11 percent of $\frac{16}{21}$. Find the fraction.

### Concept & Strategy This problem combines fraction scaling with percentage modifications. To solve it systematically, apply the initial multipliers to the variables, apply the respective percentage changes as decimal multipliers, set the result equal to the target value, and then isolate the original fraction $\frac{x}{y}$. ### Step-by-Step Solution * **Given:** * Numerator is doubled ($2x$) and then changed by $+10\%$. * Denominator is tripled ($3y$) and then changed by $-30\%$. * Result equals $11\%$ of $\frac{16}{21}$. * **Calculation:** 1. Let the original fraction be $\frac{x}{y}$. 2. Double the numerator: $2x$. Change by $+10\%$: $$ \text{New Numerator} = 2x \times 1.10 = 2.2x $$ 3. Triple the denominator: $3y$. Change by $-30\%$: $$ \text{New Denominator} = 3y \times (1 - 0.30) = 3y \times 0.70 = 2.1y $$ 4. The new fraction expression is: $$ \frac{2.2x}{2.1y} = \frac{22x}{21y} $$ 5. Calculate the target value on the right side ($11\%$ of $\frac{16}{21}$): $$ \frac{11}{100} \times \frac{16}{21} $$ 6. Set up the full equation: $$ \frac{22}{21} \times \frac{x}{y} = \frac{11}{100} \times \frac{16}{21} $$ 7. Isolate $\frac{x}{y}$ by multiplying both sides by $\frac{21}{22}$: $$ \frac{x}{y} = \left( \frac{11 \times 16}{100 \times 21} \right) \times \frac{21}{22} $$ 8. Cancel out common terms (cancel 21 with 21, and simplify $\frac{11}{22}$ to $\frac{1}{2}$): $$ \frac{x}{y} = \frac{1 \times 16}{100 \times 2} $$ $$ \frac{x}{y} = \frac{16}{200} $$ 9. Reduce the fraction by dividing the numerator and denominator by 8: $$ \frac{x}{y} = \frac{2}{25} $$ ### Exam Strategy & Shortcut **Fractional Ratios:** Work entirely in fractions instead of decimals for cleaner cancellations. $+10\%$ is an increase of $\frac{1}{10}$, so the multiplier is $\frac{11}{10}$. $-30\%$ is a decrease of $\frac{3}{10}$, so the multiplier is $\frac{7}{10}$. Setup: $$ \frac{2x \times \frac{11}{10}}{3y \times \frac{7}{10}} = \frac{11}{100} \times \frac{16}{21} $$ The denominators of 10 on the left cancel out immediately: $$ \frac{22x}{21y} = \frac{11 \times 16}{100 \times 21} $$ Cancel the 21s on the bottom across the equals sign. Cancel 11 on the right into 22 on the left (leaves 2). $$ 2x / y = \frac{16}{100} $$ Divide by 2: $$ \frac{x}{y} = \frac{8}{100} = \frac{2}{25} $$ ### Common Pitfall Students often apply the percentage change to the original variable rather than the doubled/tripled variable. For example, calculating $1.1x$ instead of $1.1(2x)$. Carefully follow the chronological sequence of operations dictated by the wording. ### Final Answer **Therefore, the correct answer is $\frac{2}{25}$.**