Problem 32 - Elementary Algebra

Question

For the following problems, show that the fractions are equivalent. $$ \frac{-2}{7} \text { and }-\frac{2}{7} $$

Step-by-Step Solution

Answer

Answer: Yes, the fractions are equivalent.

Step 1: Understand the fractions

Look at the fractions to see if there's anything special about them. Notice that the only difference between them is that one has the negative sign in the numerator while the other has the negative sign in front of the entire fraction. $$ \frac{-2}{7} \text{ and } -\frac{2}{7} $$

Step 2: Find the decimal equivalent of both fractions

Let's find the decimal representation of both fractions by dividing the numerators by their denominators. This will help us compare if they are indeed equal. $$ \frac{-2}{7} = -0.2857 \text{ (approx.)}\\ -\frac{2}{7} = -0.2857 \text{ (approx.)} $$ Since both fractions have the same decimal representation, we can say that they are equivalent.

Step 3: Cross-multiply (Alternative way)

Alternatively, we could also cross-multiply the fractions to see if the products are the same. $$ (-2)\times 7 = (-7)\times 2 $$ Which gives us: $$ -14 = -14 $$ Since the cross-multiplication products are equal, this also confirms the fractions are equivalent.

Key Concepts

Negative Fractions

When dealing with fractions, you might sometimes encounter negative signs. Negative fractions can be represented in different forms but still hold the same value. In the example given: \( \frac{-2}{7} \) and \( -\frac{2}{7} \), both fractions are negative and equivalent. Here's why:

• If the negative sign is with the numerator, the fraction appears as a negative number. For instance, \( \frac{-2}{7} \) represents \(-2\) divided by \(7\).
• If the negative sign is placed in front of the fraction, like \( -\frac{2}{7} \), it impacts the fraction as a whole. Therefore, it represents the entire fraction as negative.

So, whether it's the numerator or the whole fraction being negative, both signify the same thing: a negative value. Thus, \( \frac{-2}{7} \) and \( -\frac{2}{7} \) are truly equivalent.

Decimal Representation

Converting fractions into decimals is a useful skill that helps in comparing or understanding the fraction's value in a different form. To convert a fraction into its decimal representation, divide the numerator by the denominator.
If we convert \( \frac{-2}{7} \) and \( -\frac{2}{7} \) into decimal form, we perform the division \(-2 \div 7\), which results in \(-0.2857\) (approximately).

• This decimal value is the same for both fractions, as they represent the same point on the number line.
• By showing that both fractions equal \(-0.2857\), it confirms their equivalency despite their visual difference.

Converting fractions into decimals is a common method to verify equivalence as it provides a straightforward approach to see their numeric equality.

Cross Multiplication

Cross multiplication is an alternative method for comparing fractions and determining equivalency. It involves multiplying the numerator of one fraction by the denominator of the other fraction, and vice versa. This method can be particularly useful for avoiding direct division.
In the given example, we apply cross multiplication to \( \frac{-2}{7} \) and \( -\frac{2}{7} \):

• Multiply the numerator of the first fraction with the denominator of the second: \((-2) \times 7 = -14\).
• Multiply the numerator of the second fraction with the denominator of the first: \((-2) \times 7 = -14\).

Since both results are equal (\(-14 = -14\)), it confirms that the fractions are indeed equivalent. Cross multiplication is a powerful tool to quickly check equivalence without decimal conversion.