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Chapter 3: Problem 6

Add and simplify. $$ \frac{5}{6}+\frac{1}{9} $$

Short Answer

Expert verified
\( \frac{17}{18} \)

Step by step solution

01

- Find the Least Common Denominator (LCD)

To add fractions, we first need to find the Least Common Denominator (LCD) of the two denominators, 6 and 9. The LCD of 6 and 9 is 18.
02

- Convert Fractions to Have the LCD

Next, we convert each fraction to an equivalent fraction with the denominator of 18. For the fraction \(\frac{5}{6}\), multiply the numerator and the denominator by 3 to get \(\frac{15}{18}\). For the fraction \(\frac{1}{9}\), multiply the numerator and the denominator by 2 to get \(\frac{2}{18}\).
03

- Add the Fractions

Now, add the two fractions with the same denominators: \(\frac{15}{18} + \frac{2}{18} = \frac{17}{18}\).
04

- Simplify if Necessary

Check if the result can be simplified. In this case, \(\frac{17}{18}\) is already in its simplest form, so 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.

least common denominator
When adding fractions, the first step is to find the Least Common Denominator (LCD). The LCD is the smallest number that both denominators can divide into without leaving a remainder.

For example, consider the fractions \(\frac{5}{6}\) and \(\frac{1}{9}\). The denominators here are 6 and 9. To find their LCD:
  • List the multiples of 6: 6, 12, 18, 24, ...
  • List the multiples of 9: 9, 18, 27, ...
The first common multiple is 18, so the Least Common Denominator of 6 and 9 is 18.

We use the LCD to rewrite each fraction so that they have the same denominator, making them easier to add together.
equivalent fractions
After finding the Least Common Denominator (LCD), the next step is to convert each fraction to an equivalent fraction with this common denominator. An equivalent fraction is simply a different way of writing the same value.

Let's look at our fractions again:
simplifying fractions
The final step when adding fractions is to simplify the result if possible. Simplifying a fraction means reducing it to its simplest form. This is done by dividing the numerator and the denominator by their Greatest Common Divisor (GCD).

In our example, the result of adding \(\frac{15}{18}\) and \(\frac{2}{18}\) was \(\frac{17}{18}\).

Check if the numerator (17) and the denominator (18) have any common factors other than 1. They don't, so \(\frac{17}{18}\) is already in its simplest form.

To simplify fractions in general:
  • Find the GCD of the numerator and denominator.
  • Divide both the numerator and the denominator by their GCD.
That's it! You've successfully added and simplified the fractions.

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Most popular questions from this chapter

To cut back on their caffeine intake, Michelle and Gerry mix caffeinated and decaffeinated coffee beans before grinding for a customized mix. They mix \(\frac{3}{16} \mathrm{lb}\) of decaffeinated beans with \(\frac{5}{8} \mathrm{lb}\) of caffeinated beans. What is the total amount of coffee beans in the mixture?

Add and simplify. $$ \frac{15}{24}+\frac{7}{36}+\frac{91}{48} $$

Subtract and simplify. $$ \frac{5}{6}-\frac{1}{6} $$

Simplify. $$ \frac{3}{4} \div \frac{1}{2} \cdot\left(\frac{8}{9}-\frac{2}{3}\right) $$

Estimate each of the following as \(0, \frac{1}{2},\) or 1 . $$ \frac{19}{20} $$
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