If X = 5Y = 8Z, find X:Y:Z.

Answer

To find the ratio \( X:Y:Z \) given that \( X = 5Y = 8Z \), we can follow these steps:

### Step 1: Set a common variable
Let \( X = k \). This means we can express \( Y \) and \( Z \) in terms of \( k \).

### Step 2: Express \( Y \) in terms of \( k \)
From the equation \( X = 5Y \), we can write:
\[ Y = \frac{X}{5} = \frac{k}{5} \]

### Step 3: Express \( Z \) in terms of \( k \)
From the equation \( X = 8Z \), we can write:
\[ Z = \frac{X}{8} = \frac{k}{8} \]

### Step 4: Write the ratio \( X:Y:Z \)
Now we can express the ratio \( X:Y:Z \) as:
\[ X:Y:Z = k : \frac{k}{5} : \frac{k}{8} \]

### Step 5: Eliminate \( k \) from the ratio
To simplify the ratio, we can multiply each part by the least common multiple (LCM) of the denominators (5 and 8). The LCM of 5 and 8 is 40. Thus, we multiply each term by 40:
\[ X:Y:Z = 40k : 40 \cdot \frac{k}{5} : 40 \cdot \frac{k}{8} \]
This simplifies to:
\[ X:Y:Z = 40k : 8k : 5k \]

### Step 6: Remove \( k \)
Since \( k \) is common in all terms, we can remove it:
\[ X:Y:Z = 40 : 8 : 5 \]

### Final Answer
Thus, the ratio \( X:Y:Z \) is:
\[ \boxed{40 : 8 : 5} \]