Problem 106 - Elementary Algebra

Question

\(\frac{2}{9}+\frac{2}{9} \div\left(-\frac{1}{5}\right)\)

Step-by-Step Solution

Answer

-\(\frac{8}{9}\)

Step 1 - Evaluate the Division First

According to the order of operations (PEMDAS/BODMAS), division should be performed before addition. Compute the division: \(\frac{2}{9} \div (-\frac{1}{5})\). Dividing by a fraction is the same as multiplying by its reciprocal. Therefore, \(\frac{2}{9} \div (-\frac{1}{5}) = \frac{2}{9} \times -\frac{5}{1} = -\frac{10}{9}\).

Step 2 - Add the Fractions

Now, add \(\frac{2}{9}\) and \(-\frac{10}{9}\). When adding fractions with the same denominator, add the numerators: \(\frac{2}{9} + -\frac{10}{9} = \frac{2 - 10}{9}= \frac{-8}{9}\).

Key Concepts

Understanding Fractions

Fractions represent a part of a whole. They consist of a numerator (top number) and a denominator (bottom number). For example, in the fraction \(\frac{2}{9}\), 2 is the numerator, and 9 is the denominator. The numerator tells us how many parts we have, while the denominator tells us how many equal parts the whole is divided into.
Fractions can be further classified into:

• Proper Fractions: Where the numerator is less than the denominator, like \(\frac{2}{9}\)
• Improper Fractions: Where the numerator is greater than or equal to the denominator, like \(\frac{10}{9}\)
• Mixed Numbers: A whole number combined with a proper fraction, like 1 \(\frac{1}{3}\)

Understanding these types will help you manipulate and perform operations with fractions accurately.

Performing Division with Fractions

Division with fractions might seem tricky at first, but it's simple once you get the hang of it. The core idea is to multiply by the reciprocal of the divisor. Let's break that down:

• If you have \(\frac{2}{9} \div -\frac{1}{5}\), you first find the reciprocal of -\(\frac{1}{5}\), which is -5 (since the reciprocal of a fraction \(\frac{a}{b}\) is \(\frac{b}{a}\)).
• This changes the expression to \(\frac{2}{9} \times -5\). Multiplication is straightforward: multiply the numerators and the denominators. Therefore, \(\frac{2}{9} \times -5\) = \(\frac{2 \times -5}{9} = -\frac{10}{9}\).

Note that when multiplying fractions, whether the fraction is positive or negative, the signs will follow the rules of multiplication. Positive multiplied by negative is negative.

Adding Fractions

When adding fractions, it’s important to have a common denominator. If the fractions already have the same denominator, just add or subtract the numerators and place the result over the common denominator. For example:

• In the given problem, after computing the division, we need to add \(\frac{2}{9}\) and \(\frac{-10}{9}\).
• Both fractions have the same denominator, so we can directly add their numerators: 2 + (-10) which equals -8.
• Therefore, \(\frac{2}{9} + \frac{-10}{9} = \frac{2 - 10}{9} = \frac{-8}{9}\).

The key point here is that the operation between the fractions remains on their numerators if the denominators are the same, making addition straightforward. For cases with different denominators, ensure to find a common denominator before performing the addition.