The value of $\frac{4}{7}+\frac{5}{14}-\frac{7}{8}$ is :

The correct answer is $\frac{3}{56}$.

To evaluate the given mathematical expression, we need to add and subtract the fractions:

$\frac{4}{7}+\frac{5}{14}-\frac{7}{8}$

Step 1: Find the Least Common Multiple (LCM) of the denominators.
The denominators of the given fractions are 7, 14, and 8.
Let us find their prime factorizations:

7 = 7
14 = 2 × 7
8 = 2³ = 8

The LCM of 7, 14, and 8 is 2³ × 7 = 8 × 7 = 56.

Step 2: Convert each fraction to an equivalent fraction with a common denominator of 56.

For $\frac{4}{7}$:
$\frac{4}{7}=\frac{4\times 8}{7\times 8}=\frac{32}{56}$

For $\frac{5}{14}$:
$\frac{5}{14}=\frac{5\times 4}{14\times 4}=\frac{20}{56}$

For $\frac{7}{8}$:
$\frac{7}{8}=\frac{7\times 7}{8\times 7}=\frac{49}{56}$

Step 3: Combine the numerators over the common denominator.

$\frac{32}{56}+\frac{20}{56}-\frac{49}{56}=\frac{32+20-49}{56}$

Now perform the arithmetic operations in the numerator:

32 + 20 = 52
52 - 49 = 3

Thus, we get:

$\frac{3}{56}$

Therefore, the simplified value of the given expression is $\frac{3}{56}$.