What is the perimeter of a rectangle with length [tex]$12 \frac{2}{3} m$[/tex] and breadth [tex]$7 \frac{1}{4} m$[/tex]?
Answer (2)
To find the perimeter of the rectangle, we first convert the mixed numbers for length and breadth into improper fractions. Then, we use the formula for the perimeter of a rectangle, which yields a final answer of 6 239 meters or 39 6 5 meters. This step-by-step process involves understanding fractions and the perimeter formula.
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Convert mixed fractions of length and breadth to improper fractions: l = 3 38 and b = 4 29 .
Use the perimeter formula for a rectangle: P = 2 ( l + b ) .
Find a common denominator and add the fractions: P = 2 ( 12 152 + 12 87 ) = 2 ( 12 239 ) .
Simplify the expression to find the perimeter: P = 6 239 meters, which is 39 6 5 meters. 6 239
Explanation
Problem Analysis Let's first analyze the problem. We are given the length and breadth of a rectangle as mixed fractions, and we need to find the perimeter of the rectangle. The formula for the perimeter of a rectangle is P = 2 ( l + b ) , where l is the length and b is the breadth.
Convert to Improper Fractions Convert the mixed fractions to improper fractions. Length, l = 12 3 2 = 3 12 × 3 + 2 = 3 36 + 2 = 3 38 meters. Breadth, b = 7 4 1 = 4 7 × 4 + 1 = 4 28 + 1 = 4 29 meters.
Substitute and Simplify Now, substitute the values of l and b into the formula for the perimeter: P = 2 ( l + b ) = 2 ( 3 38 + 4 29 ) To add the fractions, we need a common denominator, which is the least common multiple (LCM) of 3 and 4. The LCM of 3 and 4 is 12. So, we rewrite the fractions with the common denominator: 3 38 = 3 × 4 38 × 4 = 12 152 4 29 = 4 × 3 29 × 3 = 12 87 Now, add the fractions: 12 152 + 12 87 = 12 152 + 87 = 12 239 Substitute this back into the perimeter formula: P = 2 ( 12 239 ) = 12 2 × 239 = 12 478
Final Answer Simplify the fraction: 12 478 = 6 239 meters. Now, convert the improper fraction to a mixed fraction: 6 239 = 39 6 5 meters.
Conclusion Therefore, the perimeter of the rectangle is 6 239 meters or 39 6 5 meters.
Examples
Understanding perimeters is crucial in many real-world applications. For instance, when fencing a garden, knowing the perimeter helps determine the amount of fencing material needed. Similarly, in construction, calculating the perimeter of a room is essential for estimating the amount of baseboard or trim required. These calculations ensure accurate material procurement and cost estimation, preventing waste and saving resources.
