Problem 68 - Introductory and Intermediate Algebra for College Students 4th

Question

Simplify each series of additions and subtractions. $$-\frac{1}{2}-\frac{2}{3}-\left(-\frac{1}{3}\right)$$

Step-by-Step Solution

Answer

The simplified form of the series is \( -\frac{5}{6} \).

Step 1: Simplify the Negatives

Handle the \( -(-\frac{1}{3}) \) part first. The minus of a negative number is a positive number, so \( - (-\frac{1}{3}) \) equals \( \frac{1}{3} \). Therefore, the equation becomes \( -\frac{1}{2} -\frac{2}{3} +\frac{1}{3} \).

Step 2: Find the Common Denominator

Before any addition or subtraction of fractions, they need to have the same denominator. The denominators here are 2 and 3. The least common multiple of 2 and 3 is 6. Therefore, convert each of the fractions to equivalents with denominators of 6. This gives \( -\frac{3}{6} -\frac{4}{6} +\frac{2}{6} \).

Step 3: Add and Subtract the Fractions

Now, add and subtract the fractions: \( -\frac{3}{6} -\frac{4}{6} +\frac{2}{6} = -\frac{5}{6} \). The series simplifies to \( -\frac{5}{6}. \)

Key Concepts

Negative Numbers

Dealing with negative numbers can be a bit tricky, but with some simple rules, they become easy to manage. A negative number is usually represented with a minus sign (-). When you see two minus signs together, like in \(-(-\frac{1}{3})\), it turns into a positive. This is because 'a minus of a minus equals a plus'. So, \(-(-x) = x\). Handling negative numbers correctly is essential for solving problems effectively and making sure arithmetic operations are carried out smoothly.

Common Denominator

When you add or subtract fractions, it's crucial to have a common denominator. This makes the denominators in the fractions the same so they can be easily added or subtracted. Think of a common denominator like a common language between fractions. Without it, you can't combine them properly. For example, if you have fractions with denominators of 2 and 3, find a number that both 2 and 3 can divide into without leaving a remainder. This involves finding the least common multiple, which in this case is 6. Once you have a common denominator, you can proceed with adding or subtracting the fractions.

Fraction Addition

Adding fractions is straightforward once they share a common denominator. You simply add the numerators and keep the common denominator. In our example, after converting the fractions to have a denominator of 6, you got \(-\frac{3}{6}, -\frac{4}{6},\) and \(+\frac{2}{6}.\) When adding and subtracting these, line them up and do the arithmetic operation by operation:

• Start with \(-\frac{3}{6} - \frac{4}{6} = -\frac{7}{6}\).
• Then, add \(+\frac{2}{6}\) to get the final sum \(-\frac{5}{6}.\)

Remember, if the fractions are negative, you're just adding negative numbers together, which means moving left on the number line.

Least Common Multiple

The least common multiple (LCM) is essential when dealing with fractions that have different denominators. The LCM of two numbers is the smallest number that is a multiple of both. In our fraction problem, we needed to combine fractions with denominators of 2 and 3.
Finding the LCM involves listing the multiples of each number:

• Multiples of 2 are 2, 4, 6, 8, 10, 12, ...
• Multiples of 3 are 3, 6, 9, 12, 15, ...

As you can see, 6 is the smallest number common to both lists. Thus, 6 is the LCM. This process allows us to convert each fraction so they share a common denominator and can be easily added or subtracted. This step is pivotal in performing many calculations involving fractions.