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Answer :
Sure, let's break down the problem step by step.
We need to determine which options are equivalent to the expression [tex]\(3 \frac{2}{5} - 1 \frac{4}{6}\)[/tex].
### Step-by-Step Breakdown:
1. Convert the mixed numbers to improper fractions:
- [tex]\(3 \frac{2}{5}\)[/tex]:
- Multiply the whole number 3 by the denominator 5: [tex]\(3 \times 5 = 15\)[/tex].
- Add the numerator 2: [tex]\(15 + 2 = 17\)[/tex].
- The improper fraction is [tex]\(\frac{17}{5}\)[/tex].
- [tex]\(1 \frac{4}{6}\)[/tex]:
- Multiply the whole number 1 by the denominator 6: [tex]\(1 \times 6 = 6\)[/tex].
- Add the numerator 4: [tex]\(6 + 4 = 10\)[/tex].
- The improper fraction is [tex]\(\frac{10}{6}\)[/tex].
2. Find a common denominator for the subtraction:
- The denominators are 5 and 6. The least common multiple of 5 and 6 is 30.
3. Convert both fractions to have the common denominator 30:
- For [tex]\(\frac{17}{5}\)[/tex]:
- Multiply both the numerator and denominator by 6: [tex]\(\frac{17 \times 6}{5 \times 6} = \frac{102}{30}\)[/tex].
- For [tex]\(\frac{10}{6}\)[/tex]:
- Multiply both the numerator and denominator by 5: [tex]\(\frac{10 \times 5}{6 \times 5} = \frac{50}{30}\)[/tex].
4. Subtract the equivalent fractions:
[tex]\[
\frac{102}{30} - \frac{50}{30} = \frac{52}{30}
\][/tex]
5. Convert back to a mixed number:
- Divide 52 by 30: 52 ÷ 30 = 1 with a remainder of 22.
- So, [tex]\(\frac{52}{30}\)[/tex] is equivalent to [tex]\(1 \frac{22}{30}\)[/tex].
Now, let's compare this result to the given options:
- [tex]$3 \frac{12}{30} - 1 \frac{20}{30}$[/tex] is equivalent to [tex]\(1 \frac{22}{30}\)[/tex], because after simplifying, it aligns with our result.
- [tex]$3 \frac{10}{30} - 1 \frac{24}{30}$[/tex] simplifies to a different value than [tex]\(1 \frac{22}{30}\)[/tex] and is not equivalent.
- [tex]$1 \frac{16}{30}$[/tex] and [tex]$1 \frac{28}{30}$[/tex] are also not equivalent to [tex]\(1 \frac{22}{30}\)[/tex].
There are two correct answers:
- [tex]\(3 \frac{12}{30} - 1 \frac{20}{30}\)[/tex]
- [tex]\(1 \frac{22}{30}\)[/tex]
These options match the calculated result of [tex]\(3 \frac{2}{5} - 1 \frac{4}{6}\)[/tex].
We need to determine which options are equivalent to the expression [tex]\(3 \frac{2}{5} - 1 \frac{4}{6}\)[/tex].
### Step-by-Step Breakdown:
1. Convert the mixed numbers to improper fractions:
- [tex]\(3 \frac{2}{5}\)[/tex]:
- Multiply the whole number 3 by the denominator 5: [tex]\(3 \times 5 = 15\)[/tex].
- Add the numerator 2: [tex]\(15 + 2 = 17\)[/tex].
- The improper fraction is [tex]\(\frac{17}{5}\)[/tex].
- [tex]\(1 \frac{4}{6}\)[/tex]:
- Multiply the whole number 1 by the denominator 6: [tex]\(1 \times 6 = 6\)[/tex].
- Add the numerator 4: [tex]\(6 + 4 = 10\)[/tex].
- The improper fraction is [tex]\(\frac{10}{6}\)[/tex].
2. Find a common denominator for the subtraction:
- The denominators are 5 and 6. The least common multiple of 5 and 6 is 30.
3. Convert both fractions to have the common denominator 30:
- For [tex]\(\frac{17}{5}\)[/tex]:
- Multiply both the numerator and denominator by 6: [tex]\(\frac{17 \times 6}{5 \times 6} = \frac{102}{30}\)[/tex].
- For [tex]\(\frac{10}{6}\)[/tex]:
- Multiply both the numerator and denominator by 5: [tex]\(\frac{10 \times 5}{6 \times 5} = \frac{50}{30}\)[/tex].
4. Subtract the equivalent fractions:
[tex]\[
\frac{102}{30} - \frac{50}{30} = \frac{52}{30}
\][/tex]
5. Convert back to a mixed number:
- Divide 52 by 30: 52 ÷ 30 = 1 with a remainder of 22.
- So, [tex]\(\frac{52}{30}\)[/tex] is equivalent to [tex]\(1 \frac{22}{30}\)[/tex].
Now, let's compare this result to the given options:
- [tex]$3 \frac{12}{30} - 1 \frac{20}{30}$[/tex] is equivalent to [tex]\(1 \frac{22}{30}\)[/tex], because after simplifying, it aligns with our result.
- [tex]$3 \frac{10}{30} - 1 \frac{24}{30}$[/tex] simplifies to a different value than [tex]\(1 \frac{22}{30}\)[/tex] and is not equivalent.
- [tex]$1 \frac{16}{30}$[/tex] and [tex]$1 \frac{28}{30}$[/tex] are also not equivalent to [tex]\(1 \frac{22}{30}\)[/tex].
There are two correct answers:
- [tex]\(3 \frac{12}{30} - 1 \frac{20}{30}\)[/tex]
- [tex]\(1 \frac{22}{30}\)[/tex]
These options match the calculated result of [tex]\(3 \frac{2}{5} - 1 \frac{4}{6}\)[/tex].
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