Problem 36 - Introductory and Intermediate Algebra for College Students 4th

Question

Find each sum without the use of a number line. $$-\frac{3}{8}+\left(-\frac{2}{3}\right)$$

Step-by-Step Solution

Answer

The sum of \(-\frac{3}{8}+\left(-\frac{2}{3}\right)\) equals to \(\frac{-25}{24}\).

Step 1: Re-write the operation with the addition of negative numbers as a subtraction

The operation \(-\frac{3}{8}-\frac{2}{3}\) can be understood as \(-\frac{3}{8}+\left(-\frac{2}{3}\right)\), where we are adding two negative numbers. This can be re-written as a subtraction problem: \(-\frac{3}{8}-\frac{2}{3}\). This makes it clearer that we are combining two quantities which are each subtracting from zero.

Step 2: Find a common denominator

Before subtracting fractions, it's necessary that they have the same denominator. The least common multiple of 8 and 3 is 24. So, the fractions need to be converted to have this common denominator: \(-\frac{3}{8} \times \frac{3}{3} = \frac{-9}{24}\), \(-\frac{2}{3} \times \frac{8}{8} = \frac{-16}{24}\).

Step 3: Conduct the subtraction

Now that the fractions have the same denominator, we can easily conduct the subtraction: \(\frac{-9}{24} - \frac{-16}{24} = \frac{-25}{24}\).

Key Concepts

Common Denominator

Understanding how to find a common denominator is crucial when tackling problems with fractions, especially when subtracting or adding them. In essence, the common denominator is a shared multiple of the denominators of two or more fractions. It allows us to compare, combine, and simplify fractions by ensuring they are talking about the same-sized parts.

For example, if we have \( -\frac{3}{8} \) and \( -\frac{2}{3} \) and want to subtract them, we need their denominators to be the same. To find a common denominator, we can list the multiples of each denominator:

• Multiples of 8: 8, 16, 24, 32, ...
• Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, ...

The smallest multiple they share is 24. Therefore, 24 becomes our common denominator, which we use to convert both fractions before subtracting them. This ensures accuracy in the final result and avoids the misconception that fractions with different denominators can be directly subtracted from one another.

Least Common Multiple

When we're working with fractions, the least common multiple (LCM) of two denominators is the smallest number into which both denominators can be divided evenly. It's essential for finding a common denominator but has uses far beyond just fraction operations, such as solving problems involving simultaneous events or when trying to determine periodical occurrences.

In our example, the denominators are 8 and 3, and we are looking for their LCM. It organically fits into the problem-solving process as follows:

• Write the multiples of each number.
• Identify the smallest multiple shared by both lists.

This method can be time-consuming for larger numbers, so alternative methods like prime factorization or using the greatest common divisor (GCD) to find the LCM are sometimes preferred. In any case, finding the LCM helps to transform fractions with different denominators into equivalent fractions with a common denominator, which is a stepping stone for any addition or subtraction operation involving fractions.

Adding and Subtracting Fractions

Adding and subtracting fractions can often intimidate students, but it's like putting together or taking apart pieces of the same sized pie. Before we add or subtract fractions, we must ensure the pieces are the same size, which means they need a common denominator. Here's how it works:

• Find the least common multiple of the denominators to get the common denominator.
• Adjust each fraction to have this common denominator by finding equivalent fractions.
• Add or subtract the numerators while keeping the common denominator fixed.
• Simplify the resulting fraction if necessary.

When working with negative fractions, keep in mind that subtracting a negative is equivalent to adding its positive counterpart. For example, \( -\frac{3}{8} - (-\frac{2}{3}) \) is equivalent to \( -\frac{3}{8} + \frac{2}{3} \) after changing the subtraction of a negative to the addition of a positive. This subtle shift is an important concept in understanding negative numbers and operations involving them.