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Chapter 0: Problem 78

Evaluate each expression. $$ \frac{2}{15}+\frac{8}{9} $$

Short Answer

Expert verified
\( \frac{46}{45} \) or \( 1 \frac{1}{45} \)

Step by step solution

01

Identify the fractions

The given fractions are \( \frac{2}{15} \) and \( \frac{8}{9} \). Our task is to add these two fractions together.
02

Find the common denominator

Determine the least common denominator (LCD) for the fractions. The denominators are 15 and 9. The LCD of 15 and 9 is 45.
03

Convert fractions to have the common denominator

Rewrite each fraction with the common denominator of 45. For \( \frac{2}{15} \): \( \frac{2 \times 3}{15 \times 3} = \frac{6}{45} \) For \( \frac{8}{9} \): \( \frac{8 \times 5}{9 \times 5} = \frac{40}{45} \)
04

Add the fractions

Now, add the fractions: \( \frac{6}{45} + \frac{40}{45} = \frac{6 + 40}{45} = \frac{46}{45} \)
05

Simplify the result

Simplify the fraction if possible. In this case, \( \frac{46}{45} \) is already in its simplest form. It can also be expressed as a mixed number: \( 1 \frac{1}{45} \)

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Key Concepts

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

Common Denominator
When adding fractions, one of the most important steps is finding a common denominator. This is because you need to have equal-sized pieces to add them together. Imagine trying to add a slice of pizza cut into 3 pieces with another slice cut into 5; it just wouldn't make sense! To find a common denominator, you look for a number that both denominators (the bottom numbers of the fractions) can divide into evenly. This number helps you bring both fractions to a comparable form so you can add them seamlessly.
Least Common Denominator (LCD)
The Least Common Denominator (LCD) is the smallest number that both denominators can evenly divide into. Finding the LCD makes it easier to work with the fractions. For example, in our problem, the denominators are 15 and 9. To find the LCD, you list the multiples of each denominator and identify the smallest common multiple. Multiples of 15 are 15, 30, 45, 60, etc. Multiples of 9 are 9, 18, 27, 36, 45, etc. The smallest common multiple here is 45. Once you find the LCD, you can rewrite each fraction with this common denominator, making them easier to add together.
Simplifying Fractions
After you add your fractions, you often need to simplify the result. Simplifying a fraction means reducing it to its simplest form, where the numerator (top number) and denominator (bottom number) have no common factors other than 1. To simplify, you divide both the numerator and the denominator by their greatest common divisor (GCD). In our example, after adding, we got \(\frac{46}{45}\). This fraction is already in its simplest form because 46 and 45 have no common factors other than 1. Simplifying fractions makes the final answer cleaner and easier to understand.

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Most popular questions from this chapter

Simplify each expression. $$25^{3 / 2}$$

Simplify each expression. Assume that all variables are positive when they appear. $$(3 \sqrt{6})(2 \sqrt{2})$$

Expressions that occur in calculus are given. Write each expression as a single quotient in which only positive exponents and radicals appear. $$\frac{\frac{1+x^{2}}{2 \sqrt{x}}-2 x \sqrt{x}}{\left(1+x^{2}\right)^{2}} \quad x>0$$

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