Answer :
To solve the problem [tex]\(\frac{14}{15} \div \frac{7}{5}\)[/tex], we need to remember that dividing by a fraction is the same as multiplying by its reciprocal. So, [tex]\(\frac{14}{15} \div \frac{7}{5}\)[/tex] is equivalent to [tex]\(\frac{14}{15} \times \frac{5}{7}\)[/tex].
Let’s evaluate each option given for correct reasoning:
A. Multiply [tex]\(\frac{14}{15}\)[/tex] by 5, and then by [tex]\(\frac{1}{7}\)[/tex].
- This statement is correct because it represents the multiplication by the reciprocal of [tex]\(\frac{7}{5}\)[/tex], which is [tex]\(\frac{5}{7}\)[/tex].
- First, multiplying [tex]\(\frac{14}{15}\)[/tex] by 5 gives us [tex]\(\frac{70}{15}\)[/tex].
- Then, multiplying by [tex]\(\frac{1}{7}\)[/tex] gives the result [tex]\(\frac{70}{105} = \frac{2}{3}\)[/tex] when simplified.
- This matches the correct reasoning because multiplying by 5 and then by [tex]\(\frac{1}{7}\)[/tex] gives the same result.
B. Divide [tex]\(\frac{14}{15}\)[/tex] by 5, then multiply by [tex]\(\frac{1}{7}\)[/tex].
- This method involves dividing first, which does not match the approach of multiplying by the reciprocal.
- Dividing by 5 gives us [tex]\(\frac{14}{75}\)[/tex], and then multiplying by [tex]\(\frac{1}{7}\)[/tex] results in [tex]\(\frac{14}{525} = \frac{2}{75}\)[/tex] after simplification.
- This result does not match the correct reasoning.
C. Multiply [tex]\(\frac{14}{15}\)[/tex] by 7, then multiply by [tex]\(\frac{1}{5}\)[/tex].
- This suggests multiplying by components of [tex]\(\frac{7}{5}\)[/tex] directly, which does not follow multiplying by the reciprocal.
- Multiplying by 7 first gives [tex]\(\frac{98}{15}\)[/tex], and then multiplying by [tex]\(\frac{1}{5}\)[/tex] gives [tex]\(\frac{98}{75}\)[/tex], which simplifies to more than 1, incorrect for this operation.
D. Multiply [tex]\(\frac{14}{15}\)[/tex] by 5, then divide by 7.
- This is correct because it represents multiplying by [tex]\(\frac{5}{7}\)[/tex].
- Multiplying by 5 first gives [tex]\(\frac{70}{15}\)[/tex], and dividing by 7 results in [tex]\(\frac{70}{105} = \frac{2}{3}\)[/tex].
- This result matches the correct calculation of the division operation.
E. Multiply [tex]\(\frac{15}{14}\)[/tex] by 7 and then divide by 5.
- This involves incorrect fractions and does not represent the division of [tex]\(\frac{14}{15}\)[/tex] by [tex]\(\frac{7}{5}\)[/tex].
- The manipulation does not correctly translate to the original problem and thus does not provide the correct method or result.
Based on the reasoning above, the correct statements for finding [tex]\(\frac{14}{15} \div \frac{7}{5}\)[/tex] are A and D, which both lead to the correct result of [tex]\(\frac{2}{3}\)[/tex].
Let’s evaluate each option given for correct reasoning:
A. Multiply [tex]\(\frac{14}{15}\)[/tex] by 5, and then by [tex]\(\frac{1}{7}\)[/tex].
- This statement is correct because it represents the multiplication by the reciprocal of [tex]\(\frac{7}{5}\)[/tex], which is [tex]\(\frac{5}{7}\)[/tex].
- First, multiplying [tex]\(\frac{14}{15}\)[/tex] by 5 gives us [tex]\(\frac{70}{15}\)[/tex].
- Then, multiplying by [tex]\(\frac{1}{7}\)[/tex] gives the result [tex]\(\frac{70}{105} = \frac{2}{3}\)[/tex] when simplified.
- This matches the correct reasoning because multiplying by 5 and then by [tex]\(\frac{1}{7}\)[/tex] gives the same result.
B. Divide [tex]\(\frac{14}{15}\)[/tex] by 5, then multiply by [tex]\(\frac{1}{7}\)[/tex].
- This method involves dividing first, which does not match the approach of multiplying by the reciprocal.
- Dividing by 5 gives us [tex]\(\frac{14}{75}\)[/tex], and then multiplying by [tex]\(\frac{1}{7}\)[/tex] results in [tex]\(\frac{14}{525} = \frac{2}{75}\)[/tex] after simplification.
- This result does not match the correct reasoning.
C. Multiply [tex]\(\frac{14}{15}\)[/tex] by 7, then multiply by [tex]\(\frac{1}{5}\)[/tex].
- This suggests multiplying by components of [tex]\(\frac{7}{5}\)[/tex] directly, which does not follow multiplying by the reciprocal.
- Multiplying by 7 first gives [tex]\(\frac{98}{15}\)[/tex], and then multiplying by [tex]\(\frac{1}{5}\)[/tex] gives [tex]\(\frac{98}{75}\)[/tex], which simplifies to more than 1, incorrect for this operation.
D. Multiply [tex]\(\frac{14}{15}\)[/tex] by 5, then divide by 7.
- This is correct because it represents multiplying by [tex]\(\frac{5}{7}\)[/tex].
- Multiplying by 5 first gives [tex]\(\frac{70}{15}\)[/tex], and dividing by 7 results in [tex]\(\frac{70}{105} = \frac{2}{3}\)[/tex].
- This result matches the correct calculation of the division operation.
E. Multiply [tex]\(\frac{15}{14}\)[/tex] by 7 and then divide by 5.
- This involves incorrect fractions and does not represent the division of [tex]\(\frac{14}{15}\)[/tex] by [tex]\(\frac{7}{5}\)[/tex].
- The manipulation does not correctly translate to the original problem and thus does not provide the correct method or result.
Based on the reasoning above, the correct statements for finding [tex]\(\frac{14}{15} \div \frac{7}{5}\)[/tex] are A and D, which both lead to the correct result of [tex]\(\frac{2}{3}\)[/tex].
