Explain why in the Fundamental Law of Fractions, $n$ must be non-zero.

Choose the correct answer below.

ADivision by 0 is undefined.
BThen, when the numerator and denominator are multiplied by 0, the resulting fraction will always equal 1.
CIf the denominator is multiplied by 0, the identity is lost.
D$n$ must be non-zero because if it were $0$, $\frac{a}{b} = \frac{a \cdot 0}{b \cdot 0} = \frac{0}{0}$. Division by 0 is undefined.

Question

01-11-2026
Mathematics

Answered

Explain why in the Fundamental Law of Fractions, $n$ must be non-zero.

Choose the correct answer below.

ADivision by 0 is undefined.
BThen, when the numerator and denominator are multiplied by 0, the resulting fraction will always equal 1.
CIf the denominator is multiplied by 0, the identity is lost.
D$n$ must be non-zero because if it were $0$, $\frac{a}{b} = \frac{a \cdot 0}{b \cdot 0} = \frac{0}{0}$. Division by 0 is undefined.

Verified Answer :

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-01T21:29:45-05:00

The Fundamental Law of Fractions requires $n eq 0$ to ensure that the fraction remains well-defined.
If $n = 0$, the expression becomes $\frac{a \cdot 0}{b \cdot 0} = \frac{0}{0}$.
Division by zero is undefined, which makes the fraction meaningless in this case.
Option D provides the correct explanation for this rule.
Final Answer: $\boxed{D}$

Explanation

The Fundamental Law of Fractions

The Fundamental Law of Fractions states that for any fraction $\frac{a}{b}$ (where $b eq 0$) and any non-zero number $n$, the fraction is equivalent to $\frac{a \cdot n}{b \cdot n}$. This law is essential for simplifying, expanding, and comparing fractions by finding common denominators. It ensures that the ratio between the numerator and the denominator remains constant.

The Consequences of $n = 0$

If we were to allow the multiplier $n$ to be zero, we would evaluate the expression $\frac{a \cdot 0}{b \cdot 0}$. This simplifies to $\frac{0}{0}$. In mathematics, division by zero is undefined because there is no unique number that, when multiplied by $0$, results in any non-zero value, and the expression $\frac{0}{0}$ is considered indeterminate in most contexts. Consequently, if $n$ were zero, the resulting expression would lose its mathematical meaning and its connection to the original fraction $\frac{a}{b}$.

Evaluating the Options

Evaluating the provided options:

Option A is an incomplete statement.
Option B is an incomplete statement.
Option C incorrectly claims that $\frac{0}{0}$ equals 1.
Option D correctly explains that using $n=0$ leads to the expression $\frac{0}{0}$ and explicitly states that division by zero is undefined.

Thus, Option D is the correct choice.

Examples

Understanding why we cannot multiply by zero in fractions is similar to understanding why we cannot divide by zero in everyday life. Imagine you have a box of 10 chocolates that you want to share equally among your friends. If you share them among 5 friends, each gets 2 chocolates. If you try to share them among 0 friends, the concept of "sharing" simply doesn't exist because there is no one to receive the chocolates. In the same way, the mathematical operation of dividing by zero is undefined, and the fundamental rules of fractions, like the Fundamental Law of Fractions, are designed to respect this constraint to keep our calculations valid and consistent.