A person ordered 4 shirts of brand A and some shirts of brand B. The price of one shirt of brand A was twice that of brand B. When the order was executed, it was found that the number of the two brands has been interchanged. This increased the bill by 40%. The ratio of the number of brand A shirts to that of brand B shirts in the original order was

Answer

To solve the problem step by step, let's define the variables and analyze the situation:

### Step 1: Define Variables
Let:
- The price of one shirt of brand B = \( x \)
- The price of one shirt of brand A = \( 2x \) (since it is twice that of brand B)
- The number of shirts ordered of brand B = \( y \)

### Step 2: Calculate the Original Bill
The original order consists of:
- 4 shirts of brand A
- \( y \) shirts of brand B

The total cost of the original order can be calculated as:
\[ \text{Original Bill} = (4 \times 2x) + (y \times x) = 8x + yx = (8 + y)x \]

### Step 3: Calculate the Bill After Interchange
After the order is executed, the number of shirts has been interchanged. Thus, the order now consists of:
- \( y \) shirts of brand A
- 4 shirts of brand B

The total cost of the new order is:
\[ \text{New Bill} = (y \times 2x) + (4 \times x) = 2yx + 4x = (2y + 4)x \]

### Step 4: Set Up the Equation for the Increase in Bill
According to the problem, the new bill is 40% more than the original bill:
\[ \text{New Bill} = \text{Original Bill} + 0.4 \times \text{Original Bill} \]
This can be expressed as:
\[ (2y + 4)x = (1 + 0.4)(8 + y)x \]
Simplifying this gives:
\[ (2y + 4)x = 1.4(8 + y)x \]

### Step 5: Cancel \( x \) and Solve for \( y \)
Assuming \( x \neq 0 \), we can cancel \( x \) from both sides:
\[ 2y + 4 = 1.4(8 + y) \]
Expanding the right side:
\[ 2y + 4 = 11.2 + 1.4y \]
Rearranging gives:
\[ 2y - 1.4y = 11.2 - 4 \]
\[ 0.6y = 7.2 \]
Dividing both sides by 0.6:
\[ y = \frac{7.2}{0.6} = 12 \]

### Step 6: Find the Ratio of Shirts
Now we have:
- Number of shirts of brand A = 4
- Number of shirts of brand B = \( y = 12 \)

The ratio of the number of brand A shirts to that of brand B shirts in the original order is:
\[ \text{Ratio} = \frac{4}{12} = \frac{1}{3} \]

### Final Answer
The ratio of the number of brand A shirts to that of brand B shirts in the original order is \( \frac{1}{3} \).

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