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

Total fruits = 187

We're told apples and mangoes are in the ratio 5:2. This means for every 5 apples, there are 2 mangoes.

When we have a ratio like 5:2, we can't directly say there are 5 apples and 2 mangoes (that would only be 7 fruits total!). Instead, we use a multiplier.

Let the common multiplier be $x$, so:

Apples = $5x$

Mangoes = $2x$

Since total fruits = 187:

Oranges = $187 - 5x - 2x = 187 - 7x$

---

The seller sells:

75 apples

26 mangoes

Half of the oranges = $\frac{187 - 7x}{2}$

If total oranges = $187 - 7x$, then half means we divide by 2.

---

Unsold = Initial - Sold

Unsold apples = $5x - 75$

Unsold mangoes = $2x - 26$

Unsold oranges = $(187 - 7x) - \frac{187 - 7x}{2}$

Simplifying unsold oranges:

$(187 - 7x) - \frac{187 - 7x}{2} = \frac{2(187 - 7x) - (187 - 7x)}{2} = \frac{187 - 7x}{2}$

This makes sense! If you sell half, you're left with half.

---

After selling, the ratio of unsold apples to unsold oranges becomes 3:2.

$\frac{\text{Unsold apples}}{\text{Unsold oranges}} = \frac{3}{2}$

Substituting our expressions:

$\frac{5x - 75}{\frac{187 - 7x}{2}} = \frac{3}{2}$

---

Cross multiplication: $(5x - 75) \times 2 = 3 \times \frac{187 - 7x}{2}$

$2(5x - 75) = \frac{3(187 - 7x)}{2}$

$4(5x - 75) = 3(187 - 7x)$

$20x - 300 = 561 - 21x$

$20x + 21x = 561 + 300$

$41x = 861$

$x = 21$

---

With $x = 21$:

Initial apples = $5(21) = 105$

Initial mangoes = $2(21) = 42$

Initial oranges = $187 - 7(21) = 187 - 147 = 40$

We can verify: $105 + 42 + 40 = 187$

Calculating unsold fruits:

Unsold apples = $105 - 75 = 30$

Unsold mangoes = $42 - 26 = 16$

Unsold oranges = $40 - 20 = 20$

We can check our ratio: $\frac{30}{20} = \frac{3}{2}$

Therefore, total unsold fruits = $30 + 16 + 20 = 66$

---

When dealing with ratio problems involving multiple constraints, we should always:

Use variables to represent the ratio parts

Set up equations for each condition

Solve systematically

Always verify the answer