Answer :
To solve the expression [tex]\(3 \frac{2}{5} - 1 \frac{4}{6}\)[/tex] and determine which options are equivalent, follow these steps:
1. Convert Mixed Numbers to Improper Fractions:
- For [tex]\(3 \frac{2}{5}\)[/tex]:
[tex]\[
3 \frac{2}{5} = \frac{3 \times 5 + 2}{5} = \frac{15 + 2}{5} = \frac{17}{5}
\][/tex]
- For [tex]\(1 \frac{4}{6}\)[/tex] (which simplifies to [tex]\(1 \frac{2}{3}\)[/tex]):
[tex]\[
1 \frac{4}{6} = \frac{1 \times 6 + 4}{6} = \frac{6 + 4}{6} = \frac{10}{6} = \frac{5}{3}
\][/tex]
2. Perform the Subtraction:
Subtract the two fractions found above:
[tex]\[
\frac{17}{5} - \frac{5}{3}
\][/tex]
To subtract these, find a common denominator, which is 15 in this case.
[tex]\[
\frac{17}{5} = \frac{17 \times 3}{5 \times 3} = \frac{51}{15}
\][/tex]
[tex]\[
\frac{5}{3} = \frac{5 \times 5}{3 \times 5} = \frac{25}{15}
\][/tex]
Subtract the fractions:
[tex]\[
\frac{51}{15} - \frac{25}{15} = \frac{26}{15}
\][/tex]
3. Compare with Given Options:
- To be equivalent, the difference after simplifying must match [tex]\(\frac{26}{15}\)[/tex].
From the listed cards:
- Option [tex]\(3 \frac{12}{30} - 1 \frac{20}{30}\)[/tex]:
Simplify and find the difference:
[tex]\[
3 \frac{12}{30} = \frac{3 \times 30 + 12}{30} = \frac{102}{30}
\][/tex]
[tex]\[
1 \frac{20}{30} = \frac{1 \times 30 + 20}{30} = \frac{50}{30}
\][/tex]
Subtract:
[tex]\[
\frac{102}{30} - \frac{50}{30} = \frac{52}{30} = \frac{26}{15}
\][/tex]
- Option [tex]\(1 \frac{22}{30}\)[/tex]:
Check the fraction:
[tex]\[
1 \frac{22}{30} = \frac{1 \times 30 + 22}{30} = \frac{52}{30} = \frac{26}{15}
\][/tex]
The cards [tex]\(3 \frac{12}{30} - 1 \frac{20}{30}\)[/tex] and [tex]\(1 \frac{22}{30}\)[/tex] match the result of [tex]\(\frac{26}{15}\)[/tex], making them equivalent expressions.
Therefore, these are the correct options:
- [tex]\(3 \frac{12}{30} - 1 \frac{20}{30}\)[/tex]
- [tex]\(1 \frac{22}{30}\)[/tex]
1. Convert Mixed Numbers to Improper Fractions:
- For [tex]\(3 \frac{2}{5}\)[/tex]:
[tex]\[
3 \frac{2}{5} = \frac{3 \times 5 + 2}{5} = \frac{15 + 2}{5} = \frac{17}{5}
\][/tex]
- For [tex]\(1 \frac{4}{6}\)[/tex] (which simplifies to [tex]\(1 \frac{2}{3}\)[/tex]):
[tex]\[
1 \frac{4}{6} = \frac{1 \times 6 + 4}{6} = \frac{6 + 4}{6} = \frac{10}{6} = \frac{5}{3}
\][/tex]
2. Perform the Subtraction:
Subtract the two fractions found above:
[tex]\[
\frac{17}{5} - \frac{5}{3}
\][/tex]
To subtract these, find a common denominator, which is 15 in this case.
[tex]\[
\frac{17}{5} = \frac{17 \times 3}{5 \times 3} = \frac{51}{15}
\][/tex]
[tex]\[
\frac{5}{3} = \frac{5 \times 5}{3 \times 5} = \frac{25}{15}
\][/tex]
Subtract the fractions:
[tex]\[
\frac{51}{15} - \frac{25}{15} = \frac{26}{15}
\][/tex]
3. Compare with Given Options:
- To be equivalent, the difference after simplifying must match [tex]\(\frac{26}{15}\)[/tex].
From the listed cards:
- Option [tex]\(3 \frac{12}{30} - 1 \frac{20}{30}\)[/tex]:
Simplify and find the difference:
[tex]\[
3 \frac{12}{30} = \frac{3 \times 30 + 12}{30} = \frac{102}{30}
\][/tex]
[tex]\[
1 \frac{20}{30} = \frac{1 \times 30 + 20}{30} = \frac{50}{30}
\][/tex]
Subtract:
[tex]\[
\frac{102}{30} - \frac{50}{30} = \frac{52}{30} = \frac{26}{15}
\][/tex]
- Option [tex]\(1 \frac{22}{30}\)[/tex]:
Check the fraction:
[tex]\[
1 \frac{22}{30} = \frac{1 \times 30 + 22}{30} = \frac{52}{30} = \frac{26}{15}
\][/tex]
The cards [tex]\(3 \frac{12}{30} - 1 \frac{20}{30}\)[/tex] and [tex]\(1 \frac{22}{30}\)[/tex] match the result of [tex]\(\frac{26}{15}\)[/tex], making them equivalent expressions.
Therefore, these are the correct options:
- [tex]\(3 \frac{12}{30} - 1 \frac{20}{30}\)[/tex]
- [tex]\(1 \frac{22}{30}\)[/tex]
