The mean proportional between 54 and 6 is x, and the fourth proportional of 2,3,12….. Is y. What is `x:y`?

Options

A. `2:1`
B. `1:1`
C. `1:2`
D. `1:3`

Answer

To solve the problem, we need to find the mean proportional \( x \) between 54 and 6, and the fourth proportional \( y \) of the numbers 2, 3, and 12. Finally, we will determine the ratio \( x:y \).

### Step 1: Finding the Mean Proportional \( x \)

The mean proportional between two numbers \( a \) and \( b \) can be found using the formula:
\[ \frac{a}{x} = \frac{x}{b} \]
For our case, \( a = 54 \) and \( b = 6 \). Therefore, we have:
\[ \frac{54}{x} = \frac{x}{6} \]

### Step 2: Cross Multiplying

Cross-multiplying gives us:
\[ 54 \cdot 6 = x^2 \]
Calculating \( 54 \cdot 6 \):
\[ 54 \cdot 6 = 324 \]
So, we have:
\[ x^2 = 324 \]

### Step 3: Solving for \( x \)

To find \( x \), we take the square root of both sides:
\[ x = \sqrt{324} = 18 \]

### Step 4: Finding the Fourth Proportional \( y \)

The fourth proportional \( y \) of the numbers \( a, b, c \) is found using the relationship:
\[ \frac{a}{b} = \frac{c}{y} \]
For our case, \( a = 2 \), \( b = 3 \), and \( c = 12 \). Thus, we have:
\[ \frac{2}{3} = \frac{12}{y} \]

### Step 5: Cross Multiplying for \( y \)

Cross-multiplying gives us:
\[ 2y = 3 \cdot 12 \]
Calculating \( 3 \cdot 12 \):
\[ 3 \cdot 12 = 36 \]
So, we have:
\[ 2y = 36 \]

### Step 6: Solving for \( y \)

To find \( y \), we divide both sides by 2:
\[ y = \frac{36}{2} = 18 \]

### Step 7: Finding the Ratio \( x:y \)

Now that we have both \( x \) and \( y \):
\[ x = 18 \quad \text{and} \quad y = 18 \]
The ratio \( x:y \) is:
\[ \frac{x}{y} = \frac{18}{18} = 1 \]
Thus, the ratio \( x:y \) can be expressed as:
\[ 1:1 \]

### Final Answer

The final answer is:
\[ \text{The ratio } x:y = 1:1 \]