Let's denote the width of the rectangle as \( w \) cm. According to the problem, the length \( l \) is given as:
\[ l = w + 4 \]
If the length is decreased by 5 cm and the width is decreased by 2 cm, the new dimensions would be:
\[ \text{New Length} = l - 5 = (w + 4) - 5 = w - 1 \] \[ \text{New Width} = w - 2 \]
The perimeter \( P \) of a rectangle is given by the formula:
\[ P = 2(\text{Length} + \text{Width}) \]
According to the problem, if we decrease the dimensions as stated, the perimeter becomes 10 cm. Therefore, we can set up the equation:
\[ 2((w - 1) + (w - 2)) = 10 \]
Now, let's simplify the equation:
\[ 2(2w - 3) = 10 \]
Dividing both sides by 2 gives:
\[ 2w - 3 = 5 \]
Adding 3 to both sides results in:
\[ 2w = 8 \]
Dividing both sides by 2 gives:
\[ w = 4 \]
Now that we have the width, we can find the length:
\[ l = w + 4 = 4 + 4 = 8 \]
Thus, the dimensions of the original rectangle are:
\[ \text{Width} = 4 \text{ cm}, \quad \text{Length} = 8 \text{ cm} \]
To verify, let's check the new dimensions after the decreases:
- New Length: \( 8 - 5 = 3 \) cm
- New Width: \( 4 - 2 = 2 \) cm
Calculating the perimeter with these new dimensions:
\[ P = 2(3 + 2) = 2 \times 5 = 10 \text{ cm} \]
This confirms our solution is correct.
Final answer: The dimensions of the original rectangle are 4 cm (width) and 8 cm (length).
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