If the length of a rectangle is increased by 12% and the width is decreased by 10%, then the ratio of the areas of the original rectangle and the changed rectangle is:

Options

A. `125: 126`
B. `126:125 `
C. `1:2 `
D. `25:26 `

Answer

To solve the problem step by step, we need to find the ratio of the areas of the original rectangle and the changed rectangle after the specified changes to its dimensions.

### Step 1: Define the original dimensions
Let the original length of the rectangle be \( L \) and the original width be \( B \).

### Step 2: Calculate the area of the original rectangle
The area \( A \) of the original rectangle is given by:
\[ A = L \times B \]

### Step 3: Calculate the new dimensions
The length is increased by 12%, so the new length \( L' \) is:
\[ L' = L + 0.12L = 1.12L \]

The width is decreased by 10%, so the new width \( B' \) is:
\[ B' = B - 0.10B = 0.90B \]

### Step 4: Calculate the area of the changed rectangle
The area \( A' \) of the changed rectangle is given by:
\[ A' = L' \times B' = (1.12L) \times (0.90B) \]
\[ A' = 1.12 \times 0.90 \times L \times B \]
\[ A' = 1.008L \times B \]

### Step 5: Find the ratio of the areas
Now, we need to find the ratio of the original area \( A \) to the new area \( A' \):
\[ \text{Ratio} = \frac{A}{A'} = \frac{L \times B}{1.008L \times B} \]

### Step 6: Simplify the ratio
Cancel out \( L \) and \( B \) from the numerator and denominator:
\[ \text{Ratio} = \frac{1}{1.008} \]

### Step 7: Convert the ratio into a simpler form
To express this ratio in a more understandable form, we can multiply both the numerator and denominator by 1000:
\[ \text{Ratio} = \frac{1000}{1008} \]

### Step 8: Simplify further
Now, we can simplify \( \frac{1000}{1008} \):
\[ \text{Ratio} = \frac{125}{126} \]

### Final Answer
Thus, the ratio of the areas of the original rectangle to the changed rectangle is:
\[ \text{Ratio} = 125 : 126 \]