If `x=1/3y` and `y=1/2 z`.then `x: y : z`, is equal to :

Options

A. `3:2:1`
B. `1:2:6`
C. `1:3:6`
D. `2:4:6`

Answer

To solve the problem of finding the ratio \( x : y : z \) given that \( x = \frac{1}{3}y \) and \( y = \frac{1}{2}z \), we can follow these steps:

### Step 1: Express \( y \) in terms of \( z \)
From the equation \( y = \frac{1}{2}z \), we can express \( z \) in terms of \( y \):
\[ z = 2y \]

### Step 2: Substitute \( y \) in the equation for \( x \)
Now, substitute \( y \) from the first equation into the equation for \( x \):
\[ x = \frac{1}{3}y \]
Substituting \( y = \frac{1}{2}z \):
\[ x = \frac{1}{3} \left(\frac{1}{2}z\right) = \frac{1}{6}z \]

### Step 3: Now we have expressions for \( x \) and \( y \) in terms of \( z \)
We have:
- \( x = \frac{1}{6}z \)
- \( y = \frac{1}{2}z \)

### Step 4: Write the ratio \( x : y : z \)
Now we can express the ratio \( x : y : z \) in terms of \( z \):
\[ x : y : z = \frac{1}{6}z : \frac{1}{2}z : z \]

### Step 5: Eliminate \( z \) from the ratio
To simplify the ratio, we can factor out \( z \):
\[ x : y : z = \frac{1}{6} : \frac{1}{2} : 1 \]

### Step 6: Convert to a common denominator
To make the ratios easier to compare, we can convert them to have a common denominator. The common denominator for 6 and 2 is 6:
\[ \frac{1}{6} : \frac{3}{6} : \frac{6}{6} = 1 : 3 : 6 \]

### Final Answer
Thus, the ratio \( x : y : z \) is:
\[ \boxed{1 : 3 : 6} \]

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