`ab : bc : ca=1 : 2 : 3`
Find `a : b : c=?`

Answer

To solve the problem where we need to find the ratio \( a : b : c \) given that \( ab : bc : ca = 1 : 2 : 3 \), we can follow these steps:

### Step 1: Set up the equations
From the given ratios, we can express \( ab \), \( bc \), and \( ca \) in terms of a common variable \( k \):
- Let \( ab = k \)
- Let \( bc = 2k \)
- Let \( ca = 3k \)

### Step 2: Express \( a \), \( b \), and \( c \) in terms of \( k \)
From the equations we have:
1. \( ab = k \) implies \( a = \frac{k}{b} \)
2. \( bc = 2k \) implies \( c = \frac{2k}{b} \)
3. \( ca = 3k \) implies \( c = \frac{3k}{a} \)

### Step 3: Substitute \( c \) from the second equation into the third
From \( c = \frac{2k}{b} \), we can substitute into the third equation:
\[ \frac{2k}{b} = \frac{3k}{a} \]
Cross-multiplying gives:
\[ 2ak = 3kb \]
Dividing both sides by \( k \) (assuming \( k \neq 0 \)):
\[ 2a = 3b \quad \Rightarrow \quad \frac{a}{b} = \frac{3}{2} \]

### Step 4: Find \( a \) in terms of \( b \)
From \( \frac{a}{b} = \frac{3}{2} \), we can express \( a \) as:
\[ a = \frac{3}{2}b \]

### Step 5: Substitute \( a \) into the expression for \( c \)
Now, we can substitute \( a \) back into the expression for \( c \):
Using \( c = \frac{2k}{b} \) and substituting \( k = ab \):
\[ c = \frac{2k}{b} = \frac{2 \cdot \frac{3}{2}b \cdot b}{b} = 3b \]

### Step 6: Write the ratios
Now we have:
- \( a = \frac{3}{2}b \)
- \( b = b \)
- \( c = 3b \)

Thus, the ratio \( a : b : c \) can be expressed as:
\[ \frac{3}{2}b : b : 3b \]

### Step 7: Normalize the ratio
To express this ratio in the simplest form, we can eliminate \( b \) (as it is common) and multiply through by 2 to eliminate the fraction:
\[ 3 : 2 : 6 \]

### Final Answer
Thus, the ratio \( a : b : c = 3 : 2 : 6 \).

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