The perimeter of a rectangular region is 340 m. The length of this region is 10 m more than thrice its breadth. Find the length and the breadth of the rectangular region.

Answer

To solve the problem step by step, we will follow these instructions:

1. **Define Variables**:
- Let the breadth of the rectangular region be \( x \) meters.
- According to the problem, the length \( L \) is given as \( L = 3x + 10 \) meters.

2. **Write the Perimeter Formula**:
- The formula for the perimeter \( P \) of a rectangle is given by:
\[ P = 2 \times (L + B) \]
- Substituting the known values, we have:
\[ 340 = 2 \times (L + x) \]

3. **Substitute the Length in the Perimeter Equation**:
- Substitute \( L \) with \( 3x + 10 \):
\[ 340 = 2 \times ((3x + 10) + x) \]

4. **Simplify the Equation**:
- Combine like terms inside the parentheses:
\[ 340 = 2 \times (4x + 10) \]

5. **Divide Both Sides by 2**:
- To simplify, divide both sides by 2:
\[ 170 = 4x + 10 \]

6. **Isolate the Variable \( x \)**:
- Subtract 10 from both sides:
\[ 170 - 10 = 4x \]
\[ 160 = 4x \]

7. **Solve for \( x \)**:
- Divide both sides by 4:
\[ x = \frac{160}{4} = 40 \]
- So, the breadth \( B \) is \( 40 \) meters.

8. **Calculate the Length**:
- Now substitute \( x \) back into the length equation:
\[ L = 3x + 10 = 3(40) + 10 = 120 + 10 = 130 \]
- So, the length \( L \) is \( 130 \) meters.

9. **Final Answer**:
- The breadth of the rectangular region is \( 40 \) meters and the length is \( 130 \) meters.