Answer :
To solve the problem of finding which statements show correct reasoning for calculating [tex]\(\frac{14}{15} \div \frac{7}{5}\)[/tex], we need to understand that dividing by a fraction is equivalent to multiplying by its reciprocal. So, the reciprocal of [tex]\(\frac{7}{5}\)[/tex] is [tex]\(\frac{5}{7}\)[/tex]. Therefore, [tex]\(\frac{14}{15} \div \frac{7}{5}\)[/tex] is the same as [tex]\(\frac{14}{15} \times \frac{5}{7}\)[/tex].
Let's check each statement to see which one correctly represents this operation:
### Statement (A):
Multiplying [tex]\(\frac{14}{15}\)[/tex] by 5 and then by [tex]\(\frac{1}{7}\)[/tex].
This can be written as:
[tex]\[
\frac{14}{15} \times 5 \times \frac{1}{7} = \frac{14 \times 5}{15 \times 7} = \frac{70}{105} = \frac{2}{3}
\][/tex]
This matches our operation [tex]\(\frac{14}{15} \times \frac{5}{7}\)[/tex]. So statement (A) is correct.
### Statement (B):
Dividing [tex]\(\frac{14}{15}\)[/tex] by 5, and then multiplying by [tex]\(\frac{1}{7}\)[/tex].
This can be written as:
[tex]\[
\left(\frac{14}{15} \div 5\right) \times \frac{1}{7} = \frac{\frac{14}{15}}{5} \times \frac{1}{7} = \frac{14}{75} \times \frac{1}{7} = \frac{14}{525} = \frac{2}{75}
\][/tex]
This is not equivalent to [tex]\(\frac{14}{15} \times \frac{5}{7}\)[/tex], so statement (B) is incorrect.
### Statement (C):
Multiplying [tex]\(\frac{14}{15}\)[/tex] by 7, and then multiplying by [tex]\(\frac{1}{5}\)[/tex].
This can be written as:
[tex]\[
\frac{14}{15} \times 7 \times \frac{1}{5} = \frac{14 \times 7}{15 \times 5} = \frac{98}{75}
\][/tex]
This does not match the operation of [tex]\(\frac{14}{15} \times \frac{5}{7}\)[/tex], so statement (C) is incorrect.
### Statement (D):
Multiplying [tex]\(\frac{14}{15}\)[/tex] by 5 and then dividing by 7.
This can be written as:
[tex]\[
\left(\frac{14}{15} \times 5\right) \div 7 = \frac{14 \times 5}{15} \div 7 = \frac{70}{15} \div 7 = \frac{70}{105} = \frac{2}{3}
\][/tex]
This correctly represents the operation [tex]\(\frac{14}{15} \times \frac{5}{7}\)[/tex], so statement (D) is correct.
### Statement (E):
Multiplying [tex]\(\frac{15}{14}\)[/tex] by 7 and then dividing by 5.
This can be written as:
[tex]\[
\left(\frac{15}{14} \times 7\right) \div 5 = \frac{105}{14} \div 5 = \frac{105}{70} = \frac{3}{2}
\][/tex]
This does not match [tex]\(\frac{14}{15} \times \frac{5}{7}\)[/tex], so statement (E) is incorrect.
Thus, the correct statements that show the reasoning for finding [tex]\(\frac{14}{15} \div \frac{7}{5}\)[/tex] are (A) and (D).
Let's check each statement to see which one correctly represents this operation:
### Statement (A):
Multiplying [tex]\(\frac{14}{15}\)[/tex] by 5 and then by [tex]\(\frac{1}{7}\)[/tex].
This can be written as:
[tex]\[
\frac{14}{15} \times 5 \times \frac{1}{7} = \frac{14 \times 5}{15 \times 7} = \frac{70}{105} = \frac{2}{3}
\][/tex]
This matches our operation [tex]\(\frac{14}{15} \times \frac{5}{7}\)[/tex]. So statement (A) is correct.
### Statement (B):
Dividing [tex]\(\frac{14}{15}\)[/tex] by 5, and then multiplying by [tex]\(\frac{1}{7}\)[/tex].
This can be written as:
[tex]\[
\left(\frac{14}{15} \div 5\right) \times \frac{1}{7} = \frac{\frac{14}{15}}{5} \times \frac{1}{7} = \frac{14}{75} \times \frac{1}{7} = \frac{14}{525} = \frac{2}{75}
\][/tex]
This is not equivalent to [tex]\(\frac{14}{15} \times \frac{5}{7}\)[/tex], so statement (B) is incorrect.
### Statement (C):
Multiplying [tex]\(\frac{14}{15}\)[/tex] by 7, and then multiplying by [tex]\(\frac{1}{5}\)[/tex].
This can be written as:
[tex]\[
\frac{14}{15} \times 7 \times \frac{1}{5} = \frac{14 \times 7}{15 \times 5} = \frac{98}{75}
\][/tex]
This does not match the operation of [tex]\(\frac{14}{15} \times \frac{5}{7}\)[/tex], so statement (C) is incorrect.
### Statement (D):
Multiplying [tex]\(\frac{14}{15}\)[/tex] by 5 and then dividing by 7.
This can be written as:
[tex]\[
\left(\frac{14}{15} \times 5\right) \div 7 = \frac{14 \times 5}{15} \div 7 = \frac{70}{15} \div 7 = \frac{70}{105} = \frac{2}{3}
\][/tex]
This correctly represents the operation [tex]\(\frac{14}{15} \times \frac{5}{7}\)[/tex], so statement (D) is correct.
### Statement (E):
Multiplying [tex]\(\frac{15}{14}\)[/tex] by 7 and then dividing by 5.
This can be written as:
[tex]\[
\left(\frac{15}{14} \times 7\right) \div 5 = \frac{105}{14} \div 5 = \frac{105}{70} = \frac{3}{2}
\][/tex]
This does not match [tex]\(\frac{14}{15} \times \frac{5}{7}\)[/tex], so statement (E) is incorrect.
Thus, the correct statements that show the reasoning for finding [tex]\(\frac{14}{15} \div \frac{7}{5}\)[/tex] are (A) and (D).
