A rectangle's length is 5 cm less than twice its width. If the length is decreased by 5 cm and width is increased by 2 cm, the perimeter of the resulting rectangular will be 74 cm. Find the length and the width of the original rectangle.

Answer

To solve the problem step by step, we will define the variables, set up equations based on the information given, and then solve for the length and width of the rectangle.

### Step 1: Define the Variables
Let:
- Width of the rectangle = \( W \) cm
- Length of the rectangle = \( L \) cm

### Step 2: Set Up the First Equation
According to the problem, the length is 5 cm less than twice the width. This can be expressed as:
\[ L = 2W - 5 \]

### Step 3: Set Up the Second Equation
The problem states that if the length is decreased by 5 cm and the width is increased by 2 cm, the perimeter of the resulting rectangle will be 74 cm. The perimeter \( P \) of a rectangle is given by the formula:
\[ P = 2(L + W) \]
Substituting the modified length and width into the perimeter formula gives us:
\[ 2((L - 5) + (W + 2)) = 74 \]

### Step 4: Simplify the Second Equation
Now, simplify the equation:
1. Substitute \( L - 5 \) and \( W + 2 \):
\[ 2((L - 5) + (W + 2)) = 74 \]
\[ 2(L + W - 3) = 74 \]
2. Divide both sides by 2:
\[ L + W - 3 = 37 \]
3. Rearranging gives us:
\[ L + W = 40 \]

### Step 5: Solve the System of Equations
Now we have two equations:
1. \( L = 2W - 5 \) (Equation 1)
2. \( L + W = 40 \) (Equation 2)

Substituting Equation 1 into Equation 2:
\[ (2W - 5) + W = 40 \]
Combine like terms:
\[ 3W - 5 = 40 \]
Adding 5 to both sides:
\[ 3W = 45 \]
Dividing by 3:
\[ W = 15 \]

### Step 6: Find the Length
Now that we have the width, we can find the length using Equation 1:
\[ L = 2W - 5 \]
Substituting \( W = 15 \):
\[ L = 2(15) - 5 \]
\[ L = 30 - 5 \]
\[ L = 25 \]

### Conclusion
The original dimensions of the rectangle are:
- Width = \( 15 \) cm
- Length = \( 25 \) cm