If `""_(3)^(X)=""_(5)^(Y)=""_(9)^(Z)`, then what is the ratio of `X:Y:Z` ?

Options

A. `9:5:6`
B. `3:5:6`
C. `15:9:5`
D. `3:5:9`

Answer

To solve the problem, we start with the given equation:

\[ \frac{X}{3} = \frac{Y}{5} = \frac{Z}{9} \]

Let's denote the common ratio as \( k \). Therefore, we can express \( X \), \( Y \), and \( Z \) in terms of \( k \):

1. From the equation \( \frac{X}{3} = k \), we can express \( X \) as:
\[ X = 3k \]

2. From the equation \( \frac{Y}{5} = k \), we can express \( Y \) as:
\[ Y = 5k \]

3. From the equation \( \frac{Z}{9} = k \), we can express \( Z \) as:
\[ Z = 9k \]

Now we have \( X \), \( Y \), and \( Z \) in terms of \( k \):
- \( X = 3k \)
- \( Y = 5k \)
- \( Z = 9k \)

Next, we can find the ratio \( X:Y:Z \):
\[ X:Y:Z = 3k : 5k : 9k \]

Since \( k \) is a common factor, we can simplify the ratio by dividing each term by \( k \):
\[ X:Y:Z = 3 : 5 : 9 \]

Thus, the final ratio of \( X:Y:Z \) is:
\[ \boxed{3 : 5 : 9} \]