If the sum of the first ten terms of an `A.P` is four times the sum of its first five terms, the ratio of the first term to the common difference is:

Explanation

To solve the problem, we need to find the ratio of the first term to the common difference of an arithmetic progression (A.P.) given that the sum of the first ten terms is four times the sum of the first five terms.

### Step-by-Step Solution:

1. **Define Variables:**
Let \( A \) be the first term and \( D \) be the common difference of the A.P.

2. **Write the Formula for the Sum of Terms:**
The sum of the first \( n \) terms of an A.P. is given by the formula:
\[ S_n = \frac{n}{2} \times (2A + (n-1)D) \]

3. **Calculate the Sum of the First 10 Terms:**
For \( n = 10 \):
\[ S_{10} = \frac{10}{2} \times (2A + (10-1)D) = 5 \times (2A + 9D) = 10A + 45D \]

4. **Calculate the Sum of the First 5 Terms:**
For \( n = 5 \):
\[ S_{5} = \frac{5}{2} \times (2A + (5-1)D) = \frac{5}{2} \times (2A + 4D) = \frac{5}{2} \times (2A + 4D) = 5A + 10D \]

5. **Set Up the Equation:**
According to the problem, the sum of the first 10 terms is four times the sum of the first 5 terms:
\[ S_{10} = 4 \times S_{5} \]
Substituting the sums we calculated:
\[ 10A + 45D = 4 \times (5A + 10D) \]

6. **Expand and Simplify:**
Expanding the right side:
\[ 10A + 45D = 20A + 40D \]
Rearranging gives:
\[ 10A + 45D - 20A - 40D = 0 \]
Simplifying further:
\[ -10A + 5D = 0 \]
This can be rewritten as:
\[ 10A = 5D \]

7. **Find the Ratio:**
Dividing both sides by \( D \) and then by 10 gives:
\[ \frac{A}{D} = \frac{5}{10} = \frac{1}{2} \]

### Final Answer:
The ratio of the first term to the common difference is:
\[ \frac{A}{D} = \frac{1}{2} \]