Question

A friend tells you that \(-\frac{a}{b}=\frac{-a}{b}=\frac{a}{-b} .\) Is your friend correct? Use examples or counterexamples to support your answer.

Short answer

Yes, your friend is correct. It doesn't matter whether the negative sign is applied to the whole fraction, to the numerator only, or to the denominator only, the overall value of the fraction remains the same.

Step-by-step solution

Analyze the given statement

In the equation \(-\frac{a}{b}=\frac{-a}{b}=\frac{a}{-b},\) a negative fraction can either be related to the numerator being negative, the denominator being negative, or the whole fraction itself being negative.

Understand the properties of negative fractions

If a negative sign is applied to the numerator, the entire fraction becomes negative. Similarly, if the negative sign is applied to the denominator only, the entire fraction also becomes negative. This leads to the equation \(-\frac{a}{b}=\frac{-a}{b}=\frac{a}{-b}\.

Provide examples

Let's take \(a = 2\) and \(b = 3\). So, \(-\frac{2}{3} = -0.67, \frac{-2}{3} = -0.67,\) and \(\frac{2}{-3} = -0.67\). Hence, they are all equal.

Apply these understanding to the given scenario

The examples show that whether the fraction is negative, the numerator is negative, or the denominator is negative, the overall value of the fraction remains the same. Therefore, the friend's statement in the exercise is indeed correct.