If a number is reduced by 40% it becomes two-thirds of another number. What is the ratio of the first number to the second number?

### Concept & Formula This problem requires translating language into a simple linear algebraic equation to find a ratio between two unknown variables. Reducing a number by a percentage means subtracting that percentage from 100%. Reducing by 40% leaves exactly 60% of the original value. ### Step-by-step Solution Let the first number be $A$ and the second number be $B$. The problem states that if $A$ is reduced by 40%, it equals two-thirds of $B$. First, express "$A$ reduced by 40%": $$ A - 40\% \text{ of } A = 60\% \text{ of } A = 0.6A $$ Now, set up the equation according to the problem: $$ 60\% \text{ of } A = \frac{2}{3} \text{ of } B $$ Convert the percentage to a fraction to make it easier to solve with the other fraction: $$ \frac{60}{100}A = \frac{2}{3}B $$ $$ \frac{3}{5}A = \frac{2}{3}B $$ To find the ratio of the first number to the second ($ \frac{A}{B} $), isolate $ \frac{A}{B} $ on one side of the equation: $$ \frac{A}{B} = \frac{\frac{2}{3}}{\frac{3}{5}} $$ To divide fractions, multiply by the reciprocal: $$ \frac{A}{B} = \frac{2}{3} \times \frac{5}{3} $$ $$ \frac{A}{B} = \frac{10}{9} $$ This means the ratio of $A$ to $B$ is 10 : 9. ### Exam Strategy & Shortcut Memorize fraction-percentage equivalents to write the equation instantly. 40% reduction means 60% remains. $ 60\% = \frac{3}{5} $. You can write $ \frac{3}{5}A = \frac{2}{3}B $ in three seconds. Cross-multiply the denominators mentally: $ 3 \times 3 = 9 $ (goes with A), $ 2 \times 5 = 10 $ (goes with B). So $ 9A = 10B $. Therefore, the ratio $ A : B $ is simply the reversed coefficients: $ 10 : 9 $. ### Common Pitfall The most frequent error is misreading "ratio of the first to the second" and accidentally giving the inverse ratio $ B : A $, which is 9 : 10 (Option D). Always explicitly write out $ \frac{A}{B} $ to ensure your final terms are correctly oriented. ### Final Answer **Therefore, the correct answer is 10 : 9.**