A fruit seller has a total of 187 fruits consisting of apples, mangoes and oranges. The number of apples and mangoes are in the ratio 5 : 2. After she sells 75 apples, 26 mangoes and half of the oranges, the ratio of number of unsold apples to number of unsold oranges becomes 3 : 2. The total number of unsold fruits is

The correct option is 66.

Let's break down the problem step-by-step to find the total number of unsold fruits.

Step 1: Set up the initial equations
Let the initial number of apples, mangoes, and oranges be $A$, $M$, and $O$ respectively.
We are given that the total number of fruits is 187:
$A+M+O=187$

The ratio of apples to mangoes is 5 : 2. This means we can express $A$ and $M$ in terms of a common variable $x$:
$A=5x$
$M=2x$

Substituting these into the total fruits equation:
$5x+2x+O=187$
$7x+O=187$
Therefore, the number of oranges is:
$O=187-7x$

Step 2: Express the unsold fruits after sales
The fruit seller sells 75 apples, 26 mangoes, and half of the oranges ($\frac{O}{2}$ sold, meaning $\frac{O}{2}$ remain unsold).
The number of unsold fruits of each type is:
Unsold apples = $A-75=5x-75$
Unsold mangoes = $M-26=2x-26$
Unsold oranges = $\frac{O}{2}=\frac{187-7x}{2}$

Step 3: Use the ratio of unsold apples to unsold oranges
The ratio of unsold apples to unsold oranges is 3 : 2:
$\frac{5x-75}{\left(\frac{187-7x}{2}\right)}=\frac{3}{2}$

Simplifying this equation:
$\frac{2(5x-75)}{187-7x}=\frac{3}{2}$
$4(5x-75)=3(187-7x)$
$20x-300=561-21x$
$41x=861$
$x=21$

Step 4: Calculate the unsold quantities
Using $x=21$:
Unsold apples = $5\left(21\right)-75=105-75=30$
Unsold mangoes = $2\left(21\right)-26=42-26=16$
Unsold oranges = $\frac{187-7\left(21\right)}{2}=\frac{187-147}{2}=\frac{40}{2}=20$

Step 5: Find the total number of unsold fruits
Total unsold fruits = Unsold apples + Unsold mangoes + Unsold oranges
Total unsold fruits = $30+16+20=66$