If X: Y = 2:3 and Y:Z = 3:4 find X:Z .

Answer

To find the ratio \( X:Z \) given the ratios \( X:Y = 2:3 \) and \( Y:Z = 3:4 \), we can follow these steps:

### Step 1: Write down the given ratios
We have:
1. \( X:Y = 2:3 \)
2. \( Y:Z = 3:4 \)

### Step 2: Express the ratios in terms of a common variable
From the first ratio \( X:Y = 2:3 \), we can express \( X \) and \( Y \) as:
- Let \( X = 2k \) (where \( k \) is a common multiplier)
- Then \( Y = 3k \)

From the second ratio \( Y:Z = 3:4 \), we can express \( Y \) and \( Z \) as:
- Let \( Y = 3m \) (where \( m \) is another common multiplier)
- Then \( Z = 4m \)

### Step 3: Set the expressions for \( Y \) equal to each other
Since both expressions represent \( Y \), we can set them equal:
\[ 3k = 3m \]

### Step 4: Solve for \( k \) and \( m \)
From \( 3k = 3m \), we can simplify to:
\[ k = m \]

### Step 5: Substitute back to find \( Z \) in terms of \( k \)
Now, substituting \( m \) back in terms of \( k \):
- From \( Z = 4m \), we have \( Z = 4k \)

### Step 6: Write \( X:Z \) in terms of \( k \)
Now we have:
- \( X = 2k \)
- \( Z = 4k \)

Thus, the ratio \( X:Z \) can be expressed as:
\[ X:Z = 2k:4k \]

### Step 7: Simplify the ratio
We can simplify \( 2k:4k \) by dividing both terms by \( 2k \):
\[ X:Z = 1:2 \]

### Final Answer
Thus, the ratio \( X:Z \) is \( 1:2 \).

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