Animated Solution for Physics - Physics and Measurement: The area of a square is $5.29{\text{ cm}}^2$. The area of $7$ such squares taking into account the significant figures is

Welcome to a classic JEE trap! At first glance, this problem looks like a trivial multiplication exercise. You are given the area of a single square, $A=5.29{\text{ cm}}^2$, and you are asked to find the total area of $7$ such squares.

Most students will immediately grab their pens, multiply $5.29$ by $7$, and then confidently apply the multiplication rule for significant figures. But physics is a subject of profound subtleties. Let's dive into why that intuitive approach is fundamentally flawed and how to navigate this conceptual minefield.

Multiplication vs. Addition

In the realm of significant figures, there is a massive philosophical difference between a measured value and an exact number.

The area $5.29{\text{ cm}}^2$ is a measured value. It has an inherent uncertainty. It possesses exactly $3$ significant figures and $2$ decimal places.

However, the number $7$ is an exact number. It is a pure count. You are not measuring $7$ squares with a ruler; you are simply counting them. Exact numbers have infinite significant figures. They do not limit the precision of your calculation.

Because $7$ is an exact count, the operation we are performing is not the multiplication of two uncertain quantities. Instead, it is the repeated addition of the same uncertain quantity:

When we perform this raw addition, we get:

Now, we must apply the rule for significant figures. But which rule? Since the fundamental operation is addition, we must strictly use the Addition Rule.

The Addition Rule states: When adding or subtracting measured quantities, the final result must have the same number of decimal places as the term with the least number of decimal places.

Let's look at our terms. Every single term in our sum is $5.29$, which has exactly $2$ decimal places. Therefore, our final answer must also be restricted to exactly $2$ decimal places.

Our raw sum is $37.03$. How many decimal places does it have? Exactly $2$.

This means our raw sum already perfectly obeys the addition rule for significant figures! No rounding or truncation is necessary.

Final Answer: $37.03{\text{ cm}}^2$

What if we had fallen for the trap? If you had treated this as a multiplication of two measured values, you would have looked at the total significant figures. $5.29$ has $3$ significant figures. You would have rounded $37.03$ to $3$ significant figures, giving you $37.0{\text{ cm}}^2$. This is option (b), a perfectly placed distractor waiting to catch you off guard.

Always remember: When dealing with integer counts or defined constants, treat them as exact numbers and rely on the addition rule if it represents repeated summation. Stay sharp!