College

1. Select all the statements that show correct reasoning for finding [tex]\frac{14}{15} \div \frac{7}{5}[/tex].

A. Multiplying [tex]\frac{14}{15}[/tex] by 5 and then by [tex]\frac{1}{7}[/tex].

B. Dividing [tex]\frac{14}{15}[/tex] by 5, and then multiplying by [tex]\frac{1}{7}[/tex].

C. Multiplying [tex]\frac{14}{15}[/tex] by 7, and then multiplying by [tex]\frac{1}{5}[/tex].

D. Multiplying [tex]\frac{14}{15}[/tex] by 5 and then dividing by 7.

E. Multiplying [tex]\frac{15}{14}[/tex] by 7 and then dividing by 5.

Answer :

To solve the division of fractions problem [tex]\(\frac{14}{15} \div \frac{7}{5}\)[/tex], we use the concept that dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of a fraction is created by swapping its numerator and denominator.

Here's the step-by-step solution:

1. Find the Reciprocal of the Divisor:
- The divisor here is [tex]\(\frac{7}{5}\)[/tex].
- The reciprocal of [tex]\(\frac{7}{5}\)[/tex] is [tex]\(\frac{5}{7}\)[/tex].

2. Change the Division to Multiplication:
- Instead of dividing by [tex]\(\frac{7}{5}\)[/tex], multiply by its reciprocal:
[tex]\[
\frac{14}{15} \times \frac{5}{7}
\][/tex]

3. Multiply the Fractions:
- To multiply two fractions, multiply the numerators together and the denominators together:
[tex]\[
\frac{14 \times 5}{15 \times 7} = \frac{70}{105}
\][/tex]

4. Simplify the Result:
- The fraction [tex]\(\frac{70}{105}\)[/tex] can be simplified by finding the greatest common divisor (GCD) of 70 and 105.
- The GCD of 70 and 105 is 35.

- Divide both the numerator and denominator by their GCD:
[tex]\[
\frac{70 \div 35}{105 \div 35} = \frac{2}{3}
\][/tex]

Therefore, [tex]\(\frac{14}{15} \div \frac{7}{5} = \frac{2}{3}\)[/tex].

Now, let's evaluate which statements correctly represent this process:

- Statement A: Multiplying [tex]\(\frac{14}{15}\)[/tex] by 5 and then by [tex]\(\frac{1}{7}\)[/tex]. This is essentially the same as multiplying by [tex]\(\frac{5}{7}\)[/tex], which is correct.
- Statement D: Multiplying [tex]\(\frac{14}{15}\)[/tex] by 5 and then dividing by 7. This is another way to think about [tex]\(\frac{14}{15} \times \frac{5}{7}\)[/tex], which is also correct.

So, the correct statements that show the right reasoning are A and D.