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

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Accepted Answer

The problem involves determining the total number of unsold fruits after a fruit seller sells a portion of her stock. Here's the step-by-step solution:

**Initial Setup:**

- Let the number of apples = $$ 5x $$
   - Let the number of mangoes = $$ 2x $$
   - Let the number of oranges = $$ 2y $$ (since half the oranges are sold, we express oranges as $$ 2y $$ for simplicity).

The total number of fruits is given as:
   $$
   5x + 2x + 2y = 187
   $$
   Simplifying:
   $$
   7x + 2y = 187 	ag{1}
   $$

**Unsold Fruits:**

- After selling:
     - Unsold apples = $$ 5x - 75 $$
     - Unsold mangoes = $$ 2x - 26 $$
     - Unsold oranges = $$ y $$ (since half of $$ 2y $$ is sold).

**Ratio Condition:**

The ratio of unsold apples to unsold oranges is given as $$ 3:2 $$. This translates to:
   $$
   \frac{5x - 75}{y} = \frac{3}{2}
   $$
   Cross-multiply:
   $$
   10x - 150 = 3y
   $$
   Rearrange:
   $$
   10x - 3y = 150 	ag{2}
   $$

**Solve Simultaneous Equations:**

From equations (1) and (2):
   - Multiply equation (1) by 3:
     $$
     21x + 6y = 561
     $$
   - Multiply equation (2) by 2:
     $$
     20x - 6y = 300
     $$

Add these two equations:
   $$
   (21x + 6y) + (20x - 6y) = 561 + 300
   $$
   Simplify:
   $$
   41x = 861
   $$
   Solve for $$ x $$:
   $$
   x = \frac{861}{41} = 21
   $$

**Find $$ y $$:**

Substitute $$ x = 21 $$ into equation (1):
   $$
   7(21) + 2y = 187
   $$
   Simplify:
   $$
   147 + 2y = 187
   $$
   Solve for $$ y $$:
   $$
   2y = 40, \quad y = 20
   $$

**Calculate Total Unsold Fruits:**

Substitute $$ x = 21, y = 20 $$ into the expressions for unsold fruits:
   - Unsold apples: $$ 5x - 75 = (5)(21) - 75 = 105 - 75 = 30 $$
   - Unsold mangoes: $$ 2x - 26 = (2)(21) - 26 = 42 - 26 = 16 $$
   - Unsold oranges: $$ y = 20 $$

Total unsold fruits:
   $$
   (30 + 16 + 20) = 66
   $$

### Final Answer:

The total number of unsold fruits is **66**.

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