Answer :
To solve the division problem [tex]\(\frac{14}{15} \div \frac{7}{5}\)[/tex], we can use the concept of dividing by a fraction, which is equivalent to multiplying by its reciprocal. So, let's go through the reasoning step by step:
1. Find the Reciprocal: The reciprocal of [tex]\(\frac{7}{5}\)[/tex] is [tex]\(\frac{5}{7}\)[/tex].
2. Change Division to Multiplication: We rewrite the division problem [tex]\(\frac{14}{15} \div \frac{7}{5}\)[/tex] as:
[tex]\[
\frac{14}{15} \times \frac{5}{7}
\][/tex]
Now, let's explore each statement to see which ones correctly follow this reasoning:
- Statement A: Multiplying [tex]\(\frac{14}{15}\)[/tex] by 5 and then by [tex]\(\frac{1}{7}\)[/tex].
- When you multiply [tex]\(\frac{14}{15}\)[/tex] by 5 and then by [tex]\(\frac{1}{7}\)[/tex], it is the same as multiplying by [tex]\(\frac{5}{7}\)[/tex].
- This is correct reasoning.
- Statement B: Dividing [tex]\(\frac{14}{15}\)[/tex] by 5, and then multiplying by [tex]\(\frac{1}{7}\)[/tex].
- Dividing by 5 and then multiplying by [tex]\(\frac{1}{7}\)[/tex] doesn’t follow the reciprocal method. This breaks the correct sequence.
- This is incorrect reasoning.
- Statement C: Multiplying [tex]\(\frac{14}{15}\)[/tex] by 7 and then multiplying by [tex]\(\frac{1}{5}\)[/tex].
- This means [tex]\( \frac{14}{15} \times 7 \times \frac{1}{5}\)[/tex] can be rewritten as [tex]\( \frac{14}{15} \times \frac{7}{5} \times \frac{1}{7} \)[/tex], which simplifies back to [tex]\(\frac{14}{15} \times \frac{1}{5}\)[/tex] or [tex]\(\frac{14 \times 1}{15 \times 5}\)[/tex]. They mistakenly reversed the multiplication and division order.
- This is incorrect reasoning.
- Statement D: Multiplying [tex]\(\frac{14}{15}\)[/tex] by 5 and then dividing by 7.
- This is equivalent to [tex]\(\frac{14}{15} \times \frac{5}{7}\)[/tex].
- This is correct reasoning.
- Statement E: Multiplying [tex]\(\frac{15}{14}\)[/tex] by 7 and then dividing by 5.
- This calculates something entirely different, not equivalent to multiplying by [tex]\(\frac{5}{7}\)[/tex].
- This is incorrect reasoning.
Therefore, the statements that show correct reasoning are A and D.
1. Find the Reciprocal: The reciprocal of [tex]\(\frac{7}{5}\)[/tex] is [tex]\(\frac{5}{7}\)[/tex].
2. Change Division to Multiplication: We rewrite the division problem [tex]\(\frac{14}{15} \div \frac{7}{5}\)[/tex] as:
[tex]\[
\frac{14}{15} \times \frac{5}{7}
\][/tex]
Now, let's explore each statement to see which ones correctly follow this reasoning:
- Statement A: Multiplying [tex]\(\frac{14}{15}\)[/tex] by 5 and then by [tex]\(\frac{1}{7}\)[/tex].
- When you multiply [tex]\(\frac{14}{15}\)[/tex] by 5 and then by [tex]\(\frac{1}{7}\)[/tex], it is the same as multiplying by [tex]\(\frac{5}{7}\)[/tex].
- This is correct reasoning.
- Statement B: Dividing [tex]\(\frac{14}{15}\)[/tex] by 5, and then multiplying by [tex]\(\frac{1}{7}\)[/tex].
- Dividing by 5 and then multiplying by [tex]\(\frac{1}{7}\)[/tex] doesn’t follow the reciprocal method. This breaks the correct sequence.
- This is incorrect reasoning.
- Statement C: Multiplying [tex]\(\frac{14}{15}\)[/tex] by 7 and then multiplying by [tex]\(\frac{1}{5}\)[/tex].
- This means [tex]\( \frac{14}{15} \times 7 \times \frac{1}{5}\)[/tex] can be rewritten as [tex]\( \frac{14}{15} \times \frac{7}{5} \times \frac{1}{7} \)[/tex], which simplifies back to [tex]\(\frac{14}{15} \times \frac{1}{5}\)[/tex] or [tex]\(\frac{14 \times 1}{15 \times 5}\)[/tex]. They mistakenly reversed the multiplication and division order.
- This is incorrect reasoning.
- Statement D: Multiplying [tex]\(\frac{14}{15}\)[/tex] by 5 and then dividing by 7.
- This is equivalent to [tex]\(\frac{14}{15} \times \frac{5}{7}\)[/tex].
- This is correct reasoning.
- Statement E: Multiplying [tex]\(\frac{15}{14}\)[/tex] by 7 and then dividing by 5.
- This calculates something entirely different, not equivalent to multiplying by [tex]\(\frac{5}{7}\)[/tex].
- This is incorrect reasoning.
Therefore, the statements that show correct reasoning are A and D.
