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To solve the addition of the two mixed fractions, [tex]\(8 \frac{1}{5}\)[/tex] and [tex]\(9 \frac{1}{3}\)[/tex], let's follow these steps:
### Step 1: Convert the Mixed Fractions to Improper Fractions
1. For [tex]\(8 \frac{1}{5}\)[/tex]:
- Multiply the whole number by the denominator: [tex]\(8 \times 5 = 40\)[/tex].
- Add the numerator to this product: [tex]\(40 + 1 = 41\)[/tex].
- The improper fraction is [tex]\(\frac{41}{5}\)[/tex].
2. For [tex]\(9 \frac{1}{3}\)[/tex]:
- Multiply the whole number by the denominator: [tex]\(9 \times 3 = 27\)[/tex].
- Add the numerator to this product: [tex]\(27 + 1 = 28\)[/tex].
- The improper fraction is [tex]\(\frac{28}{3}\)[/tex].
### Step 2: Find a Common Denominator
The denominators are 5 and 3. The least common multiple (LCM) of 5 and 3 is 15.
### Step 3: Convert Each Fraction to Have the Common Denominator
1. For [tex]\(\frac{41}{5}\)[/tex]:
- To convert [tex]\(\frac{41}{5}\)[/tex] to a denominator of 15, multiply both the numerator and the denominator by 3:
- [tex]\(\frac{41 \times 3}{5 \times 3} = \frac{123}{15}\)[/tex].
2. For [tex]\(\frac{28}{3}\)[/tex]:
- To convert [tex]\(\frac{28}{3}\)[/tex] to a denominator of 15, multiply both the numerator and the denominator by 5:
- [tex]\(\frac{28 \times 5}{3 \times 5} = \frac{140}{15}\)[/tex].
### Step 4: Add the Fractions
Now, add the fractions [tex]\(\frac{123}{15}\)[/tex] and [tex]\(\frac{140}{15}\)[/tex]:
- [tex]\(\frac{123}{15} + \frac{140}{15} = \frac{263}{15}\)[/tex].
### Step 5: Convert the Result Back to a Mixed Fraction
Convert the improper fraction [tex]\(\frac{263}{15}\)[/tex] back to a mixed fraction:
1. Divide the numerator by the denominator:
- [tex]\(263 \div 15 = 17\)[/tex] (the quotient is the whole number part).
2. Find the remainder:
- [tex]\(263 - (17 \times 15) = 263 - 255 = 8\)[/tex].
So, [tex]\(\frac{263}{15} = 17 \frac{8}{15}\)[/tex].
### Final Answer
[tex]\[ 8 \frac{1}{5} + 9 \frac{1}{3} = 17 \frac{8}{15} \][/tex]
### Summary of Steps
1. Convert [tex]\(8 \frac{1}{5}\)[/tex] and [tex]\(9 \frac{1}{3}\)[/tex] to improper fractions: [tex]\(\frac{41}{5}\)[/tex] and [tex]\(\frac{28}{3}\)[/tex].
2. Find a common denominator: 15.
3. Convert the fractions: [tex]\(\frac{123}{15}\)[/tex] and [tex]\(\frac{140}{15}\)[/tex].
4. Add the fractions: [tex]\(\frac{263}{15}\)[/tex].
5. Convert back to a mixed fraction: [tex]\(17 \frac{8}{15}\)[/tex].
So, the process yields the final mixed fraction result: [tex]\(17 \frac{8}{15}\)[/tex].
### Step 1: Convert the Mixed Fractions to Improper Fractions
1. For [tex]\(8 \frac{1}{5}\)[/tex]:
- Multiply the whole number by the denominator: [tex]\(8 \times 5 = 40\)[/tex].
- Add the numerator to this product: [tex]\(40 + 1 = 41\)[/tex].
- The improper fraction is [tex]\(\frac{41}{5}\)[/tex].
2. For [tex]\(9 \frac{1}{3}\)[/tex]:
- Multiply the whole number by the denominator: [tex]\(9 \times 3 = 27\)[/tex].
- Add the numerator to this product: [tex]\(27 + 1 = 28\)[/tex].
- The improper fraction is [tex]\(\frac{28}{3}\)[/tex].
### Step 2: Find a Common Denominator
The denominators are 5 and 3. The least common multiple (LCM) of 5 and 3 is 15.
### Step 3: Convert Each Fraction to Have the Common Denominator
1. For [tex]\(\frac{41}{5}\)[/tex]:
- To convert [tex]\(\frac{41}{5}\)[/tex] to a denominator of 15, multiply both the numerator and the denominator by 3:
- [tex]\(\frac{41 \times 3}{5 \times 3} = \frac{123}{15}\)[/tex].
2. For [tex]\(\frac{28}{3}\)[/tex]:
- To convert [tex]\(\frac{28}{3}\)[/tex] to a denominator of 15, multiply both the numerator and the denominator by 5:
- [tex]\(\frac{28 \times 5}{3 \times 5} = \frac{140}{15}\)[/tex].
### Step 4: Add the Fractions
Now, add the fractions [tex]\(\frac{123}{15}\)[/tex] and [tex]\(\frac{140}{15}\)[/tex]:
- [tex]\(\frac{123}{15} + \frac{140}{15} = \frac{263}{15}\)[/tex].
### Step 5: Convert the Result Back to a Mixed Fraction
Convert the improper fraction [tex]\(\frac{263}{15}\)[/tex] back to a mixed fraction:
1. Divide the numerator by the denominator:
- [tex]\(263 \div 15 = 17\)[/tex] (the quotient is the whole number part).
2. Find the remainder:
- [tex]\(263 - (17 \times 15) = 263 - 255 = 8\)[/tex].
So, [tex]\(\frac{263}{15} = 17 \frac{8}{15}\)[/tex].
### Final Answer
[tex]\[ 8 \frac{1}{5} + 9 \frac{1}{3} = 17 \frac{8}{15} \][/tex]
### Summary of Steps
1. Convert [tex]\(8 \frac{1}{5}\)[/tex] and [tex]\(9 \frac{1}{3}\)[/tex] to improper fractions: [tex]\(\frac{41}{5}\)[/tex] and [tex]\(\frac{28}{3}\)[/tex].
2. Find a common denominator: 15.
3. Convert the fractions: [tex]\(\frac{123}{15}\)[/tex] and [tex]\(\frac{140}{15}\)[/tex].
4. Add the fractions: [tex]\(\frac{263}{15}\)[/tex].
5. Convert back to a mixed fraction: [tex]\(17 \frac{8}{15}\)[/tex].
So, the process yields the final mixed fraction result: [tex]\(17 \frac{8}{15}\)[/tex].
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