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Chapter 7: Problem 64

In the following exercises, simplify. $$ \frac{5}{13}+\frac{3}{4}+\frac{1}{4} $$

Short Answer

Expert verified
The simplified expression is \(\frac{18}{13}\).

Step by step solution

01

Combine like terms

First, add the fractions with the same denominator. Combine \(\frac{3}{4}\) and \(\frac{1}{4}\) by adding their numerators: \(3 + 1 = 4\). Therefore, \(\frac{3}{4} + \frac{1}{4} = \frac{4}{4} = 1\). The expression now becomes: \(\frac{5}{13} + 1\).
02

Add the simplified fractions

The simplified expression is \(\frac{5}{13} + 1\). To add these, convert \(1\) to a fraction with a denominator of \(13\). Thus, \(1 = \frac{13}{13}\). Now the expression is \(\frac{5}{13} + \frac{13}{13}\).
03

Add the fractions

Combine the fractions by adding their numerators: \(\frac{5}{13} + \frac{13}{13} = \frac{5 + 13}{13} = \frac{18}{13}\).

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Key Concepts

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

Adding Fractions
Adding fractions involves combining their numerators while keeping the denominator the same. For example, if we have \(\frac{3}{4}\) and \(\frac{1}{4}\), we can add them because they have the same denominator. Just add the numerators: \3 + 1 = 4\. Therefore, \(\frac{3}{4} + \frac{1}{4} = \frac{4}{4} = 1\). This process is straightforward when the denominators are the same. However, when they are different, we need to find a common denominator first. This will be discussed in the next section. Remember, practice makes perfect in mastering fraction addition!
Common Denominators
When adding fractions with different denominators, like \(\frac{5}{13} + 1\) from our problem, we need to find a common denominator. The first step is to convert all the fractions so they share the same denominator. For example, the fraction \1\ can be written as \(\frac{13}{13}\) since any number divided by itself equals \1\. This makes it possible to add \(\frac{5}{13}\) and \(\frac{13}{13}\) by adding their numerators together because now the denominators are the same. Hence, the expression becomes: \(\frac{5}{13} + \frac{13}{13} = \frac{18}{13}\). Always ensure that the fractions have the same denominator before adding their numerators.
Fraction Simplification
Simplifying fractions involves reducing them to their simplest form. A fraction is considered simplified when the numerator and denominator have no common factors other than \1\. In the final step of our problem, we got \(\frac{18}{13}\). Sometimes you may need to simplify further but here there are no common factors between 18 and 13, so this fraction is already in its simplest form. Simplification makes it easier to understand and use the fractions in further calculations. To simplify, always look for the greatest common divisor (GCD) of the numerator and the denominator, and then divide both by that number.

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Most popular questions from this chapter

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