Answer :
To solve the problem of finding [tex]\(\frac{14}{15} \div \frac{7}{5}\)[/tex], we need to divide one fraction by another. Dividing by a fraction is the same as multiplying by its reciprocal. So we can rewrite the division problem as:
[tex]\[
\frac{14}{15} \div \frac{7}{5} = \frac{14}{15} \times \frac{5}{7}
\][/tex]
Now, let's evaluate the statements to see which ones represent correct reasoning for finding this result:
a. Multiplying [tex]\(\frac{14}{15}\)[/tex] by 5 and then by [tex]\(\frac{1}{7}\)[/tex]:
This means calculating:
[tex]\[
\left(\frac{14}{15} \times 5 \right) \times \frac{1}{7}
\][/tex]
This is equivalent to multiplying by [tex]\(\frac{5}{7}\)[/tex], since multiplying [tex]\(\frac{14}{15}\)[/tex] by 5 and then by [tex]\(\frac{1}{7}\)[/tex] gives:
[tex]\[
\frac{14 \times 5}{15} \times \frac{1}{7} = \frac{70}{15} \times \frac{1}{7} = \frac{70}{105} = \frac{2}{3}
\][/tex]
So, statement (a) corresponds to the correct calculation.
b. Dividing [tex]\(\frac{14}{15}\)[/tex] by 5, and then multiplying by [tex]\(\frac{1}{7}\)[/tex]:
This means calculating:
[tex]\[
\left(\frac{14}{15} \div 5 \right) \times \frac{1}{7} = \frac{14}{15 \times 5} \times \frac{1}{7} = \frac{14}{75} \times \frac{1}{7} = \frac{14}{525}
\][/tex]
This does not match the calculation of [tex]\(\frac{2}{3}\)[/tex], so this statement is not correct.
c. Multiplying [tex]\(\frac{14}{15}\)[/tex] by 7, and then multiplying by [tex]\(\frac{1}{5}\)[/tex]:
This means calculating:
[tex]\[
\left(\frac{14}{15} \times 7 \right) \times \frac{1}{5} = \frac{98}{15} \times \frac{1}{5} = \frac{98}{75}
\][/tex]
This does not correctly represent dividing by [tex]\(\frac{7}{5}\)[/tex], so it is not correct.
d. Multiplying [tex]\(\frac{14}{15}\)[/tex] by 5 and then dividing by 7:
This means calculating:
[tex]\[
\left(\frac{14}{15} \times 5 \right) \div 7 = \frac{70}{15} \div 7 = \frac{70}{15 \times 7} = \frac{70}{105} = \frac{2}{3}
\][/tex]
This reproduction is indeed equivalent to multiplying by the reciprocal [tex]\(\frac{5}{7}\)[/tex], so it is correct.
e. Multiplying [tex]\(\frac{15}{14}\)[/tex] by 7 and then dividing by 5:
This approach is not involved in correctly calculating [tex]\(\frac{14}{15} \times \frac{5}{7}\)[/tex] as it seems to reverse the order of fractions, and approaches a reciprocal form incorrectly.
Thus, the statements with correct reasoning are:
- a. Multiplying [tex]\(\frac{14}{15}\)[/tex] by 5 and then by [tex]\(\frac{1}{7}\)[/tex]
- d. Multiplying [tex]\(\frac{14}{15}\)[/tex] by 5 and then dividing by 7
[tex]\[
\frac{14}{15} \div \frac{7}{5} = \frac{14}{15} \times \frac{5}{7}
\][/tex]
Now, let's evaluate the statements to see which ones represent correct reasoning for finding this result:
a. Multiplying [tex]\(\frac{14}{15}\)[/tex] by 5 and then by [tex]\(\frac{1}{7}\)[/tex]:
This means calculating:
[tex]\[
\left(\frac{14}{15} \times 5 \right) \times \frac{1}{7}
\][/tex]
This is equivalent to multiplying by [tex]\(\frac{5}{7}\)[/tex], since multiplying [tex]\(\frac{14}{15}\)[/tex] by 5 and then by [tex]\(\frac{1}{7}\)[/tex] gives:
[tex]\[
\frac{14 \times 5}{15} \times \frac{1}{7} = \frac{70}{15} \times \frac{1}{7} = \frac{70}{105} = \frac{2}{3}
\][/tex]
So, statement (a) corresponds to the correct calculation.
b. Dividing [tex]\(\frac{14}{15}\)[/tex] by 5, and then multiplying by [tex]\(\frac{1}{7}\)[/tex]:
This means calculating:
[tex]\[
\left(\frac{14}{15} \div 5 \right) \times \frac{1}{7} = \frac{14}{15 \times 5} \times \frac{1}{7} = \frac{14}{75} \times \frac{1}{7} = \frac{14}{525}
\][/tex]
This does not match the calculation of [tex]\(\frac{2}{3}\)[/tex], so this statement is not correct.
c. Multiplying [tex]\(\frac{14}{15}\)[/tex] by 7, and then multiplying by [tex]\(\frac{1}{5}\)[/tex]:
This means calculating:
[tex]\[
\left(\frac{14}{15} \times 7 \right) \times \frac{1}{5} = \frac{98}{15} \times \frac{1}{5} = \frac{98}{75}
\][/tex]
This does not correctly represent dividing by [tex]\(\frac{7}{5}\)[/tex], so it is not correct.
d. Multiplying [tex]\(\frac{14}{15}\)[/tex] by 5 and then dividing by 7:
This means calculating:
[tex]\[
\left(\frac{14}{15} \times 5 \right) \div 7 = \frac{70}{15} \div 7 = \frac{70}{15 \times 7} = \frac{70}{105} = \frac{2}{3}
\][/tex]
This reproduction is indeed equivalent to multiplying by the reciprocal [tex]\(\frac{5}{7}\)[/tex], so it is correct.
e. Multiplying [tex]\(\frac{15}{14}\)[/tex] by 7 and then dividing by 5:
This approach is not involved in correctly calculating [tex]\(\frac{14}{15} \times \frac{5}{7}\)[/tex] as it seems to reverse the order of fractions, and approaches a reciprocal form incorrectly.
Thus, the statements with correct reasoning are:
- a. Multiplying [tex]\(\frac{14}{15}\)[/tex] by 5 and then by [tex]\(\frac{1}{7}\)[/tex]
- d. Multiplying [tex]\(\frac{14}{15}\)[/tex] by 5 and then dividing by 7
