Answer :
To solve the problem of determining which cards are equivalent to the expression [tex]\(3 \frac{2}{5} - 1 \frac{4}{6}\)[/tex], we'll break down the steps needed to find the correct equivalent cards.
1. Convert Mixed Numbers to Improper Fractions:
- First mixed number: [tex]\(3 \frac{2}{5}\)[/tex] can be converted:
[tex]\[ 3 \frac{2}{5} = 3 + \frac{2}{5} = \frac{15}{5} + \frac{2}{5} = \frac{17}{5} \][/tex]
- Second mixed number: [tex]\(1 \frac{4}{6}\)[/tex] simplifies to:
[tex]\[ 1 \frac{4}{6} = 1 + \frac{4}{6} = \frac{6}{6} + \frac{4}{6} = \frac{10}{6} \][/tex]
2. Subtract the Fractions:
- We need a common denominator to subtract [tex]\(\frac{17}{5}\)[/tex] and [tex]\(\frac{10}{6}\)[/tex].
- The least common multiple of 5 and 6 is 30, so convert both fractions:
[tex]\(\frac{17}{5}\)[/tex] converted to a denominator of 30:
[tex]\[ \frac{17}{5} = \frac{17 \times 6}{30} = \frac{102}{30} \][/tex]
[tex]\(\frac{10}{6}\)[/tex] converted to a denominator of 30:
[tex]\[ \frac{10}{6} = \frac{10 \times 5}{30} = \frac{50}{30} \][/tex]
- Subtract the fractions:
[tex]\[ \frac{102}{30} - \frac{50}{30} = \frac{102 - 50}{30} = \frac{52}{30} \][/tex]
3. Convert the Result Back to a Mixed Number:
- Simplify [tex]\(\frac{52}{30}\)[/tex] by dividing both numerator and denominator by their greatest common divisor 2:
[tex]\[ \frac{52}{30} = \frac{26}{15} \][/tex]
- Convert [tex]\(\frac{26}{15}\)[/tex] to a mixed number:
[tex]\[ 26 \div 15 = 1 \text{ remainder } 11 \][/tex]
So, [tex]\(\frac{26}{15} = 1 \frac{11}{15}\)[/tex].
4. Scale to a Common Denominator of 30 for Comparison:
- Convert [tex]\(1 \frac{11}{15}\)[/tex] to have a denominator of 30:
[tex]\[ 1 \frac{11}{15} = 1 + \frac{11 \times 2}{30} = 1 \frac{22}{30} \][/tex]
5. Identify Equivalent Options:
- Comparing with given options: [tex]\((1, 22, 30)\)[/tex] matches the mixed number [tex]\(1 \frac{22}{30}\)[/tex].
Thus, the card that is equivalent to [tex]\(3 \frac{2}{5} - 1 \frac{4}{6}\)[/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 mixed number: [tex]\(3 \frac{2}{5}\)[/tex] can be converted:
[tex]\[ 3 \frac{2}{5} = 3 + \frac{2}{5} = \frac{15}{5} + \frac{2}{5} = \frac{17}{5} \][/tex]
- Second mixed number: [tex]\(1 \frac{4}{6}\)[/tex] simplifies to:
[tex]\[ 1 \frac{4}{6} = 1 + \frac{4}{6} = \frac{6}{6} + \frac{4}{6} = \frac{10}{6} \][/tex]
2. Subtract the Fractions:
- We need a common denominator to subtract [tex]\(\frac{17}{5}\)[/tex] and [tex]\(\frac{10}{6}\)[/tex].
- The least common multiple of 5 and 6 is 30, so convert both fractions:
[tex]\(\frac{17}{5}\)[/tex] converted to a denominator of 30:
[tex]\[ \frac{17}{5} = \frac{17 \times 6}{30} = \frac{102}{30} \][/tex]
[tex]\(\frac{10}{6}\)[/tex] converted to a denominator of 30:
[tex]\[ \frac{10}{6} = \frac{10 \times 5}{30} = \frac{50}{30} \][/tex]
- Subtract the fractions:
[tex]\[ \frac{102}{30} - \frac{50}{30} = \frac{102 - 50}{30} = \frac{52}{30} \][/tex]
3. Convert the Result Back to a Mixed Number:
- Simplify [tex]\(\frac{52}{30}\)[/tex] by dividing both numerator and denominator by their greatest common divisor 2:
[tex]\[ \frac{52}{30} = \frac{26}{15} \][/tex]
- Convert [tex]\(\frac{26}{15}\)[/tex] to a mixed number:
[tex]\[ 26 \div 15 = 1 \text{ remainder } 11 \][/tex]
So, [tex]\(\frac{26}{15} = 1 \frac{11}{15}\)[/tex].
4. Scale to a Common Denominator of 30 for Comparison:
- Convert [tex]\(1 \frac{11}{15}\)[/tex] to have a denominator of 30:
[tex]\[ 1 \frac{11}{15} = 1 + \frac{11 \times 2}{30} = 1 \frac{22}{30} \][/tex]
5. Identify Equivalent Options:
- Comparing with given options: [tex]\((1, 22, 30)\)[/tex] matches the mixed number [tex]\(1 \frac{22}{30}\)[/tex].
Thus, the card that is equivalent to [tex]\(3 \frac{2}{5} - 1 \frac{4}{6}\)[/tex] is:
- [tex]\(1 \frac{22}{30}\)[/tex]
Therefore, the correct answer is [tex]\(1 \frac{22}{30}\)[/tex].
