Problem 35 - Introductory and Intermediate Algebra for College Students 4th

Question

Find each sum without the use of a number line. $$-\frac{3}{7}+\left(-\frac{4}{5}\right)$$

Step-by-Step Solution

Answer

The sum of -3/7 and -4/5 is -43/35.

Step 1: Find a Common Denominator

It's essential to find a common denominator when adding fractions. In this case, the least common multiple (LCM) of 7 and 5 is 35, which will be the common denominator.

Step 2: Convert Each Fraction

Convert the fractions to their equivalents with the common denominator. The fraction -3/7 becomes -15/35 (-3 multiplied by 5 for the numerator and 7 multiplied by 5 for the denominator). The fraction -4/5 becomes -28/35 (-4 multiplied by 7 for the numerator and 5 multiplied by 7 for the denominator).

Step 3: Sum the Fractions

Now that both fractions have the same denominator, they can be added together. The numerical part of the fractions is added, while the denominator stays the same: -15/35 + -28/35 = -43/35.

Key Concepts

Understanding Common Denominators

Before you can add fractions, they need to share the same denominator, which is the bottom part of a fraction. Without a common denominator, you can't do simple addition because the fractions are like pieces of different wholes. Imagine trying to combine pieces of a puzzle from two separate boxes—they just won't fit together!

To remedy this, you find a common denominator, which allows you to "resize" the fractions so they become compatible. Think of it as ensuring both fractions are built from the same "size" pieces. This is the key first step when working towards adding any two fractions efficiently.

Finding the Least Common Multiple (LCM)

To establish a common denominator, you first need to determine the least common multiple, or LCM, of the fractions' denominators. The LCM is the smallest number that both denominators can divide into evenly.

For instance, in the exercise with fractions \(-\frac{3}{7}\) and \(-\frac{4}{5}\), the denominators are 7 and 5. Here, you find the LCM by listing the multiples of each number:

• Multiples of 7: 7, 14, 21, 28, 35...
• Multiples of 5: 5, 10, 15, 20, 25, 30, 35...

The least common multiple is 35, which is the smallest number shared by the two lists of multiples. Hence, 35 becomes the common denominator for these fractions.

Adding Fractions with the Same Denominator

Once both fractions are expressed with the same denominator, you can proceed to add them. It's important to remember that while the numerators (the top parts of fractions) are added or subtracted, the denominator remains unchanged.

In our example, \(-\frac{3}{7}\) becomes \(-\frac{15}{35}\) and \(-\frac{4}{5}\) transforms into \(-\frac{28}{35}\). Now, you simply add the numerators:

• -15 (from \(-\frac{15}{35}\))
• -28 (from \(-\frac{28}{35}\))

Adding these, we get \(-15 + (-28) = -43\). So, the sum is \(-\frac{43}{35}\). Simple, right? Just add the tops, leave the bottom alone, and your fractions are successfully added.

Handling Negative Fractions

Negative fractions can sometimes trip us up. Nevertheless, the basic rules remain the same. A negative sign in front of a fraction means the whole value is negative—which can occur with either or both the numerator and denominator.

In our example, \(-\frac{3}{7}\) and \(-\frac{4}{5}\) are both negative, so when added, the negative values amplify. Think of negative numbers as directions pointing left on a number line; when combined, they simply take you further in that direction.

When you sum \(-\frac{15}{35}\) and \(-\frac{28}{35}\), you add \(-15\) and \(-28\), leading to \(-43\), keeping the negative sign intact. Therefore, the final result, \(-\frac{43}{35}\), is a negative fraction.