Express each of the following ratios in simplest form :
`3 (1)/(4):6(1)/(2)`

Answer

To express the ratio \(3 \frac{1}{4} : 6 \frac{1}{2}\) in simplest form, we can follow these steps:

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

- For \(3 \frac{1}{4}\):
\[ 3 \frac{1}{4} = \frac{(3 \times 4) + 1}{4} = \frac{12 + 1}{4} = \frac{13}{4} \]

- For \(6 \frac{1}{2}\):
\[ 6 \frac{1}{2} = \frac{(6 \times 2) + 1}{2} = \frac{12 + 1}{2} = \frac{13}{2} \]

So, we can rewrite the ratio as:
\[ \frac{13}{4} : \frac{13}{2} \]

### Step 2: Rewrite the Ratio as a Division of Fractions
Next, we can express the ratio as a division of fractions:
\[ \frac{13}{4} \div \frac{13}{2} \]

### Step 3: Multiply by the Reciprocal
To divide by a fraction, we multiply by its reciprocal:
\[ \frac{13}{4} \times \frac{2}{13} \]

### Step 4: Simplify the Expression
Now, we can simplify the expression:
- The \(13\) in the numerator and denominator cancels out:
\[ \frac{13 \times 2}{4 \times 13} = \frac{2}{4} \]

### Step 5: Reduce to Simplest Form
Now, we need to reduce \(\frac{2}{4}\) to its simplest form. We can find the highest common factor (HCF) of 2 and 4.

- The HCF of 2 and 4 is 2. Now we divide both the numerator and the denominator by their HCF:
\[ \frac{2 \div 2}{4 \div 2} = \frac{1}{2} \]

Thus, the simplest form of the ratio \(3 \frac{1}{4} : 6 \frac{1}{2}\) is:
\[ 1 : 2 \]

### Final Answer:
The simplest form of the ratio \(3 \frac{1}{4} : 6 \frac{1}{2}\) is \(1 : 2\).

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