Find triplicate ratio of :
(i) `1 : 3`
(ii) `(m)/(2) : (n)/(3)`

Answer

To find the triplicate ratio of the given ratios, we will follow these steps:

### Step 1: Understand the concept of triplicate ratio
The triplicate ratio of two quantities \( A \) and \( B \) is given by \( A^3 : B^3 \).

### Step 2: Calculate the triplicate ratio for \( 1 : 3 \)
Given the ratio \( 1 : 3 \):
- We need to find \( 1^3 : 3^3 \).
- Calculate \( 1^3 = 1 \).
- Calculate \( 3^3 = 27 \).
- Therefore, the triplicate ratio of \( 1 : 3 \) is \( 1 : 27 \).

### Step 3: Calculate the triplicate ratio for \( \frac{m}{2} : \frac{n}{3} \)
Given the ratio \( \frac{m}{2} : \frac{n}{3} \):
- We need to find \( \left(\frac{m}{2}\right)^3 : \left(\frac{n}{3}\right)^3 \).
- Calculate \( \left(\frac{m}{2}\right)^3 = \frac{m^3}{2^3} = \frac{m^3}{8} \).
- Calculate \( \left(\frac{n}{3}\right)^3 = \frac{n^3}{3^3} = \frac{n^3}{27} \).
- Therefore, the triplicate ratio of \( \frac{m}{2} : \frac{n}{3} \) is \( \frac{m^3}{8} : \frac{n^3}{27} \).

### Step 4: Write the final answer for the second part
To express the triplicate ratio in a more standard form:
- We can write it as \( 27m^3 : 8n^3 \).

### Final Answers:
1. The triplicate ratio of \( 1 : 3 \) is \( 1 : 27 \).
2. The triplicate ratio of \( \frac{m}{2} : \frac{n}{3} \) is \( 27m^3 : 8n^3 \).

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