Express each of the following rations in simplest form :
` 4 (1)/(2): 1 (1)/(8)`

Answer

To express the ratio \( 4 \frac{1}{2} : 1 \frac{1}{8} \) in simplest form, follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
First, we need to convert the mixed numbers into improper fractions.

1. For \( 4 \frac{1}{2} \):
- Multiply the whole number (4) by the denominator (2) and add the numerator (1):
\[ 4 \times 2 + 1 = 8 + 1 = 9 \]
- Therefore, \( 4 \frac{1}{2} = \frac{9}{2} \).

2. For \( 1 \frac{1}{8} \):
- Multiply the whole number (1) by the denominator (8) and add the numerator (1):
\[ 1 \times 8 + 1 = 8 + 1 = 9 \]
- Therefore, \( 1 \frac{1}{8} = \frac{9}{8} \).

### Step 2: Write the Ratio in Fraction Form
Now we can express the ratio as:
\[ \frac{9}{2} : \frac{9}{8} \]

### Step 3: Simplify the Ratio
To simplify the ratio \( \frac{9}{2} : \frac{9}{8} \), we can write it as a single fraction:
\[ \frac{9}{2} \div \frac{9}{8} \]
This is the same as multiplying by the reciprocal:
\[ \frac{9}{2} \times \frac{8}{9} \]

### Step 4: Cancel Common Terms
Now, we can cancel the common terms (9):
\[ \frac{8}{2} = 4 \]
So, the simplified ratio is:
\[ 4 : 1 \]

### Final Answer
Thus, the simplest form of the ratio \( 4 \frac{1}{2} : 1 \frac{1}{8} \) is:
\[ 4 : 1 \]