Introduction to the Problem

A rectangular piece of land has an area of (frac{1}{2}). One of its dimensions (either the length or the breadth) is given as (frac{7}{8}). The task is to find the other dimension.

Understanding the Properties of a Rectangle

A rectangle has two dimensions: length (l) and breadth (b). The area (A) of a rectangle is given by the formula:

[A l times b]

Given the area is (frac{1}{2}), we can write:

[l times b frac{1}{2}]

Finding the Other Dimension

If one dimension (let's say the length) is (frac{7}{8}), we can find the other dimension (breadth) by rearranging the formula:

[b frac{frac{1}{2}}{frac{7}{8}}]

To divide by a fraction, we multiply by its reciprocal. Therefore:

[b frac{1}{2} times frac{8}{7} frac{8}{14} frac{4}{7}]

Verifying the Solution

We can verify the solution by substituting the value back into the area formula:

[frac{7}{8} times frac{4}{7} frac{28}{56} frac{1}{2}]

This confirms that the area of the rectangle is indeed (frac{1}{2}).

Explanation of Multiplication by Reciprocal

Multiplying a fraction by its reciprocal ensures the product is 1. In this context, we need to find a fraction that, when multiplied with (frac{4}{7}), gives us the area (frac{1}{2}). The reciprocal of (frac{4}{7}) is (frac{7}{4}).

To get (frac{1}{2}) from (frac{4}{7}), we multiply by the reciprocal of (frac{4}{7}), which is (frac{7}{4}), but since we are trying to get half of the reciprocal, we directly use (frac{8}{7}) to reach the desired area.

Conclusion

The other dimension of the rectangular piece of land, when one dimension is (frac{7}{8}), is (frac{4}{7}).

This problem showcases the application of basic arithmetic operations on fractions and the concept of reciprocals in solving practical geometry problems.