If bc:ac:ab=1:3:5, then find `(a)/(bc):(b)/(ca)`.

Answer

To solve the problem where we need to find \(\frac{a}{bc} : \frac{b}{ca}\) given the ratio \(bc : ac : ab = 1 : 3 : 5\), we can follow these steps:

### Step 1: Assign Variables Based on the Given Ratios
From the ratio \(bc : ac : ab = 1 : 3 : 5\), we can assign:
- \(bc = 1k\)
- \(ac = 3k\)
- \(ab = 5k\)

where \(k\) is a common multiplier.

### Step 2: Express \(a\), \(b\), and \(c\) in Terms of \(k\)
We can express \(a\), \(b\), and \(c\) using the relationships from the ratios:
- From \(bc = 1k\), we can write \(c = \frac{1k}{b}\).
- From \(ac = 3k\), we can write \(a = \frac{3k}{c}\).
- From \(ab = 5k\), we can write \(b = \frac{5k}{a}\).

### Step 3: Substitute to Find Relationships
Now, substituting \(c\) from the first equation into the second:
\[ a = \frac{3k}{\frac{1k}{b}} = \frac{3kb}{1} = 3kb \]
Now substituting \(a\) into the third equation:
\[ b = \frac{5k}{3kb} \implies b^2 = \frac{5k}{3k} \implies b^2 = \frac{5}{3} \implies b = \sqrt{\frac{5}{3}} \]

### Step 4: Find \(a\) and \(c\)
Now we can find \(a\) and \(c\):
- From \(b = \sqrt{\frac{5}{3}}\):
\[ c = \frac{1k}{b} = \frac{1k}{\sqrt{\frac{5}{3}}} = \sqrt{\frac{3}{5}}k \]
- Substitute \(b\) into \(a = 3kb\):
\[ a = 3k\sqrt{\frac{5}{3}} = 3\sqrt{\frac{5}{3}}k \]

### Step 5: Calculate \(\frac{a}{bc}\) and \(\frac{b}{ca}\)
Now we calculate:
\[ bc = 1k = k \]
\[ ca = c \cdot a = \left(\sqrt{\frac{3}{5}}k\right) \cdot \left(3\sqrt{\frac{5}{3}}k\right) = 3k^2 \]

Now we can find:
\[ \frac{a}{bc} = \frac{3\sqrt{\frac{5}{3}}k}{k} = 3\sqrt{\frac{5}{3}} \]
\[ \frac{b}{ca} = \frac{\sqrt{\frac{5}{3}}k}{3k^2} = \frac{\sqrt{\frac{5}{3}}}{3k} \]

### Step 6: Form the Final Ratio
Now we need to find the ratio:
\[ \frac{a}{bc} : \frac{b}{ca} = 3\sqrt{\frac{5}{3}} : \frac{\sqrt{\frac{5}{3}}}{3k} \]

To simplify, we can multiply both sides by \(3k\):
\[ 3\sqrt{\frac{5}{3}} \cdot 3k : \sqrt{\frac{5}{3}} = 9k : 1 \]

### Final Answer
Thus, the final answer is:
\[ \frac{a}{bc} : \frac{b}{ca} = 9k : 1 \]