Two numbers are respectively 20 percent and 50 percent more than a third number. These two numbers are in the ratio

### Concept & Relative Percentages When multiple distinct quantities are compared to a single common base using percentages, you can assign a value of $100$ to that base. This allows you to rapidly express the other quantities as integer values before forming a ratio. ### Step-by-Step Solution 1. Let the third number (the common base) be $100$. 2. The first number is $20\%$ more than the third number: $$ \text{First Number} = 100 + (20\% \text{ of } 100) = 100 + 20 = 120 $$ 3. The second number is $50\%$ more than the third number: $$ \text{Second Number} = 100 + (50\% \text{ of } 100) = 100 + 50 = 150 $$ 4. The question asks for the ratio of these two numbers (first number to second number): $$ \text{Ratio} = 120 : 150 $$ 5. Simplify the ratio by dividing both sides by their greatest common factor, which is $30$: $$ \frac{120}{30} : \frac{150}{30} = 4 : 5 $$ ### Exam Strategy & Shortcut Since both numbers are scaled from the same base, you can bypass the base entirely and just use the scale factors. The first number is $1.2$ times the base, and the second is $1.5$ times the base. Ratio = $1.2 : 1.5$. Multiply by $10$ to get $12 : 15$, then divide by $3$ to get $4 : 5$. ### Common Pitfall Students often confuse "20% more than" with just "20% of". If they simply use $20$ and $50$ as the numbers, they will get a ratio of $20 : 50 = 2 : 5$, which is an incorrect distractor (Option A). Always add the percentage to the base value ($100$). ### Final Answer Therefore, the correct answer is **4 : 5**.