Answer :
To solve the problem of finding which cards are equivalent to [tex]\(3 \frac{2}{5} - 1 \frac{4}{6}\)[/tex], let's go through the solution step by step:
1. Convert Mixed Numbers to Improper Fractions:
- [tex]\(3 \frac{2}{5}\)[/tex] can be written as an improper fraction:
- Multiply the whole number by the denominator and add the numerator:
[tex]\[
3 \frac{2}{5} = \frac{3 \times 5 + 2}{5} = \frac{15 + 2}{5} = \frac{17}{5}
\][/tex]
- [tex]\(1 \frac{4}{6}\)[/tex] can be simplified first:
- The fraction [tex]\(\frac{4}{6}\)[/tex] simplifies to [tex]\(\frac{2}{3}\)[/tex].
- Convert the mixed number to an improper fraction:
[tex]\[
1 \frac{4}{6} = 1 \frac{2}{3} = \frac{1 \times 3 + 2}{3} = \frac{3 + 2}{3} = \frac{5}{3}
\][/tex]
2. Find a Common Denominator:
- To subtract these fractions, convert them to have a common denominator. The least common denominator between 5 and 3 is 15.
- Convert [tex]\(\frac{17}{5}\)[/tex] to an equivalent fraction with 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 an equivalent fraction with a denominator of 15:
[tex]\[
\frac{5}{3} = \frac{5 \times 5}{3 \times 5} = \frac{25}{15}
\][/tex]
3. Subtract the Fractions:
Now we subtract the two fractions:
[tex]\[
\frac{51}{15} - \frac{25}{15} = \frac{51 - 25}{15} = \frac{26}{15}
\][/tex]
4. Convert the Result to a Mixed Number:
Divide the numerator by the denominator to get a mixed number:
[tex]\[
\frac{26}{15} = 1 \frac{11}{15}
\][/tex]
5. Identify Equivalent Cards:
We need to find the card expressing the same value as [tex]\(1 \frac{11}{15}\)[/tex] using a denominator of 30, since that's used in the cards:
- Convert [tex]\(1 \frac{11}{15}\)[/tex] to a fraction with denominator 30:
- Multiply the numerator and denominator by 2:
[tex]\[
1 \frac{11}{15} = 1 + \frac{11 \times 2}{15 \times 2} = 1 \frac{22}{30}
\][/tex]
Therefore, the card equivalent to [tex]\(1 \frac{11}{15}\)[/tex] is:
- [tex]\(1 \frac{22}{30}\)[/tex]
The correct equivalent card from the options is [tex]\(1 \frac{22}{30}\)[/tex].
1. Convert Mixed Numbers to Improper Fractions:
- [tex]\(3 \frac{2}{5}\)[/tex] can be written as an improper fraction:
- Multiply the whole number by the denominator and add the numerator:
[tex]\[
3 \frac{2}{5} = \frac{3 \times 5 + 2}{5} = \frac{15 + 2}{5} = \frac{17}{5}
\][/tex]
- [tex]\(1 \frac{4}{6}\)[/tex] can be simplified first:
- The fraction [tex]\(\frac{4}{6}\)[/tex] simplifies to [tex]\(\frac{2}{3}\)[/tex].
- Convert the mixed number to an improper fraction:
[tex]\[
1 \frac{4}{6} = 1 \frac{2}{3} = \frac{1 \times 3 + 2}{3} = \frac{3 + 2}{3} = \frac{5}{3}
\][/tex]
2. Find a Common Denominator:
- To subtract these fractions, convert them to have a common denominator. The least common denominator between 5 and 3 is 15.
- Convert [tex]\(\frac{17}{5}\)[/tex] to an equivalent fraction with 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 an equivalent fraction with a denominator of 15:
[tex]\[
\frac{5}{3} = \frac{5 \times 5}{3 \times 5} = \frac{25}{15}
\][/tex]
3. Subtract the Fractions:
Now we subtract the two fractions:
[tex]\[
\frac{51}{15} - \frac{25}{15} = \frac{51 - 25}{15} = \frac{26}{15}
\][/tex]
4. Convert the Result to a Mixed Number:
Divide the numerator by the denominator to get a mixed number:
[tex]\[
\frac{26}{15} = 1 \frac{11}{15}
\][/tex]
5. Identify Equivalent Cards:
We need to find the card expressing the same value as [tex]\(1 \frac{11}{15}\)[/tex] using a denominator of 30, since that's used in the cards:
- Convert [tex]\(1 \frac{11}{15}\)[/tex] to a fraction with denominator 30:
- Multiply the numerator and denominator by 2:
[tex]\[
1 \frac{11}{15} = 1 + \frac{11 \times 2}{15 \times 2} = 1 \frac{22}{30}
\][/tex]
Therefore, the card equivalent to [tex]\(1 \frac{11}{15}\)[/tex] is:
- [tex]\(1 \frac{22}{30}\)[/tex]
The correct equivalent card from the options is [tex]\(1 \frac{22}{30}\)[/tex].
