To solve the problem step by step, we will first determine the cost price (CP) and selling price (SP) for apples, oranges, and mangoes, and then find the net profit percentage on the sale of 1 apple, 2 oranges, and 2 mangoes.
### Step 1: Determine the Cost Price and Selling Price of Apples
- Given: The cost price of 16 apples is equal to the selling price of 10 apples.
- Let the cost price of 1 apple be \( CP_a \).
- Therefore, the cost price of 16 apples = \( 16 \times CP_a \).
- Let the selling price of 1 apple be \( SP_a \).
- Therefore, the selling price of 10 apples = \( 10 \times SP_a \).
From the problem, we have:
\[ 16 \times CP_a = 10 \times SP_a \]
Rearranging gives us:
\[ \frac{CP_a}{SP_a} = \frac{10}{16} = \frac{5}{8} \]
This means:
\[ SP_a = \frac{8}{5} CP_a \]
### Step 2: Calculate Profit Percentage for Apples
- Profit = Selling Price - Cost Price
\[ \text{Profit on 1 apple} = SP_a - CP_a = \frac{8}{5} CP_a - CP_a = \left(\frac{8}{5} - 1\right) CP_a = \frac{3}{5} CP_a \]
- Profit Percentage:
\[ \text{Profit Percentage} = \left(\frac{\text{Profit}}{CP_a}\right) \times 100 = \left(\frac{\frac{3}{5} CP_a}{CP_a}\right) \times 100 = 60\% \]
### Step 3: Determine the Cost Price and Selling Price of Oranges
- Given: The cost price of 12 oranges is equal to the selling price of 16 oranges.
- Let the cost price of 1 orange be \( CP_o \).
- Therefore, the cost price of 12 oranges = \( 12 \times CP_o \).
- Let the selling price of 1 orange be \( SP_o \).
- Therefore, the selling price of 16 oranges = \( 16 \times SP_o \).
From the problem, we have:
\[ 12 \times CP_o = 16 \times SP_o \]
Rearranging gives us:
\[ \frac{CP_o}{SP_o} = \frac{16}{12} = \frac{4}{3} \]
This means:
\[ SP_o = \frac{3}{4} CP_o \]
### Step 4: Calculate Loss Percentage for Oranges
- Loss = Cost Price - Selling Price
\[ \text{Loss on 1 orange} = CP_o - SP_o = CP_o - \frac{3}{4} CP_o = \frac{1}{4} CP_o \]
- Loss Percentage:
\[ \text{Loss Percentage} = \left(\frac{\text{Loss}}{CP_o}\right) \times 100 = \left(\frac{\frac{1}{4} CP_o}{CP_o}\right) \times 100 = 25\% \]
### Step 5: Determine the Cost Price and Selling Price of Mangoes
- Given: The cost price of 6 mangoes is equal to the selling price of 4 mangoes.
- Let the cost price of 1 mango be \( CP_m \).
- Therefore, the cost price of 6 mangoes = \( 6 \times CP_m \).
- Let the selling price of 1 mango be \( SP_m \).
- Therefore, the selling price of 4 mangoes = \( 4 \times SP_m \).
From the problem, we have:
\[ 6 \times CP_m = 4 \times SP_m \]
Rearranging gives us:
\[ \frac{CP_m}{SP_m} = \frac{4}{6} = \frac{2}{3} \]
This means:
\[ SP_m = \frac{3}{2} CP_m \]
### Step 6: Calculate Profit Percentage for Mangoes
- Profit = Selling Price - Cost Price
\[ \text{Profit on 1 mango} = SP_m - CP_m = \frac{3}{2} CP_m - CP_m = \left(\frac{3}{2} - 1\right) CP_m = \frac{1}{2} CP_m \]
- Profit Percentage:
\[ \text{Profit Percentage} = \left(\frac{\text{Profit}}{CP_m}\right) \times 100 = \left(\frac{\frac{1}{2} CP_m}{CP_m}\right) \times 100 = 50\% \]
### Step 7: Calculate the Cost Price of 1 Apple, 2 Oranges, and 2 Mangoes
- Given the ratio of the cost price of 1 apple, 1 orange, and 1 mango is 1:1:2.
- Let \( CP_a = x \), \( CP_o = x \), and \( CP_m = 2x \).
Total cost price:
\[ CP_{\text{total}} = CP_a + 2 \times CP_o + 2 \times CP_m = x + 2x + 4x = 7x \]
### Step 8: Calculate Selling Price of 1 Apple, 2 Oranges, and 2 Mangoes
- Selling price of 1 apple:
\[ SP_a = \frac{8}{5} x \]
- Selling price of 2 oranges:
\[ SP_o = \frac{3}{4} x \]
\[ SP_{\text{oranges}} = 2 \times SP_o = 2 \times \frac{3}{4} x = \frac{3}{2} x \]
- Selling price of 2 mangoes:
\[ SP_m = \frac{3}{2} \times 2x = 3x \]
Total selling price:
\[ SP_{\text{total}} = SP_a + SP_{\text{oranges}} + SP_{\text{mangoes}} = \frac{8}{5} x + \frac{3}{2} x + 3x \]
### Step 9: Find a Common Denominator and Calculate Total Selling Price
- The common denominator for \( \frac{8}{5} \), \( \frac{3}{2} \), and \( 3 \) is 10.
- Convert each term:
\[ SP_a = \frac{16}{10} x \]
\[ SP_{\text{oranges}} = \frac{15}{10} x \]
\[ SP_{\text{mangoes}} = \frac{30}{10} x \]
Total selling price:
\[ SP_{\text{total}} = \frac{16}{10} x + \frac{15}{10} x + \frac{30}{10} x = \frac{61}{10} x \]
### Step 10: Calculate Net Profit and Profit Percentage
- Net Profit:
\[ \text{Net Profit} = SP_{\text{total}} - CP_{\text{total}} = \frac{61}{10} x - 7x = \left(\frac{61}{10} - \frac{70}{10}\right) x = -\frac{9}{10} x \]
- Profit Percentage:
\[ \text{Profit Percentage} = \left(\frac{\text{Net Profit}}{CP_{\text{total}}}\right) \times 100 = \left(\frac{-\frac{9}{10} x}{7x}\right) \times 100 = -\frac{9}{70} \times 100 \approx -12.86\% \]
### Conclusion
The net profit percentage on the sale of 1 apple, 2 oranges, and 2 mangoes is approximately **-12.86%**, indicating a loss.
