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Chapter 3: Problem 5

Add and simplify. $$ \frac{2}{3}+\frac{5}{6} $$

Short Answer

Expert verified
\frac{3}{2}

Step by step solution

01

Find a Common Denominator

To add the fractions, first find a common denominator. The denominators are 3 and 6. The least common multiple of 3 and 6 is 6.
02

Convert Fractions to Common Denominator

Convert \(\frac{2}{3}\) to a fraction with a denominator of 6. By multiplying both the numerator and the denominator by 2, \(\frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6}\).
03

Add the Fractions

Now, add the fractions with common denominators: \[ \frac{4}{6} + \frac{5}{6} = \frac{4 + 5}{6} = \frac{9}{6} \]
04

Simplify the Result

Simplify \(\frac{9}{6}\) by dividing the numerator and the denominator by their greatest common divisor, which is 3: \[ \frac{9 \, \div \, 3}{6 \, \div \, 3} = \frac{3}{2} \]

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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, the first step is finding a common denominator. This means identifying a shared denominator that both fractions can use. Fractions must have the same denominator to be added or subtracted. Without a common denominator, you can't directly combine the fractions.
To find a common denominator, look at the denominators of the fractions you're working with. For the exercise \(\frac{2}{3} + \frac{5}{6}\), the denominators are 3 and 6. The common denominator should be a number that both 3 and 6 can divide into without leaving a remainder. This brings us to the next important concept, the least common multiple.
least common multiple
The least common multiple (LCM) of two numbers is the smallest number that they both divide into evenly. When adding fractions, finding the LCM of the denominators will help you find the least common denominator. For instance, in our exercise, the denominators are 3 and 6.
First, consider the multiples of each number:
  • Multiples of 3: 3, 6, 9, 12, ...
  • Multiples of 6: 6, 12, 18, 24, ...
You can see that the smallest number both sets share is 6, so the LCM of 3 and 6 is 6. This is our common denominator. With the common denominator found, you can convert each fraction to have this same denominator before adding them.
simplifying fractions
After finding a common denominator and adding the fractions, you might need to simplify the result. Simplifying a fraction means reducing it to its simplest form, where the numerator and denominator have no common factors other than 1. To simplify the fraction \(\frac{9}{6}\) from our exercise:
- Find the greatest common divisor (GCD) of the numerator and denominator. For 9 and 6, the GCD is 3.
- Divide both the numerator and the denominator by their GCD:
$$\frac{9 \ \div\ \ 3}{6 \ \div\ \ 3} \ =\ \ \frac{3}{2}$$
This gives you the simplified fraction \(\frac{3}{2}\). Simplifying fractions often makes them easier to understand and work with in future calculations.

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Most popular questions from this chapter

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