Answer :
Let's work through each part of the question step-by-step:
Part 1: Finding [tex]\(\frac{14}{15} \div \frac{7}{5}\)[/tex]
To divide by a fraction, you multiply by its reciprocal. So, to find [tex]\(\frac{14}{15} \div \frac{7}{5}\)[/tex], you multiply [tex]\(\frac{14}{15}\)[/tex] by the reciprocal of [tex]\(\frac{7}{5}\)[/tex], which is [tex]\(\frac{5}{7}\)[/tex].
Thus, [tex]\(\frac{14}{15} \div \frac{7}{5} = \frac{14}{15} \times \frac{5}{7}\)[/tex].
Now, let's consider each statement:
- A. Multiplying [tex]\(\frac{14}{15}\)[/tex] by 5 and then by [tex]\(\frac{1}{7}\)[/tex].
This corresponds to [tex]\(\frac{14}{15} \times 5 \times \frac{1}{7}\)[/tex]. It is the same as multiplying by the reciprocal [tex]\(\frac{5}{7}\)[/tex]. Therefore, this is correct.
- B. Dividing [tex]\(\frac{14}{15}\)[/tex] by 5, and then multiplying by [tex]\(\frac{1}{7}\)[/tex].
This means [tex]\(\left(\frac{14}{15} \div 5\right) \times \frac{1}{7}\)[/tex], which does not represent the correct operation for division by a fraction. So, this is incorrect.
- C. Multiplying [tex]\(\frac{14}{15}\)[/tex] by 7, and then multiplying by [tex]\(\frac{1}{5}\)[/tex].
This corresponds to [tex]\(\frac{14}{15} \times 7 \times \frac{1}{5}\)[/tex] and does not lead to the correct calculation. So, this is incorrect.
- D. Multiplying [tex]\(\frac{14}{15}\)[/tex] by 5 and then dividing by 7.
This is the same as [tex]\(\left(\frac{14}{15} \times 5\right) \div 7\)[/tex], which can be rearranged as the correct operation [tex]\(\frac{14}{15} \times \frac{5}{7}\)[/tex]. Therefore, this is correct.
Correct options are: A and D.
Part 2: Clare's division [tex]\(\frac{4}{3} \div \frac{5}{2}\)[/tex]
Clare's method was incorrect. Let's examine why:
Clare said that [tex]\(\frac{4}{3} \div \frac{5}{2}\)[/tex] is [tex]\(\frac{10}{3}\)[/tex]. Here is what Clare did:
1. She calculated [tex]\(\frac{4}{3} \times 5 = \frac{20}{3}\)[/tex].
2. Then, she divided [tex]\(\frac{20}{3}\)[/tex] by 2 to get [tex]\(\frac{10}{3}\)[/tex].
The mistake here is that Clare should have multiplied by the reciprocal of [tex]\(\frac{5}{2}\)[/tex], which is [tex]\(\frac{2}{5}\)[/tex].
The correct steps are:
1. [tex]\(\frac{4}{3} \times \frac{2}{5} = \frac{8}{15}\)[/tex].
The correct quotient is [tex]\(\frac{8}{15}\)[/tex].
In conclusion, Clare's answer was incorrect because she failed to multiply by the reciprocal. The correct quotient of [tex]\(\frac{4}{3} \div \frac{5}{2}\)[/tex] is [tex]\(\frac{8}{15}\)[/tex].
Part 1: Finding [tex]\(\frac{14}{15} \div \frac{7}{5}\)[/tex]
To divide by a fraction, you multiply by its reciprocal. So, to find [tex]\(\frac{14}{15} \div \frac{7}{5}\)[/tex], you multiply [tex]\(\frac{14}{15}\)[/tex] by the reciprocal of [tex]\(\frac{7}{5}\)[/tex], which is [tex]\(\frac{5}{7}\)[/tex].
Thus, [tex]\(\frac{14}{15} \div \frac{7}{5} = \frac{14}{15} \times \frac{5}{7}\)[/tex].
Now, let's consider each statement:
- A. Multiplying [tex]\(\frac{14}{15}\)[/tex] by 5 and then by [tex]\(\frac{1}{7}\)[/tex].
This corresponds to [tex]\(\frac{14}{15} \times 5 \times \frac{1}{7}\)[/tex]. It is the same as multiplying by the reciprocal [tex]\(\frac{5}{7}\)[/tex]. Therefore, this is correct.
- B. Dividing [tex]\(\frac{14}{15}\)[/tex] by 5, and then multiplying by [tex]\(\frac{1}{7}\)[/tex].
This means [tex]\(\left(\frac{14}{15} \div 5\right) \times \frac{1}{7}\)[/tex], which does not represent the correct operation for division by a fraction. So, this is incorrect.
- C. Multiplying [tex]\(\frac{14}{15}\)[/tex] by 7, and then multiplying by [tex]\(\frac{1}{5}\)[/tex].
This corresponds to [tex]\(\frac{14}{15} \times 7 \times \frac{1}{5}\)[/tex] and does not lead to the correct calculation. So, this is incorrect.
- D. Multiplying [tex]\(\frac{14}{15}\)[/tex] by 5 and then dividing by 7.
This is the same as [tex]\(\left(\frac{14}{15} \times 5\right) \div 7\)[/tex], which can be rearranged as the correct operation [tex]\(\frac{14}{15} \times \frac{5}{7}\)[/tex]. Therefore, this is correct.
Correct options are: A and D.
Part 2: Clare's division [tex]\(\frac{4}{3} \div \frac{5}{2}\)[/tex]
Clare's method was incorrect. Let's examine why:
Clare said that [tex]\(\frac{4}{3} \div \frac{5}{2}\)[/tex] is [tex]\(\frac{10}{3}\)[/tex]. Here is what Clare did:
1. She calculated [tex]\(\frac{4}{3} \times 5 = \frac{20}{3}\)[/tex].
2. Then, she divided [tex]\(\frac{20}{3}\)[/tex] by 2 to get [tex]\(\frac{10}{3}\)[/tex].
The mistake here is that Clare should have multiplied by the reciprocal of [tex]\(\frac{5}{2}\)[/tex], which is [tex]\(\frac{2}{5}\)[/tex].
The correct steps are:
1. [tex]\(\frac{4}{3} \times \frac{2}{5} = \frac{8}{15}\)[/tex].
The correct quotient is [tex]\(\frac{8}{15}\)[/tex].
In conclusion, Clare's answer was incorrect because she failed to multiply by the reciprocal. The correct quotient of [tex]\(\frac{4}{3} \div \frac{5}{2}\)[/tex] is [tex]\(\frac{8}{15}\)[/tex].
