The cost of an apple is twice that of a banana and the cost of a banana is 25% less than that of an orange. If the cost of each type of fruit increases by 10%, then the percentage increase in the cost of 4 bananas, 2 apples and 3 orange is

Answer

To solve the problem step by step, we will first define the costs of the fruits based on the given relationships, then calculate the total cost before and after the price increase, and finally determine the percentage increase.

### Step 1: Define the costs of the fruits
Let:
- The cost of a banana be \( B \).
- The cost of an orange be \( O \).
- The cost of an apple be \( A \).

From the problem, we know:
1. The cost of an apple is twice that of a banana:
\[ A = 2B \]
2. The cost of a banana is 25% less than that of an orange:
\[ B = O - 0.25O = 0.75O \]

### Step 2: Express costs in terms of one fruit
Let's express the costs in terms of the cost of the orange \( O \):
- From \( B = 0.75O \), we can substitute this into the equation for \( A \):
\[ A = 2B = 2(0.75O) = 1.5O \]

Now we have:
- \( A = 1.5O \)
- \( B = 0.75O \)
- \( O = O \)

### Step 3: Calculate the total cost of 4 bananas, 2 apples, and 3 oranges
The total cost before the price increase is:
\[ \text{Total Cost} = 4B + 2A + 3O \]
Substituting the values of \( A \) and \( B \):
\[ \text{Total Cost} = 4(0.75O) + 2(1.5O) + 3O \]
Calculating each term:
\[ = 3O + 3O + 3O = 9O \]

### Step 4: Calculate the new costs after a 10% increase
Each fruit's cost increases by 10%:
- New cost of an apple:
\[ A' = A + 0.1A = 1.1A = 1.1(1.5O) = 1.65O \]
- New cost of a banana:
\[ B' = B + 0.1B = 1.1B = 1.1(0.75O) = 0.825O \]
- New cost of an orange:
\[ O' = O + 0.1O = 1.1O \]

### Step 5: Calculate the total cost after the price increase
The total cost after the increase is:
\[ \text{New Total Cost} = 4B' + 2A' + 3O' \]
Substituting the new costs:
\[ = 4(0.825O) + 2(1.65O) + 3(1.1O) \]
Calculating each term:
\[ = 3.3O + 3.3O + 3.3O = 9.9O \]

### Step 6: Calculate the increase in total cost
The increase in total cost is:
\[ \text{Increase} = \text{New Total Cost} - \text{Old Total Cost} = 9.9O - 9O = 0.9O \]

### Step 7: Calculate the percentage increase
The percentage increase is given by:
\[ \text{Percentage Increase} = \left( \frac{\text{Increase}}{\text{Old Total Cost}} \right) \times 100 = \left( \frac{0.9O}{9O} \right) \times 100 = 10\% \]

### Final Answer
The percentage increase in the cost of 4 bananas, 2 apples, and 3 oranges is **10%**.