Express each of the following ratios in simplest form,
(i) `(1)/(4) : (1)/(6)`
(ii) `8(1)/(3) : 6(1)/(4)`
(iii) `8.5 : 10`

Answer

Let's solve each part of the question step by step.

### (i) Express `(1)/(4) : (1)/(6)` in simplest form

1. **Write the ratio as a fraction**:
\[ \frac{1/4}{1/6} \]

2. **Simplify the fraction**:
To divide by a fraction, we multiply by its reciprocal. Thus, we have:
\[ \frac{1}{4} \div \frac{1}{6} = \frac{1}{4} \times \frac{6}{1} = \frac{6}{4} \]

3. **Reduce the fraction**:
Simplifying \(\frac{6}{4}\) gives:
\[ \frac{6 \div 2}{4 \div 2} = \frac{3}{2} \]

4. **Express as a ratio**:
Therefore, the simplest form of the ratio is:
\[ 3 : 2 \]

### (ii) Express `8(1)/(3) : 6(1)/(4)` in simplest form

1. **Convert mixed numbers to improper fractions**:
\[ 8 \frac{1}{3} = \frac{8 \times 3 + 1}{3} = \frac{25}{3} \]
\[ 6 \frac{1}{4} = \frac{6 \times 4 + 1}{4} = \frac{25}{4} \]

2. **Write the ratio as a fraction**:
\[ \frac{25/3}{25/4} \]

3. **Simplify the fraction**:
This becomes:
\[ \frac{25}{3} \div \frac{25}{4} = \frac{25}{3} \times \frac{4}{25} = \frac{4}{3} \]

4. **Express as a ratio**:
Therefore, the simplest form of the ratio is:
\[ 4 : 3 \]

### (iii) Express `8.5 : 10` in simplest form

1. **Convert the decimal to a fraction**:
\[ 8.5 = \frac{85}{10} \]

2. **Write the ratio as a fraction**:
\[ \frac{85/10}{10/1} \]

3. **Simplify the fraction**:
This becomes:
\[ \frac{85}{10} \div 10 = \frac{85}{10} \times \frac{1}{10} = \frac{85}{100} \]

4. **Reduce the fraction**:
Simplifying \(\frac{85}{100}\) gives:
\[ \frac{85 \div 5}{100 \div 5} = \frac{17}{20} \]

5. **Express as a ratio**:
Therefore, the simplest form of the ratio is:
\[ 17 : 20 \]

### Summary of Results
- (i) \(3 : 2\)
- (ii) \(4 : 3\)
- (iii) \(17 : 20\)