If x is the fourth proportional to 36, 48 and 15, and 4.2, y, 9 and 13.5 are in proportion, then the difference between y and x is:

Options

A. 13.7
B. 16.4
C. 15.6
D. 14.8

Answer

To solve the problem step by step, we will first find the value of \( x \) and then the value of \( y \). Finally, we will calculate the difference between \( y \) and \( x \).

### Step 1: Find the value of \( x \)
We know that \( x \) is the fourth proportional to 36, 48, and 15. This means we can set up the proportion as follows:

\[ \frac{36}{48} = \frac{15}{x} \]

Now, we will cross-multiply to solve for \( x \):

\[ 36 \cdot x = 48 \cdot 15 \]

Calculating the right side:

\[ 48 \cdot 15 = 720 \]

Now, we can rewrite the equation:

\[ 36x = 720 \]

Next, divide both sides by 36 to isolate \( x \):

\[ x = \frac{720}{36} = 20 \]

### Step 2: Find the value of \( y \)
Next, we know that 4.2, \( y \), 9, and 13.5 are in proportion. We can set up the proportion as follows:

\[ \frac{4.2}{y} = \frac{9}{13.5} \]

Cross-multiplying gives us:

\[ 4.2 \cdot 13.5 = 9 \cdot y \]

Calculating the left side:

\[ 4.2 \cdot 13.5 = 56.7 \]

Now we can rewrite the equation:

\[ 56.7 = 9y \]

Next, divide both sides by 9 to isolate \( y \):

\[ y = \frac{56.7}{9} = 6.3 \]

### Step 3: Calculate the difference between \( y \) and \( x \)
Now that we have both values, \( x = 20 \) and \( y = 6.3 \), we can find the difference:

\[ \text{Difference} = x - y = 20 - 6.3 = 13.7 \]

### Final Answer
The difference between \( y \) and \( x \) is:

\[ \boxed{13.7} \]