Problem 18 - Prealgebra

Question

Find the multiplicative inverse of each number. $$-\frac{1}{5}$$

Step-by-Step Solution

Answer

The multiplicative inverse of \(-\frac{1}{5}\) is \(-5\).

Step 1: Understand the Multiplicative Inverse

The multiplicative inverse of a number is another number that, when multiplied with the original number, results in 1. For the fraction \(-\frac{1}{5}\), the goal is to find a value that makes the product equal to 1.

Step 2: Define the Inverse for a Negative Fraction

For a fraction, its multiplicative inverse is simply obtained by swapping its numerator and denominator, even if it's negative. Therefore, the multiplicative inverse of \(-\frac{1}{5}\) is \(-5\), since multiplying these gives 1.

Step 3: Verify the Result

Multiply \(-\frac{1}{5}\) by \(-5\) to ensure the result is 1. Doing the multiplication: \(-\frac{1}{5} \times -5 = \frac{1 \times 5}{5} = \frac{5}{5} = 1\). The result confirms that \(-5\) is the multiplicative inverse.

Key Concepts

Negative Fractions

Negative fractions are essentially fractions that have a negative sign in front of them, indicating that the value of the fraction is less than zero. They can be a little confusing at first, but let's break it down.
In a negative fraction, the negative sign can be placed in front of the entire fraction, or it can be attached to either the numerator or the denominator.

• For example, \(-\frac{1}{5}\) is the same as \(\frac{-1}{5}\) and \(\frac{1}{-5}\).

The value doesn't change because multiplication and division apply the same rules to negative numbers.
This means that when working with negative fractions, you are dealing with negative multiplication and division.
In these cases, ensure to apply the rules of multiplying and dividing negative numbers correctly — this is the key to avoiding mistakes.

Numerator and Denominator

In any fraction, the numerator and denominator play integral roles. The fraction \(-\frac{1}{5}\) consists of two components:

• The **numerator**: This is the top number in the fraction, which for \-\frac{1}{5}\ is \-1\. It indicates how many parts we have.
• The **denominator**: This is the bottom number in the fraction, \5\ in our example, showing into how many parts the whole is divided.

To find the multiplicative inverse—or reciprocal—of a fraction, you swap the positions of the numerator and denominator. For \(-\frac{1}{5}\), this means our inverse becomes \(-5\) because the same fraction as \[\frac{-5}{1}\] simplifies to \-5\.
Remember, swapping the numerator and denominator always yields the fraction's reciprocal, which is crucial when finding the multiplicative inverse.

Verification of Result

Verifying the result of a calculation ensures that you've arrived at the correct answer. To confirm that \-5\ is the correct multiplicative inverse of \(-\frac{1}{5}\), you can multiply them together.
Conducting the multiplication looks like this:

• Multiply the numerators: \-1 \times (-5) = 5\.
• Multiply the denominators: \5 \times 1 = 5\.

So the fraction becomes \frac{5}{5}\, which simplifies to 1.
This confirms that the multiplication of \(-\frac{1}{5}\) by \-5\ indeed results in 1, verifying that \-5\ is the correct multiplicative inverse.
Checking your work like this is a good habit in math, ensuring precision and understanding.