Step 1: Write the given ratio of fractions

The apples are distributed among Ajay, Amit, and Amar in the fractional ratio: $\frac{1}{5}: \frac{1}{7}: \frac{1}{6}$

$\text{Ratio} = \frac{1}{5}: \frac{1}{7}: \frac{1}{6}$

Step 2: Convert fractional ratio to whole number ratio

To eliminate fractions, find the Least Common Multiple (LCM) of the denominators $5, 7,$ and $6$. Since they share no common factors, $\text{LCM} = 5 \times 7 \times 6 = 210$. Multiply each term by $210$: $\text{Ajay} = 210 \times \frac{1}{5} = 42$, $\text{Amit} = 210 \times \frac{1}{7} = 30$, $\text{Amar} = 210 \times \frac{1}{6} = 35$

$\text{Simplified Ratio} = 42: 30: 35$

Step 3: Calculate the total sum of ratio parts

Add the simplified ratio parts together to get the total number of parts: $\text{Total parts} = 42 + 30 + 35 = 107$

$\text{Total parts} = 107$

Step 4: Find Amar's share of apples

Using the total number of apples ($214$), calculate Amar's share using his ratio proportion: $\text{Amar's apples} = 214 \times \frac{35}{107} = 2 \times 35 = 70$

$\text{Amar's apples} = 70$