If in an A.P `(a_(7))/(a_(10))=(5)/(7)`,find` (a_(4))/(a_(7))`

Answer

To solve the problem, we need to find the ratio \( \frac{a_4}{a_7} \) given that \( \frac{a_7}{a_{10}} = \frac{5}{7} \) in an arithmetic progression (A.P.).

### Step-by-Step Solution:

1. **Understanding the Terms in A.P.**:
In an arithmetic progression, the \( n \)-th term \( a_n \) can be expressed as:
\[ a_n = a + (n - 1)d \]
where \( a \) is the first term and \( d \) is the common difference.

2. **Expressing \( a_7 \) and \( a_{10} \)**:
- For \( a_7 \):
\[ a_7 = a + (7 - 1)d = a + 6d \]
- For \( a_{10} \):
\[ a_{10} = a + (10 - 1)d = a + 9d \]

3. **Setting up the Equation**:
We know from the problem statement that:
\[ \frac{a_7}{a_{10}} = \frac{5}{7} \]
Substituting the expressions for \( a_7 \) and \( a_{10} \):
\[ \frac{a + 6d}{a + 9d} = \frac{5}{7} \]

4. **Cross Multiplying**:
Cross multiplying gives us:
\[ 7(a + 6d) = 5(a + 9d) \]
Expanding both sides:
\[ 7a + 42d = 5a + 45d \]

5. **Rearranging the Equation**:
Bringing all terms involving \( a \) to one side and all terms involving \( d \) to the other side:
\[ 7a - 5a = 45d - 42d \]
Simplifying this leads to:
\[ 2a = 3d \]

6. **Finding \( \frac{a_4}{a_7} \)**:
Now we need to express \( a_4 \):
- For \( a_4 \):
\[ a_4 = a + (4 - 1)d = a + 3d \]
- We already have \( a_7 = a + 6d \).

Now, we can find the ratio \( \frac{a_4}{a_7} \):
\[ \frac{a_4}{a_7} = \frac{a + 3d}{a + 6d} \]

7. **Substituting \( a \) in terms of \( d \)**:
From \( 2a = 3d \), we can express \( a \) as:
\[ a = \frac{3d}{2} \]
Substituting this into the ratio:
\[ \frac{a + 3d}{a + 6d} = \frac{\frac{3d}{2} + 3d}{\frac{3d}{2} + 6d} \]
Simplifying the numerator and denominator:
\[ = \frac{\frac{3d}{2} + \frac{6d}{2}}{\frac{3d}{2} + \frac{12d}{2}} = \frac{\frac{9d}{2}}{\frac{15d}{2}} = \frac{9d}{15d} = \frac{9}{15} = \frac{3}{5} \]

### Final Answer:
Thus, the value of \( \frac{a_4}{a_7} \) is:
\[ \frac{3}{5} \]