Answer :
Sure, let's solve this step by step.
We need to determine which of the given options are equivalent to [tex]\( 2 \frac{7}{15} \)[/tex].
### Step 1: Convert the Mixed Number to an Improper Fraction
First, let's convert [tex]\( 2 \frac{7}{15} \)[/tex] to an improper fraction.
[tex]\[
2 \frac{7}{15} = \frac{2 \times 15 + 7}{15} = \frac{30 + 7}{15} = \frac{37}{15}
\][/tex]
So, [tex]\( 2 \frac{7}{15} \)[/tex] is equivalent to [tex]\( \frac{37}{15} \)[/tex].
### Step 2: Convert the Improper Fraction to a Decimal
Next, let's convert [tex]\( \frac{37}{15} \)[/tex] to a decimal.
[tex]\[
\frac{37}{15} \approx 2.46666\ldots
\][/tex]
Since the decimal repeats, it can be written as [tex]\( 2.\overline{466} \)[/tex].
### Step 3: Convert the Decimal to a Percentage
Now, let's convert the decimal [tex]\( 2.\overline{466} \)[/tex] to a percentage.
[tex]\[
2.\overline{466} \times 100 = 246.\overline{666} \%
\][/tex]
### Step 4: Compare with Given Options
We need to see which of the provided options matches [tex]\( \frac{37}{15} \)[/tex], [tex]\( 2.\overline{466} \)[/tex], and [tex]\( 246.\overline{666} \% \)[/tex]:
1. [tex]\(\frac{14}{15}\)[/tex], [tex]\(2.4 \overline{6}\)[/tex], [tex]\(246.6\%\)[/tex]
[tex]\(\frac{14}{15}\)[/tex] is not equal to [tex]\(\frac{37}{15}\)[/tex], so option 1 is incorrect.
2. [tex]\(\frac{37}{15}\)[/tex], [tex]\(2.4 \overline{6}\)[/tex], [tex]\(246.6 \overline{6} \%\)[/tex]
- Improper fraction: [tex]\(\frac{37}{15}\)[/tex] – correct
- Decimal: [tex]\(2.4\overline{6}\)[/tex] – matches our calculated decimal
- Percentage: [tex]\(246.6 \overline{6} \%\)[/tex] – matches our calculated percentage
So, option 2 looks correct.
3. [tex]\(\frac{37}{15}\)[/tex], [tex]\(2.715\)[/tex], [tex]\(271.5\%\)[/tex]
- Improper fraction: [tex]\(\frac{37}{15}\)[/tex] – correct
- Decimal: [tex]\(2.715\)[/tex] – does not match our [tex]\(2.\overline{466}\)[/tex]
- Percentage: [tex]\(271.5\%\)[/tex] – does not match our [tex]\(246.\overline{666}\%\)[/tex]
So, option 3 is incorrect.
4. [tex]\(\frac{37}{15}\)[/tex], [tex]\(2.715\)[/tex], [tex]\(271.5\%\)[/tex]
Same as option 3, so it is incorrect as well.
### Conclusion
The correct choice that matches [tex]\(\frac{37}{15}\)[/tex], [tex]\(2.\overline{466}\)[/tex], and [tex]\(246.\overline{666} \%\)[/tex] is:
[tex]\[
\boxed{\frac{37}{15}, 2.4 \overline{6}, 246.6 \overline{6} \%}
\][/tex] which corresponds to the second option.
We need to determine which of the given options are equivalent to [tex]\( 2 \frac{7}{15} \)[/tex].
### Step 1: Convert the Mixed Number to an Improper Fraction
First, let's convert [tex]\( 2 \frac{7}{15} \)[/tex] to an improper fraction.
[tex]\[
2 \frac{7}{15} = \frac{2 \times 15 + 7}{15} = \frac{30 + 7}{15} = \frac{37}{15}
\][/tex]
So, [tex]\( 2 \frac{7}{15} \)[/tex] is equivalent to [tex]\( \frac{37}{15} \)[/tex].
### Step 2: Convert the Improper Fraction to a Decimal
Next, let's convert [tex]\( \frac{37}{15} \)[/tex] to a decimal.
[tex]\[
\frac{37}{15} \approx 2.46666\ldots
\][/tex]
Since the decimal repeats, it can be written as [tex]\( 2.\overline{466} \)[/tex].
### Step 3: Convert the Decimal to a Percentage
Now, let's convert the decimal [tex]\( 2.\overline{466} \)[/tex] to a percentage.
[tex]\[
2.\overline{466} \times 100 = 246.\overline{666} \%
\][/tex]
### Step 4: Compare with Given Options
We need to see which of the provided options matches [tex]\( \frac{37}{15} \)[/tex], [tex]\( 2.\overline{466} \)[/tex], and [tex]\( 246.\overline{666} \% \)[/tex]:
1. [tex]\(\frac{14}{15}\)[/tex], [tex]\(2.4 \overline{6}\)[/tex], [tex]\(246.6\%\)[/tex]
[tex]\(\frac{14}{15}\)[/tex] is not equal to [tex]\(\frac{37}{15}\)[/tex], so option 1 is incorrect.
2. [tex]\(\frac{37}{15}\)[/tex], [tex]\(2.4 \overline{6}\)[/tex], [tex]\(246.6 \overline{6} \%\)[/tex]
- Improper fraction: [tex]\(\frac{37}{15}\)[/tex] – correct
- Decimal: [tex]\(2.4\overline{6}\)[/tex] – matches our calculated decimal
- Percentage: [tex]\(246.6 \overline{6} \%\)[/tex] – matches our calculated percentage
So, option 2 looks correct.
3. [tex]\(\frac{37}{15}\)[/tex], [tex]\(2.715\)[/tex], [tex]\(271.5\%\)[/tex]
- Improper fraction: [tex]\(\frac{37}{15}\)[/tex] – correct
- Decimal: [tex]\(2.715\)[/tex] – does not match our [tex]\(2.\overline{466}\)[/tex]
- Percentage: [tex]\(271.5\%\)[/tex] – does not match our [tex]\(246.\overline{666}\%\)[/tex]
So, option 3 is incorrect.
4. [tex]\(\frac{37}{15}\)[/tex], [tex]\(2.715\)[/tex], [tex]\(271.5\%\)[/tex]
Same as option 3, so it is incorrect as well.
### Conclusion
The correct choice that matches [tex]\(\frac{37}{15}\)[/tex], [tex]\(2.\overline{466}\)[/tex], and [tex]\(246.\overline{666} \%\)[/tex] is:
[tex]\[
\boxed{\frac{37}{15}, 2.4 \overline{6}, 246.6 \overline{6} \%}
\][/tex] which corresponds to the second option.
