Answer :
To solve the problem of finding [tex]\(\frac{14}{15} \div \frac{7}{5}\)[/tex], we need to understand that dividing by a fraction is the same as multiplying by its reciprocal. Therefore, we can rewrite the division problem:
[tex]\[
\frac{14}{15} \div \frac{7}{5} = \frac{14}{15} \times \frac{5}{7}
\][/tex]
Now, let's go through each option to see which statements show correct reasoning for this calculation:
A. Multiplying [tex]\(\frac{14}{15}\)[/tex] by 5 and then by [tex]\(\frac{1}{7}\)[/tex].
This is correct because multiplying by 5 is part of finding the reciprocal of [tex]\(\frac{7}{5}\)[/tex], then [tex]\(\frac{1}{7}\)[/tex] adjusts the second fraction correctly.
B. Dividing [tex]\(\frac{14}{15}\)[/tex] by 5, and then multiplying by [tex]\(\frac{1}{7}\)[/tex].
This isn't correct. Dividing by 5 is not part of getting the reciprocal of [tex]\(\frac{7}{5}\)[/tex].
C. Multiplying [tex]\(\frac{14}{15}\)[/tex] by 7, and then multiplying by [tex]\(\frac{1}{5}\)[/tex].
This isn't correct because it doesn't represent multiplying by the reciprocal [tex]\(\frac{5}{7}\)[/tex].
D. Multiplying [tex]\(\frac{14}{15}\)[/tex] by 5 and then dividing by 7.
This is correct, as multiplying by 5 and dividing by 7 is equivalent to multiplying by [tex]\(\frac{5}{7}\)[/tex].
E. Multiplying [tex]\(\frac{15}{14}\)[/tex] by 7 and then dividing by 5.
This isn't correct. This statement involves flipping the first fraction, which is not how division is solved here.
Based on the correct reasoning, the solution shows that option D is correct for solving [tex]\(\frac{14}{15} \div \frac{7}{5}\)[/tex].
[tex]\[
\frac{14}{15} \div \frac{7}{5} = \frac{14}{15} \times \frac{5}{7}
\][/tex]
Now, let's go through each option to see which statements show correct reasoning for this calculation:
A. Multiplying [tex]\(\frac{14}{15}\)[/tex] by 5 and then by [tex]\(\frac{1}{7}\)[/tex].
This is correct because multiplying by 5 is part of finding the reciprocal of [tex]\(\frac{7}{5}\)[/tex], then [tex]\(\frac{1}{7}\)[/tex] adjusts the second fraction correctly.
B. Dividing [tex]\(\frac{14}{15}\)[/tex] by 5, and then multiplying by [tex]\(\frac{1}{7}\)[/tex].
This isn't correct. Dividing by 5 is not part of getting the reciprocal of [tex]\(\frac{7}{5}\)[/tex].
C. Multiplying [tex]\(\frac{14}{15}\)[/tex] by 7, and then multiplying by [tex]\(\frac{1}{5}\)[/tex].
This isn't correct because it doesn't represent multiplying by the reciprocal [tex]\(\frac{5}{7}\)[/tex].
D. Multiplying [tex]\(\frac{14}{15}\)[/tex] by 5 and then dividing by 7.
This is correct, as multiplying by 5 and dividing by 7 is equivalent to multiplying by [tex]\(\frac{5}{7}\)[/tex].
E. Multiplying [tex]\(\frac{15}{14}\)[/tex] by 7 and then dividing by 5.
This isn't correct. This statement involves flipping the first fraction, which is not how division is solved here.
Based on the correct reasoning, the solution shows that option D is correct for solving [tex]\(\frac{14}{15} \div \frac{7}{5}\)[/tex].
