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Answer :
To solve the problem of finding which cards are equivalent to [tex]\(3 \frac{2}{5} - 1 \frac{4}{6}\)[/tex], we need to perform a few steps.
1. Convert Mixed Numbers to Improper Fractions:
- First, convert [tex]\(3 \frac{2}{5}\)[/tex]:
[tex]\[
3 \frac{2}{5} = 3 + \frac{2}{5} = \frac{15}{5} + \frac{2}{5} = \frac{17}{5}
\][/tex]
- Convert [tex]\(1 \frac{4}{6}\)[/tex]:
[tex]\[
1 \frac{4}{6} = 1 + \frac{4}{6} = \frac{6}{6} + \frac{4}{6} = \frac{10}{6}
\][/tex]
- Simplify [tex]\(\frac{10}{6}\)[/tex]:
[tex]\[
\frac{10}{6} = \frac{5}{3}
\][/tex]
2. Perform the Subtraction:
- Calculate [tex]\(\frac{17}{5} - \frac{5}{3}\)[/tex]. To subtract these fractions, they need a common denominator:
- The least common multiple of 5 and 3 is 15.
- Convert [tex]\(\frac{17}{5}\)[/tex] to a denominator of 15:
[tex]\[
\frac{17}{5} = \frac{17 \times 3}{5 \times 3} = \frac{51}{15}
\][/tex]
- Convert [tex]\(\frac{5}{3}\)[/tex] to a denominator of 15:
[tex]\[
\frac{5}{3} = \frac{5 \times 5}{3 \times 5} = \frac{25}{15}
\][/tex]
- Subtract:
[tex]\[
\frac{51}{15} - \frac{25}{15} = \frac{51 - 25}{15} = \frac{26}{15}
\][/tex]
3. Convert Back to a Mixed Number:
- [tex]\(\frac{26}{15}\)[/tex] as a mixed number:
[tex]\[
1 \frac{11}{15}
\][/tex]
4. Compare with Options:
- We need to see which options match or are equivalent to [tex]\(1 \frac{11}{15}\)[/tex].
- Check the given options, converted to equivalent fractions with a denominator of 30:
- [tex]\(1 \frac{16}{30} = 1 + \frac{16}{30} = 1 + \frac{8}{15} = 1 \frac{8}{15}\)[/tex]
- [tex]\(1 \frac{22}{30} = 1 + \frac{22}{30} = 1 + \frac{11}{15} = 1 \frac{11}{15}\)[/tex]
- [tex]\(1 \frac{28}{30} = 1 + \frac{28}{30} = 1 + \frac{14}{15} = 1 \frac{14}{15}\)[/tex]
- Other options are not mixed numbers, and we have already calculated valid differences.
5. Result:
- The correct card that matches our calculated result of [tex]\(1 \frac{11}{15}\)[/tex] is [tex]\(1 \frac{22}{30}\)[/tex].
Therefore, the correct answer is: [tex]\(1 \frac{22}{30}\)[/tex].
1. Convert Mixed Numbers to Improper Fractions:
- First, convert [tex]\(3 \frac{2}{5}\)[/tex]:
[tex]\[
3 \frac{2}{5} = 3 + \frac{2}{5} = \frac{15}{5} + \frac{2}{5} = \frac{17}{5}
\][/tex]
- Convert [tex]\(1 \frac{4}{6}\)[/tex]:
[tex]\[
1 \frac{4}{6} = 1 + \frac{4}{6} = \frac{6}{6} + \frac{4}{6} = \frac{10}{6}
\][/tex]
- Simplify [tex]\(\frac{10}{6}\)[/tex]:
[tex]\[
\frac{10}{6} = \frac{5}{3}
\][/tex]
2. Perform the Subtraction:
- Calculate [tex]\(\frac{17}{5} - \frac{5}{3}\)[/tex]. To subtract these fractions, they need a common denominator:
- The least common multiple of 5 and 3 is 15.
- Convert [tex]\(\frac{17}{5}\)[/tex] to a denominator of 15:
[tex]\[
\frac{17}{5} = \frac{17 \times 3}{5 \times 3} = \frac{51}{15}
\][/tex]
- Convert [tex]\(\frac{5}{3}\)[/tex] to a denominator of 15:
[tex]\[
\frac{5}{3} = \frac{5 \times 5}{3 \times 5} = \frac{25}{15}
\][/tex]
- Subtract:
[tex]\[
\frac{51}{15} - \frac{25}{15} = \frac{51 - 25}{15} = \frac{26}{15}
\][/tex]
3. Convert Back to a Mixed Number:
- [tex]\(\frac{26}{15}\)[/tex] as a mixed number:
[tex]\[
1 \frac{11}{15}
\][/tex]
4. Compare with Options:
- We need to see which options match or are equivalent to [tex]\(1 \frac{11}{15}\)[/tex].
- Check the given options, converted to equivalent fractions with a denominator of 30:
- [tex]\(1 \frac{16}{30} = 1 + \frac{16}{30} = 1 + \frac{8}{15} = 1 \frac{8}{15}\)[/tex]
- [tex]\(1 \frac{22}{30} = 1 + \frac{22}{30} = 1 + \frac{11}{15} = 1 \frac{11}{15}\)[/tex]
- [tex]\(1 \frac{28}{30} = 1 + \frac{28}{30} = 1 + \frac{14}{15} = 1 \frac{14}{15}\)[/tex]
- Other options are not mixed numbers, and we have already calculated valid differences.
5. Result:
- The correct card that matches our calculated result of [tex]\(1 \frac{11}{15}\)[/tex] is [tex]\(1 \frac{22}{30}\)[/tex].
Therefore, the correct answer is: [tex]\(1 \frac{22}{30}\)[/tex].
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