Express each of the following rations in simplest form :
`(1)/(6) : (1)/(9)`

Answer

To express the ratio \( \frac{1}{6} : \frac{1}{9} \) in simplest form, follow these steps:

### Step 1: Write the ratio in fraction form
The ratio \( \frac{1}{6} : \frac{1}{9} \) can be expressed as a fraction:
\[ \frac{\frac{1}{6}}{\frac{1}{9}} \]

### Step 2: Simplify the fraction
When dividing fractions, we multiply by the reciprocal of the denominator:
\[ \frac{\frac{1}{6}}{\frac{1}{9}} = \frac{1}{6} \times \frac{9}{1} = \frac{9}{6} \]

### Step 3: Reduce the fraction to simplest form
To simplify \( \frac{9}{6} \), we find the highest common factor (HCF) of 9 and 6. The HCF of 9 and 6 is 3. Now, divide both the numerator and the denominator by their HCF:
\[ \frac{9 \div 3}{6 \div 3} = \frac{3}{2} \]

### Step 4: Write the ratio in simplest form
Thus, the ratio \( \frac{1}{6} : \frac{1}{9} \) in simplest form is:
\[ 3 : 2 \]

### Final Answer
The simplest form of the ratio \( \frac{1}{6} : \frac{1}{9} \) is \( 3 : 2 \).

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