Context: In Fig. 8.3, the length and breadth of the shaded rectangle are unit and unit, and its area is square units. Q. Do you see any relation between the area and the product of length and breadth?

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Question 1

Context: In Fig. 8.3, the length and breadth of the shaded rectangle are $\dfrac{1}{2}$ unit and $\dfrac{1}{4}$ unit, and its area is $\dfrac{1}{8}$ square units.

Q. Do you see any relation between the area and the product of length and breadth?

Solution

Understand the Question

The area of a rectangle is found by multiplying its length and breadth: $\text{Area} = \text{Length} \times \text{Breadth}$ .
Here, we are given a rectangle with length $\dfrac{1}{2}\text{ unit}$ , breadth $\dfrac{1}{4}\text{ unit}$ , and area $\dfrac{1}{8}\text{ square units}$ .
We multiply the length and breadth to verify whether their product equals the given area.

Step 1 · Calculate the Product of Length and Breadth

Diagram 1

Given dimensions: $\text{Length } (L) = \dfrac{1}{2} \text{ unit}$ $\text{Breadth } (B) = \dfrac{1}{4} \text{ unit}$ $\text{Area } (A) = \dfrac{1}{8} \text{ square units}$

Multiplying length and breadth:

\begin{aligned} L \times B &= \dfrac{1}{2} \times \dfrac{1}{4} \\[0.6em] &= \dfrac{1 \times 1}{2 \times 4} \\[0.6em] &= \dfrac{1}{8} \text{ square units} \end{aligned}

Step 2 · Compare Product with the Given Area

Comparing the calculated product with the given area: $\text{Product of length and breadth} = \dfrac{1}{8} \text{ square units}$ $\text{Given area} = \dfrac{1}{8} \text{ square units}$

Since both values are equal: $\text{Area} = \text{Length} \times \text{Breadth}$

Answer

Yes, the area is equal to the product of length and breadth ( $\text{Area} = \text{Length} \times \text{Breadth}$ ).

Common Mistakes

Fraction Multiplication Error: Adding the denominators instead of multiplying them (e.g., writing $\dfrac{1}{2} \times \dfrac{1}{4} = \dfrac{1}{6}$ instead of $\dfrac{1}{8}$ ).
Units Confusion: Forgetting that multiplying linear units ( $\text{unit} \times \text{unit}$ ) results in $\text{square units}$ for area.

Question 1

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