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Answer :
To solve the problem of determining which cards are equivalent to [tex]\(3 \frac{2}{5} - 1 \frac{4}{6}\)[/tex], we need to break down the mixed numbers into fractions, perform the subtraction, and simplify the result.
### Step 1: Convert Mixed Numbers to Improper Fractions
1. Convert [tex]\(3 \frac{2}{5}\)[/tex] to an improper fraction:
- Multiply the whole number by the denominator and add the numerator:
[tex]\[
3 \times 5 + 2 = 15 + 2 = 17
\][/tex]
- So, [tex]\(3 \frac{2}{5}\)[/tex] becomes [tex]\(\frac{17}{5}\)[/tex].
2. Convert [tex]\(1 \frac{4}{6}\)[/tex] to an improper fraction:
- Multiply the whole number by the denominator and add the numerator:
[tex]\[
1 \times 6 + 4 = 6 + 4 = 10
\][/tex]
- So, [tex]\(1 \frac{4}{6}\)[/tex] becomes [tex]\(\frac{10}{6}\)[/tex].
### Step 2: Find a Common Denominator
To subtract [tex]\(\frac{17}{5}\)[/tex] and [tex]\(\frac{10}{6}\)[/tex], we need a common denominator.
- The least common multiple (LCM) of 5 and 6 is 30.
### Step 3: Convert Fractions to the Common Denominator
1. Convert [tex]\(\frac{17}{5}\)[/tex]:
- Multiply the numerator and the denominator by 6:
[tex]\[
\frac{17 \times 6}{5 \times 6} = \frac{102}{30}
\][/tex]
2. Convert [tex]\(\frac{10}{6}\)[/tex]:
- Multiply the numerator and the denominator by 5:
[tex]\[
\frac{10 \times 5}{6 \times 5} = \frac{50}{30}
\][/tex]
### Step 4: Subtract the Fractions
Subtract the fractions:
[tex]\[
\frac{102}{30} - \frac{50}{30} = \frac{52}{30}
\][/tex]
### Step 5: Simplify the Result
Simplify [tex]\(\frac{52}{30}\)[/tex]:
- Find the greatest common divisor (GCD) of 52 and 30, which is 2.
- Divide the numerator and the denominator by 2:
[tex]\[
\frac{52 \div 2}{30 \div 2} = \frac{26}{15}
\][/tex]
Convert [tex]\(\frac{26}{15}\)[/tex] to a mixed number:
- Divide 26 by 15. The quotient is 1 and the remainder is 11.
- So, [tex]\(\frac{26}{15}\)[/tex] becomes [tex]\(1 \frac{11}{15}\)[/tex].
### Step 6: Match with the Given Answers
We need to see which of the given options match this result. We have [tex]\(1 \frac{11}{15}\)[/tex], which is equivalent to:
- Rewriting [tex]\( \frac{11}{15} \)[/tex] as [tex]\( \frac{22}{30} \)[/tex], because [tex]\(\frac{11}{15} = \frac{22}{30}\)[/tex] (by multiplying both the numerator and the denominator by 2).
Hence, the card that matches this result is:
- [tex]\(1 \frac{22}{30}\)[/tex]
So, the correct answer is:
- [tex]\(1 \frac{22}{30}\)[/tex]
### Step 1: Convert Mixed Numbers to Improper Fractions
1. Convert [tex]\(3 \frac{2}{5}\)[/tex] to an improper fraction:
- Multiply the whole number by the denominator and add the numerator:
[tex]\[
3 \times 5 + 2 = 15 + 2 = 17
\][/tex]
- So, [tex]\(3 \frac{2}{5}\)[/tex] becomes [tex]\(\frac{17}{5}\)[/tex].
2. Convert [tex]\(1 \frac{4}{6}\)[/tex] to an improper fraction:
- Multiply the whole number by the denominator and add the numerator:
[tex]\[
1 \times 6 + 4 = 6 + 4 = 10
\][/tex]
- So, [tex]\(1 \frac{4}{6}\)[/tex] becomes [tex]\(\frac{10}{6}\)[/tex].
### Step 2: Find a Common Denominator
To subtract [tex]\(\frac{17}{5}\)[/tex] and [tex]\(\frac{10}{6}\)[/tex], we need a common denominator.
- The least common multiple (LCM) of 5 and 6 is 30.
### Step 3: Convert Fractions to the Common Denominator
1. Convert [tex]\(\frac{17}{5}\)[/tex]:
- Multiply the numerator and the denominator by 6:
[tex]\[
\frac{17 \times 6}{5 \times 6} = \frac{102}{30}
\][/tex]
2. Convert [tex]\(\frac{10}{6}\)[/tex]:
- Multiply the numerator and the denominator by 5:
[tex]\[
\frac{10 \times 5}{6 \times 5} = \frac{50}{30}
\][/tex]
### Step 4: Subtract the Fractions
Subtract the fractions:
[tex]\[
\frac{102}{30} - \frac{50}{30} = \frac{52}{30}
\][/tex]
### Step 5: Simplify the Result
Simplify [tex]\(\frac{52}{30}\)[/tex]:
- Find the greatest common divisor (GCD) of 52 and 30, which is 2.
- Divide the numerator and the denominator by 2:
[tex]\[
\frac{52 \div 2}{30 \div 2} = \frac{26}{15}
\][/tex]
Convert [tex]\(\frac{26}{15}\)[/tex] to a mixed number:
- Divide 26 by 15. The quotient is 1 and the remainder is 11.
- So, [tex]\(\frac{26}{15}\)[/tex] becomes [tex]\(1 \frac{11}{15}\)[/tex].
### Step 6: Match with the Given Answers
We need to see which of the given options match this result. We have [tex]\(1 \frac{11}{15}\)[/tex], which is equivalent to:
- Rewriting [tex]\( \frac{11}{15} \)[/tex] as [tex]\( \frac{22}{30} \)[/tex], because [tex]\(\frac{11}{15} = \frac{22}{30}\)[/tex] (by multiplying both the numerator and the denominator by 2).
Hence, the card that matches this result is:
- [tex]\(1 \frac{22}{30}\)[/tex]
So, the correct answer is:
- [tex]\(1 \frac{22}{30}\)[/tex]
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