The area of a field in the shape of a trapezium is 1440 square metres. The perpendicular distance between the two parallel sides is 24 metres, and the ratio of the lengths of the parallel sides is 5 : 3. What is the length of the longer parallel side in metres?

Introduction:
This question is about the area of a trapezium, also known as a trapezoid in some curricula. We are given the total area, the height (perpendicular distance between the parallel sides), and the ratio in which the two parallel sides are related. From this information, we need to find the length of the longer parallel side. The problem checks whether students can correctly use the area formula for a trapezium and work with ratios and simple algebra.

Given Data / Assumptions:

• Area of the trapezium = 1440 sq m. • Perpendicular distance between the parallel sides (height) = 24 m. • Ratio of the lengths of the two parallel sides = 5 : 3. • Let the parallel sides be 5x and 3x metres. • We are asked to find the length of the longer parallel side, that is 5x.

Concept / Approach:
The area A of a trapezium with parallel sides a and b and height h is given by A = (1/2) * (a + b) * h. Here, a and b are the lengths of the two parallel sides. Using the given ratio, we can express the sides as 5x and 3x. Then we substitute A = 1440 and h = 24 into the formula and solve for x. Once x is known, the longer side 5x can be found by simple multiplication.

Step-by-Step Solution:
Step 1: Represent the parallel sides using the ratio. Let the longer side be 5x and the shorter side be 3x. Step 2: Write the area formula for a trapezium. Area = (1/2) * (sum of parallel sides) * height. So, 1440 = (1/2) * (5x + 3x) * 24. Step 3: Simplify inside the brackets. 5x + 3x = 8x. Step 4: Substitute back. 1440 = (1/2) * 8x * 24. Step 5: Simplify the right-hand side. (1/2) * 8x = 4x. So, 1440 = 4x * 24 = 96x. Step 6: Solve for x. x = 1440 / 96 = 15. Step 7: Compute the longer parallel side. Longer side = 5x = 5 * 15 = 75 m.

Verification / Alternative check:
We can verify by computing the shorter side and checking if the original area is obtained. Shorter side = 3x = 3 * 15 = 45 m. Then sum of parallel sides = 75 + 45 = 120 m. Using the formula: Area = (1/2) * 120 * 24 = 60 * 24 = 1440 sq m, which matches the given area. This confirms that the value x = 15 and longer side = 75 m are correct.

Why Other Options Are Wrong:
Option 85 m: If the longer side were 85 m, with the same ratio, the shorter side would be 51 m, which would not produce an area of exactly 1440 sq m with height 24 m. Option 95 m and 105 m: These values would require correspondingly larger shorter sides based on the 5 : 3 ratio, resulting in areas much greater than 1440 sq m. Option 65 m: This would be too small compared to the required side lengths that satisfy the area and height conditions.

Common Pitfalls:
A typical mistake is to forget to apply the factor (1/2) in the trapezium area formula, leading to an incorrect equation and a wrong value for x. Many students also mis-handle the ratio, for example using 5x and x instead of 5x and 3x. Another common error is in arithmetic while solving 1440 = 96x, especially in division. Writing out each step carefully reduces the chance of error.

Final Answer:
The length of the longer parallel side of the trapezium is 75 metres.