College

Which cards are equivalent to [tex]3 \frac{2}{5} - 1 \frac{4}{6}[/tex]? Choose ALL the correct answers.

A. [tex]3 \frac{2}{30} - 1 \frac{4}{30}[/tex]
B. [tex]3 \frac{12}{30} - 1 \frac{20}{30}[/tex]
C. [tex]3 \frac{10}{30} - 1 \frac{24}{30}[/tex]
D. [tex]1 \frac{16}{30}[/tex]
E. [tex]1 \frac{22}{30}[/tex]
F. [tex]1 \frac{28}{30}[/tex]

Answer :

Sure, let's solve this problem step-by-step:

To determine which cards are equivalent to [tex]\(3 \frac{2}{5} - 1 \frac{4}{6}\)[/tex], we first need to convert the mixed numbers to improper fractions and perform the subtraction.

1. Convert to Improper Fractions:

- [tex]\(3 \frac{2}{5} = \frac{15}{5} + \frac{2}{5} = \frac{17}{5}\)[/tex]
- [tex]\(1 \frac{4}{6} = \frac{6}{6} + \frac{4}{6} = \frac{10}{6}\)[/tex]

2. Perform the Subtraction:

To subtract the improper fractions, we need a common denominator:

- The least common denominator (LCD) for 5 and 6 is 30.

- Convert [tex]\(\frac{17}{5}\)[/tex] to a fraction with a denominator of 30:
[tex]\[
\frac{17}{5} = \frac{17 \times 6}{5 \times 6} = \frac{102}{30}
\][/tex]

- Convert [tex]\(\frac{10}{6}\)[/tex] to a fraction with a denominator of 30:
[tex]\[
\frac{10}{6} = \frac{10 \times 5}{6 \times 5} = \frac{50}{30}
\][/tex]

- Subtract the fractions:
[tex]\[
\frac{102}{30} - \frac{50}{30} = \frac{52}{30}
\][/tex]

3. Simplify the Difference:

[tex]\(\frac{52}{30}\)[/tex] can be simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 2:
[tex]\[
\frac{52}{30} = \frac{26}{15}
\][/tex]

Convert [tex]\(\frac{26}{15}\)[/tex] back to a mixed number:
[tex]\[
26 \div 15 = 1 \quad \text{remainder: } 11
\][/tex]
This gives [tex]\(1 \frac{11}{15}\)[/tex].

4. Compare with Given Cards:

Now, look for the equivalent result [tex]\(1 \frac{11}{15}\)[/tex] among the options provided. We convert each option to a decimal to check:

- [tex]\(3 \frac{2}{30}-1 \frac{4}{30} = 1 \frac{28}{30}\)[/tex]
- [tex]\(3 \frac{12}{30}-1 \frac{20}{30} = 1 \frac{22}{30}\)[/tex]
- [tex]\(3 \frac{10}{30}-1 \frac{24}{30} = 1 \frac{16}{30}\)[/tex]
- [tex]\(1 \frac{16}{30}\)[/tex]
- [tex]\(1 \frac{22}{30}\)[/tex]
- [tex]\(1 \frac{28}{30}\)[/tex]

None of these options match with [tex]\(1 \frac{11}{15}\)[/tex], so it looks like no card given in the options is equivalent to the subtraction result [tex]\(1 \frac{11}{15}\)[/tex].

I hope this helps! If you need further assistance, feel free to ask.