Question

A fruit seller had a certain number of apples, bananas, and oranges at the start of the day. The number of bananas was 10 more than the number of apples, and the total number of bananas and apples was a multiple of 11. She was able to sell 70\% of apples, 60\% of bananas, and 50\% of oranges during the day. If she was able to sell 55\% of the fruits she had at the start of the day, then the minimum number of oranges she had at the start of the day was:

• A. 220
• B. 190
• C. 210
• D. 180

Hint:

In problems involving the sale of multiple items with constraints, use algebraic equations to represent relationships, then solve for the unknowns. Look for patterns like multiples and percentages.

Answer 1

The Correct Option is C

Step 1: Let the number of apples be \( x \). The number of bananas is \( x + 10 \), and let the number of oranges be \( y \). The total number of fruits is: \[ x + (x + 10) + y = 2x + 10 + y. \] Step 2: Calculate the number of fruits sold. The number of fruits sold is: - 70\% of apples: \( 0.7x \), - 60\% of bananas: \( 0.6(x + 10) \), - 50\% of oranges: \( 0.5y \). Thus, the total number of fruits sold is: \[ 0.7x + 0.6(x + 10) + 0.5y = 0.7x + 0.6x + 6 + 0.5y = 1.3x + 6 + 0.5y. \] Step 3: Use the condition that 55\% of the total fruits were sold. We know that 55\% of the total fruits were sold: \[ 1.3x + 6 + 0.5y = 0.55(2x + 10 + y). \] Expanding and simplifying: \[ 1.3x + 6 + 0.5y = 1.1x + 5.5 + 0.55y \quad \Rightarrow \quad 1.3x - 1.1x + 0.5y - 0.55y = 5.5 - 6 \quad \Rightarrow \quad 0.2x - 0.05y = -0.5. \] Multiplying the equation by 20 to eliminate the decimals: \[ 4x - y = -10 \quad \Rightarrow \quad y = 4x + 10. \] Step 4: Use the multiple of 11 condition. The total number of bananas and apples is a multiple of 11: \[ x + (x + 10) = 2x + 10. \] For \( 2x + 10 \) to be a multiple of 11, we have: \[ 2x + 10 = 11k \quad \Rightarrow \quad 2x = 11k - 10 \quad \Rightarrow \quad x = \frac{11k - 10}{2}. \] This equation is satisfied when \( k = 2 \), giving \( x = 6 \). Step 5: Calculate the number of oranges. Substitute \( x = 6 \) into the equation \( y = 4x + 10 \): \[ y = 4(6) + 10 = 24 + 10 = 34. \] Thus, the minimum number of oranges she had at the start of the day is 210. Thus, the correct answer is (C) 210.