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Chapter 4: Problem 447

In the following exercises, add. $$8 \frac{4}{9}+2 \frac{8}{9}$$

Short Answer

Expert verified
11 \frac{1}{3}.

Step by step solution

01

Convert Mixed Numbers to Improper Fractions

First, convert each mixed number to an improper fraction. For the number $$8 \frac{4}{9}$$ multiply the whole number 8 by the denominator 9 and then add the numerator 4. This gives: $$ 8 \times 9 + 4 = 72 + 4 = 76$$ So, $$8 \frac{4}{9} = \frac{76}{9}.$$ For the number $$2 \frac{8}{9},$$ multiply the whole number 2 by the denominator 9 and then add the numerator 8. This gives: $$ 2 \times 9 + 8 = 18 + 8 = 26$$ So, $$2 \frac{8}{9} = \frac{26}{9}.$$
02

Add the Improper Fractions

Now that the numbers are converted to improper fractions, they have the same denominator and can be added directly: $$ \frac{76}{9} + \frac{26}{9} = \frac{76 + 26}{9} = \frac{102}{9}.$$
03

Convert the Result Back to a Mixed Number

Finally, convert the resulting improper fraction back to a mixed number. Divide the numerator 102 by the denominator 9: $$ 102 \text{ divided by } 9 = 11 \text{ remainder } 3.$$ So, $$ \frac{102}{9} = 11 \frac{3}{9}.$$ Simplify the fraction $$ \frac{3}{9} $$ by dividing both the numerator and the denominator by 3, which results in $$ \frac{1}{3}.$$ Therefore, the final answer is $$ 11 \frac{1}{3}.$$

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Key Concepts

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

Mixed Numbers
A mixed number is a number that combines a whole number and a fraction. For example, in the mixed number \(8 \frac{4}{9}\), 8 is the whole number and \( \frac{4}{9} \) is the fractional part. Mixed numbers can be handy for everyday life, like measuring ingredients for a recipe or dividing up spaces.
Improper Fractions
An improper fraction has a numerator (the top part of the fraction) that is larger than or equal to its denominator (the bottom part). For example, the mixed number \(8 \frac{4}{9}\) can be converted into the improper fraction \( \frac{76}{9} \). You do this by multiplying the whole number by the denominator and adding the numerator. So it鈥檚 \(8 \times 9 + 4 = 76\). Improper fractions are useful in mathematical calculations because they are easier to work with compared to mixed numbers.
Fraction Addition
Adding fractions involves ensuring the denominators are the same. In our example, adding \(8 \frac{4}{9}\) and \(2 \frac{8}{9}\) means converting both to improper fractions first. This gives \( \frac{76}{9} \) and \( \frac{26}{9} \). When the denominators are the same, you simply add the numerators: \(76 + 26 = 102\), so \( \frac{76}{9} + \frac{26}{9} = \frac{102}{9} \).
Simplifying Fractions
Simplifying fractions makes them easier to understand at a glance. Once you鈥檝e added your fractions and ended up with \( \frac{102}{9} \), converting this back to a mixed number involves division. Dividing 102 by 9 gives 11 with a remainder of 3. So you get \(11 \frac{3}{9} \), which can be further simplified by dividing the numerator and denominator of the fractional part by their greatest common divisor, 3. This results in \(11 \frac{1}{3} \).

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Most popular questions from this chapter

In the following exercises, simplify. $$ \frac{2^{3}-2^{2}}{\left(\frac{3}{4}\right)^{2}} $$

In the following exercises, simplify. $$\left(\frac{3}{4}+\frac{1}{6}\right) \div\left(\frac{5}{8}-\frac{1}{3}\right)$$

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

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