If A = `(1)/(3)B` and `B = (1)/(2)C, then `A : B : C`?

Options

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

Answer

To solve the problem where \( A = \frac{1}{3}B \) and \( B = \frac{1}{2}C \), we need to find the ratio \( A : B : C \).

### Step 1: Express A in terms of B
Given:
\[ A = \frac{1}{3}B \]

### Step 2: Express B in terms of C
Given:
\[ B = \frac{1}{2}C \]

### Step 3: Substitute B in the expression for A
Now, substitute the expression for \( B \) into the equation for \( A \):
\[ A = \frac{1}{3} \left( \frac{1}{2}C \right) \]
\[ A = \frac{1}{6}C \]

### Step 4: Write the ratios
Now we have:
- \( A = \frac{1}{6}C \)
- \( B = \frac{1}{2}C \)
- \( C = C \)

Now we can express the ratios:
- \( A : B : C = \frac{1}{6}C : \frac{1}{2}C : C \)

### Step 5: Eliminate C from the ratios
To simplify the ratio, we can divide each term by \( C \):
\[ A : B : C = \frac{1}{6} : \frac{1}{2} : 1 \]

### Step 6: Find a common denominator
The common denominator for \( 6 \), \( 2 \), and \( 1 \) is \( 6 \). So we can express each term with this common denominator:
- \( \frac{1}{6} = 1 \)
- \( \frac{1}{2} = \frac{3}{6} \)
- \( 1 = \frac{6}{6} \)

Thus, we can write:
\[ A : B : C = 1 : 3 : 6 \]

### Final Answer
The final ratio is:
\[ A : B : C = 1 : 3 : 6 \]

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