The length and breadth of a rectangular field are in the ratio 8 : 7. If the…

For any rectangle the perimeter is the total boundary length, given by P = 2(l + b); rearranged, this means the sum of one length and one breadth is always exactly half the perimeter. Separately, when two quantities are known only through a fixed ratio a : b, they can be written as ax and bx for a common multiplier x, so the pair collapses to a single unknown that one equation can determine.

Applying this to the given field:

1. Let the length be 8x m and the breadth be 7x m, so that 8x : 7x reduces to 8 : 7 for every value of x.

2. Substitute into the perimeter formula: P = 2(8x + 7x) = 2(15x) = 30x.

3. The perimeter is given as 90 m, so 30x = 90.

4. Divide both sides by 30 to get x = 3, so one ratio part equals 3 m.

5. The length is 8x = 8 × 3 = 24 m.

Cross-check: with x = 3 the breadth is 7 × 3 = 21 m, and 2(24 + 21) = 2 × 45 = 90 m, which matches the given perimeter; dividing 24 : 21 by 3 gives 8 : 7, which matches the given ratio. Shortcut for the exam: half the perimeter is 45 m and the ratio parts total 8 + 7 = 15, so one part is 45 ÷ 15 = 3 m and the length is 8 parts.