College

Select all 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 7 and then dividing by 5.

Answer :

To find [tex]\(\frac{14}{15} \div \frac{7}{5}\)[/tex], we can follow these steps:

1. Understand the Division of Fractions: Dividing by a fraction is the same as multiplying by its reciprocal. So, [tex]\(\frac{14}{15} \div \frac{7}{5}\)[/tex] becomes [tex]\(\frac{14}{15} \times \frac{5}{7}\)[/tex].

2. Multiply the Fractions: When multiplying two fractions, you multiply the numerators together and the denominators together:

[tex]\[
\text{Numerator: } 14 \times 5 = 70
\][/tex]
[tex]\[
\text{Denominator: } 15 \times 7 = 105
\][/tex]

This gives us [tex]\(\frac{70}{105}\)[/tex].

3. Simplify the Fraction: To simplify [tex]\(\frac{70}{105}\)[/tex], find the greatest common divisor (GCD) of 70 and 105:

- Both 70 and 105 are divisible by 35.
- Divide both the numerator and the denominator by 35:

[tex]\[
\frac{70 \div 35}{105 \div 35} = \frac{2}{3}
\][/tex]

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

Let's connect this reasoning to the given statements:

- [A] Multiplying [tex]\(\frac{14}{15}\)[/tex] by 5 and then by [tex]\(\frac{1}{7}\)[/tex]: This statement is correct because it follows the process of multiplying by the reciprocal. First, multiply [tex]\(\frac{14}{15}\)[/tex] by 5, then by [tex]\(\frac{1}{7}\)[/tex], simplifying the expression as [tex]\(\frac{2}{3}\)[/tex].

- [C] Multiplying [tex]\(\frac{14}{15}\)[/tex] by 7, and then multiplying by [tex]\(\frac{1}{5}\)[/tex]: This statement is incorrect since it doesn't involve the correct steps of multiplying by the reciprocal of [tex]\(\frac{7}{5}\)[/tex].

- [E] Multiplying [tex]\(\frac{14}{15}\)[/tex] by 7 and then dividing by 5: This statement is incorrect as it would not follow the correct operation of division by a fraction through multiplication with the reciprocal.

Thus, the correct statements that show the proper reasoning are [A] and none of the others.