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.
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.
