The ratio of the arithmetic mean of two numbers to one of the numbers is 3 : 5. What is the ratio of the smaller number to the larger one?

### Concept & Arithmetic Mean The arithmetic mean (average) of two numbers $a$ and $b$ is given by the formula $\frac{a + b}{2}$. ### Step-by-Step Solution * Let the two numbers be $x$ and $y$. Their arithmetic mean is $\frac{x + y}{2}$. * The ratio of the arithmetic mean to one of the numbers is $3 : 5$. Let's assume the ratio is with $y$. $$\frac{\frac{x + y}{2}}{y} = \frac{3}{5}$$ * Cross-multiply to solve for the relationship between $x$ and $y$: $$5 \times \left(\frac{x + y}{2}\right) = 3y$$ * $5x + 5y = 6y$ * $5x = y$ * This means $y$ is 5 times $x$. So, $x$ is the smaller number and $y$ is the larger number. * The ratio of the smaller number to the larger one is $x : y = x : 5x = 1 : 5$. ### Exam Strategy & Shortcut Assume the number given in the ratio is 5. If the mean is 3, the sum of the two numbers must be $2 \times 3 = 6$. Since one number is 5, the other number must be $6 - 5 = 1$. The two numbers are 1 and 5. The ratio of the smaller to the larger is $1 : 5$. This avoids algebra entirely. ### Common Pitfall Setting up the ratio with the wrong number and concluding an impossible negative relationship if one doesn't realize the mean is always strictly between two distinct positive numbers. ### Final Answer Therefore, the correct answer is **1 : 5**.