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Chapter 4: Problem 694

In the following exercises, simplify. $$ -\frac{3}{10}+\left(-\frac{5}{8}\right) $$

Short Answer

Expert verified
-\frac{37}{40}

Step by step solution

01

Identify common denominators

To add the fractions, find a common denominator. The denominators are 10 and 8. The least common multiple (LCM) of 10 and 8 is 40.
02

Convert fractions to have common denominator

Convert \(-\frac{3}{10}\) and \(-\frac{5}{8}\) to have the common denominator of 40. Multiply \(-\frac{3}{10}\) by \frac{4}{4}\ to get \(-\frac{12}{40}\). Multiply \(-\frac{5}{8}\) by \frac{5}{5}\ to get \(-\frac{25}{40}\).
03

Add the fractions

Now add the two fractions: \(-\frac{12}{40} + -\frac{25}{40}= -\frac{12+25}{40}= -\frac{37}{40}\).
04

Simplify

Since \(-\frac{37}{40}\) is already in its simplest form, no further simplification is needed.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common Denominators
When adding fractions, it is crucial to have a common denominator. A common denominator allows you to combine fractions easily. For example, if you have the fractions \(-\frac{3}{10}\) and \(-\frac{5}{8}\), the denominators are 10 and 8. To add these, you need a common denominator. This denominator must be a multiple of both 10 and 8. Once you find this, you can proceed with fraction addition. Without a common denominator, direct addition isn't possible.
Least Common Multiple
The Least Common Multiple (LCM) of two numbers is the smallest multiple that is evenly divisible by both numbers. Finding the LCM of the denominators is the first step in adding fractions. For our example with denominators 10 and 8, list the multiples of both:
  • Multiples of 10: 10, 20, 30, 40, 50...
  • Multiples of 8: 8, 16, 24, 32, 40...
The smallest common multiple is 40. Thus, 40 will be our common denominator. Knowing how to find the LCM makes it easier to perform operations with fractions.
Fraction Addition
Once you have a common denominator, you can add the fractions. Convert each fraction to an equivalent fraction with this common denominator. In our example, convert \(-\frac{3}{10}\) and \(-\frac{5}{8}\) as follows:
Multiply \(-\frac{3}{10}\) by \frac{4}{4}\ to get \(-\frac{12}{40}\).
Multiply \(-\frac{5}{8}\) by \frac{5}{5}\ to get \(-\frac{25}{40}\).
Now add these two fractions:
\-\frac{12}{40} + -\frac{25}{40} = -\frac{37}{40}\.
Note that the numerators are added while the denominator remains the same.
Simplifying Fractions
After adding the fractions, you should simplify the result if possible. Simplifying means reducing the fraction to its lowest terms. Check if the numerator and denominator have a common factor. If they do, divide both by that factor. In our example, \(-\frac{37}{40}\) cannot be simplified further because 37 and 40 share no common factors other than 1. Therefore, \(-\frac{37}{40}\) is already in its simplest form. Simplifying fractions makes them easier to understand and work with.

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Most popular questions from this chapter

Explain why it is necessary to have a common denominator to add or subtract fractions.

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