Problem 31 - Elementary Algebra

Question

For the following problems, show that the fractions are equivalent. $$ \frac{-3}{5} \text { and }-\frac{3}{5} $$

Step-by-Step Solution

Answer

Answer: Yes, the fractions are equivalent.

Step 1: Check if the fractions has the same denominator

In this case, both fractions have the same denominator of 5. So, we don't need to find a common denominator.

Step 2: Simplify both fractions if necessary

Both fractions are already in their simplest form. So, we can directly compare their numerators.

Step 3: Compare the numerators of both fractions

The numerators of both fractions are -3. Thus, the fractions are equivalent. To sum it up, we have shown that the given fractions are equivalent by comparing their simplified forms. The fractions $$ \frac{-3}{5} \text { and } -\frac{3}{5} $$ are indeed equivalent.

Key Concepts

Simplifying Fractions

Simplifying fractions is like tidying up a room. We try to reduce the numbers within the fraction to their smallest possible values while maintaining the same ratio. This means that the fraction is in its simplest form. To simplify, you need to find the greatest common divisor (GCD) of the numerator and the denominator.

• The GCD is the largest number that can divide both the numerator and the denominator without leaving any leftover.
• Once you find it, divide both the numerator and the denominator by this number to simplify the fraction.

When you simplify a fraction, you make it easier to understand and calculate. In our exercise, there was no need to simplify because
\( \frac{-3}{5} \) and \( -\frac{3}{5} \) are already in their simplest forms. Fractions in simplest form are easier to read and work with, as they are reduced to the lowest possible terms without changing their value.

Common Denominator

A common denominator is a shared multiple of two or more denominators. It is crucial when you need to perform operations such as addition or subtraction with fractions. Having a common denominator allows you to compare fractions with different denominators by rewriting them with the same denominator. However, in the case of checking if two fractions are equivalent, if the denominators are already the same, like in our exercise with \( \frac{-3}{5} \) and \( -\frac{3}{5} \), you do not need to find another.

• If fractions have the same denominator, you only need to compare the numerators to determine equivalency.
• This makes the process straightforward and direct.

For fractions like our example, the denominator is already common, so there's no extra step. It simplifies your task and makes comparing fractions a breeze.

Numerators

The numerator is the number above the line in a fraction. It tells you how many parts of the whole you have. When comparing fractions, the numerator plays a vital role. If fractions have the same denominator, as seen with \( \frac{-3}{5} \) and \( -\frac{3}{5} \), checking the numerators will show if the fractions are equivalent or not.

• The numerator is crucial in determining the size of the fraction.
• When two fractions have the same denominator, simply look at the numerators to tell the difference.

In our exercise, both numerators are -3. This equality confirms they are equivalent fractions. Understanding numerators helps you quickly solve problems like this and reinforces basic fraction concepts.