Question

There are 187 fruits in total. The ratio of apples to mangoes is $2 : 3$, and the rest are oranges. After selling 67 apples, 26 mangoes, and half of the oranges, the ratio of the remaining apples to oranges becomes $1 : 2$. Determine how many unsold fruits remain.

Answer 1

Step 1: Define variables
Let the number of apples = $2x$, mangoes = $3x$, and oranges = $187 - 5x$ (since the total number of fruits is 187).
Step 2: Remaining fruits after sales
Apples sold = 67 $\rightarrow$ Remaining apples = $2x - 67$,
Mangoes sold = 26 $\rightarrow$ Remaining mangoes = $3x - 26$,
Half of the oranges sold $\rightarrow$ Remaining oranges = $\frac{187 - 5x}{2}$.
**Step 3: Ratio condition **
After sales, the ratio of remaining apples to remaining oranges is $1 : 2$:
$\frac{2x - 67}{\frac{187 - 5x}{2}} = \frac{1}{2}.$
**Step 4: Solve for $x$ **
Cross-multiply to simplify:
$2 \cdot (2x - 67) = 187 - 5x.$
$4x - 134 = 187 - 5x.$
$4x + 5x = 187 + 134 \quad \rightarrow \quad 9x = 321.$
$x = 35.7 \quad \text{(round down to nearest integer $x = 35$)}.$
Step 5: Calculate original fruit counts
Apples = $2x = 2 \times 35 = 70$,
Mangoes = $3x = 3 \times 35 = 105$,
Oranges = $187 - 5x = 187 - 175 = 12$.
Step 6: Remaining fruits after sales
Remaining apples = $70 - 67 = 3$,
Remaining mangoes = $105 - 26 = 79$,
Remaining oranges = $\frac{12}{2} = 6$.
**Step 7: Total unsold fruits **
$\text{Total unsold fruits} = 3 + 79 + 6 = 88.$
Conclusion: The total number of unsold fruits is 88.

Answer 2

Step 1: Define variables
Let the number of apples = $2x$, mangoes = $3x$, and oranges = $187 - 5x$ (since the total number of fruits is 187).
Step 2: Remaining fruits after sales
Apples sold = 67 $\rightarrow$ Remaining apples = $2x - 67$,
Mangoes sold = 26 $\rightarrow$ Remaining mangoes = $3x - 26$,
Half of the oranges sold $\rightarrow$ Remaining oranges = $\frac{187 - 5x}{2}$.
**Step 3: Ratio condition **
After sales, the ratio of remaining apples to remaining oranges is $1 : 2$:
$\frac{2x - 67}{\frac{187 - 5x}{2}} = \frac{1}{2}.$
**Step 4: Solve for $x$ **
Cross-multiply to simplify:
$2 \cdot (2x - 67) = 187 - 5x.$
$4x - 134 = 187 - 5x.$
$4x + 5x = 187 + 134 \quad \rightarrow \quad 9x = 321.$
$x = 35.7 \quad \text{(round down to nearest integer $x = 35$)}.$
Step 5: Calculate original fruit counts
Apples = $2x = 2 \times 35 = 70$,
Mangoes = $3x = 3 \times 35 = 105$,
Oranges = $187 - 5x = 187 - 175 = 12$.
Step 6: Remaining fruits after sales
Remaining apples = $70 - 67 = 3$,
Remaining mangoes = $105 - 26 = 79$,
Remaining oranges = $\frac{12}{2} = 6$.
**Step 7: Total unsold fruits **
$\text{Total unsold fruits} = 3 + 79 + 6 = 88.$
Conclusion: The total number of unsold fruits is 88.