Three-fourths of a number is equal to 60% of another number and the difference between these two numbers is 20. What is the sum of these two numbers?

### Concept & Logic Equating a fraction of one variable to a percentage of another establishes a fixed ratio between the two variables. Once the ratio is found, the given difference allows us to calculate the exact multiplier, leading to the sum. $$ \text{Fraction} \times A = \text{Percentage} \times B \implies \text{Ratio of } A:B $$ ### Step-by-Step Solution * Let the first number be $A$ and the second number be $B$. * **Given:** $\frac{3}{4}$ of $A$ is equal to $60\%$ of $B$. * Convert $60\%$ to a fraction: $60\% = \frac{60}{100} = \frac{3}{5}$. * Set up the equation: $$ \frac{3}{4} \times A = \frac{3}{5} \times B $$ * Find the ratio of $A$ to $B$ by isolating the variables on one side: $$ \frac{A}{B} = \frac{\frac{3}{5}}{\frac{3}{4}} = \frac{3}{5} \times \frac{4}{3} = \frac{4}{5} $$ * This means $A$ is $4k$ and $B$ is $5k$ for some constant multiplier $k$. * We are given that the difference between the numbers is 20. $$ B - A = 20 $$ $$ 5k - 4k = 20 \implies k = 20 $$ * Now, calculate the sum of the two numbers. $$ \text{Sum} = A + B = 4k + 5k = 9k $$ * Substitute the value of $k$: $$ \text{Sum} = 9 \times 20 = 180 $$ ### Exam Strategy & Shortcut Once you find the ratio $\frac{A}{B} = \frac{4}{5}$, you know the numbers are multiples of 4 and 5. The sum of the numbers is $4 + 5 = 9$ parts. Therefore, the correct answer MUST be a multiple of 9. Look at the options: (a) 170 (Not divisible by 9) (b) 180 (Divisible by 9) (c) 220 (Not divisible by 9) Without calculating the exact values, you can confidently mark 180. ### Common Pitfall Setting up the difference backwards as $A - B = 20$ (which leads to negative numbers) or getting confused between the fractions and percentages while crossing over terms. Relying heavily on equations without noticing the "multiple of 9" shortcut costs valuable time. ### Final Answer **Therefore, the correct answer is 180.**