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Quantitative Aptitude Question on Ratio and Proportion

A fruit seller has a stock of mangoes, bananas, and apples with at least one fruit of each type. At the beginning of a day, the number of mangoes makes up 40% of his stock. That day, he sells half of the mangoes,96 bananas, and 40% of the apples. At the end of the day, he ends up selling 50% of the fruits. What is the smallest possible total number of fruits in the stock at the beginning of the day?

Answer

Step 1 : Let the total number of fruits at the beginning of the day be TT

Since mangoes make up 40%40\% of the total stock:
Mangoes=0.4×T.\text{Mangoes} = 0.4 \times T.
The remaining 60%60\% of the stock consists of bananas and apples:
Bananas+Apples=0.6⋅T.\text{Bananas} + \text{Apples} = 0.6 \cdot T.

Step 2 : Fruits sold during the day
During the day:

  • Mangoes sold = 12⋅Mangoes=12⋅0.4⋅T=0.2⋅T\frac{1}{2} \cdot \text{Mangoes} = \frac{1}{2} \cdot 0.4 \cdot T = 0.2 \cdot T
  • Bananas sold = 96.
  • Apples sold = 40%40\% of the apples. Let the apples be AA. Then:
    Apples sold=0.4⋅A.\text{Apples sold} = 0.4 \cdot A.

Step 3: Total fruits sold

At the end of the day, 50%50\% of the total fruits were sold. Thus:
Total fruits sold=0.5⋅T.\text{Total fruits sold} = 0.5 \cdot T.

Substitute the components:
Total fruits sold=(Mangoes sold)+(Bananas sold)+(Apples sold).\text{Total fruits sold} = (\text{Mangoes sold}) + (\text{Bananas sold}) + (\text{Apples sold}).

Substitute the values:
0.5⋅T=0.2⋅T+96+0.4⋅A.(Equation 1)0.5 \cdot T = 0.2 \cdot T + 96 + 0.4 \cdot A. \quad \text{(Equation 1)}

Step 4 : Express apples in terms of TT

From the stock composition:
A+Bananas=0.6⋅T.A + \text{Bananas} = 0.6 \cdot T.

Let bananas B=96B = 96. Substitute:
A+96=0.6⋅T.A + 96 = 0.6 \cdot T.

Solve for AA:
A=0.6⋅T−96.(Equation 2)A = 0.6 \cdot T - 96. \quad \text{(Equation 2)}

Step 5 : Substitute AA into Equation 1

Substitute A=0.6×T−96A = 0.6 \times T - 96 into Equation 1:
0.5⋅T=0.2⋅T+96+0.4⋅(0.6⋅T−96).0.5 \cdot T = 0.2 \cdot T + 96 + 0.4 \cdot (0.6 \cdot T - 96).

Simplify:
0.5⋅T=0.2⋅T+96+0.4⋅0.6⋅T−0.4⋅96.0.5 \cdot T = 0.2 \cdot T + 96 + 0.4 \cdot 0.6 \cdot T - 0.4 \cdot 96.

0.5⋅T=0.2⋅T+96+0.24⋅T−38.4.0.5 \cdot T = 0.2 \cdot T + 96 + 0.24 \cdot T - 38.4.

Step 6 : Combine terms

Combine terms:
0.5⋅T=0.44⋅T+57.6.0.5 \cdot T = 0.44 \cdot T + 57.6.

Simplify further:
0.5⋅T−0.44⋅T=57.6.0.5 \cdot T - 0.44 \cdot T = 57.6.

0.06⋅T=57.6.0.06 \cdot T = 57.6.

Solve for TT:
T=57.60.06=960.T = \frac{57.6}{0.06} = 960.

Final Answer
The smallest possible total number of fruits in the stock at the beginning of the day is: 960.\boxed{960}.