Divide. Write Your Answer In Lowest Terms As A Proper Or Improper Fraction. ( − 5 14 ) ÷ 15 7 \left(-\frac{5}{14}\right) \div \frac{15}{7} ( − 14 5 ​ ) ÷ 7 15 ​

Understanding the Problem

When dividing fractions, we need to invert the second fraction (i.e., flip the numerator and denominator) and then multiply. This is a fundamental concept in mathematics, and it's essential to understand it to solve problems like the one presented.

Inverting the Second Fraction

To divide $\left(-\frac{5}{14}\right)$ by $\frac{15}{7}$, we need to invert the second fraction. This means we flip the numerator and denominator of $\frac{15}{7}$ to get $\frac{7}{15}$.

Multiplying the Fractions

Now that we have the inverted fraction, we can multiply the two fractions together. To do this, we multiply the numerators and denominators separately.

$\left(-\frac{5}{14}\right) \div \frac{15}{7} = \left(-\frac{5}{14}\right) \times \frac{7}{15}$

Multiplying the Numerators and Denominators

Now, let's multiply the numerators and denominators separately.

Numerators: $-5 \times 7 = -35$ Denominators: $14 \times 15 = 210$

Writing the Result as a Fraction

Now that we have the product of the numerators and denominators, we can write the result as a fraction.

$\left(-\frac{5}{14}\right) \div \frac{15}{7} = -\frac{35}{210}$

Simplifying the Fraction

To simplify the fraction, we need to find the greatest common divisor (GCD) of the numerator and denominator. The GCD of 35 and 210 is 35.

Dividing the Numerator and Denominator by the GCD

Now, let's divide the numerator and denominator by the GCD.

Numerator: $-35 \div 35 = -1$ Denominator: $210 \div 35 = 6$

Writing the Result in Lowest Terms

Now that we have the result of the division, we can write it in lowest terms.

$\left(-\frac{5}{14}\right) \div \frac{15}{7} = -\frac{1}{6}$

Conclusion

In conclusion, to divide $\left(-\frac{5}{14}\right)$ by $\frac{15}{7}$, we need to invert the second fraction and then multiply. The result of the division is $-\frac{1}{6}$.

Frequently Asked Questions

• What is the rule for dividing fractions?
  • To divide fractions, we need to invert the second fraction and then multiply.
• How do we invert a fraction?
  • To invert a fraction, we flip the numerator and denominator.
• What is the result of dividing $\left(-\frac{5}{14}\right)$ by $\frac{15}{7}$?
  • The result of dividing $\left(-\frac{5}{14}\right)$ by $\frac{15}{7}$ is $-\frac{1}{6}$.

Example Problems

• Divide $\frac{3}{4}$ by $\frac{5}{6}$.
  • To divide $\frac{3}{4}$ by $\frac{5}{6}$, we need to invert the second fraction and then multiply. The result of the division is $\frac{9}{20}$.
• Divide $\left(-\frac{2}{3}\right)$ by $\frac{3}{4}$.
  • To divide $\left(-\frac{2}{3}\right)$ by $\frac{3}{4}$, we need to invert the second fraction and then multiply. The result of the division is $-\frac{8}{9}$.

Practice Problems

• Divide $\frac{1}{2}$ by $\frac{3}{4}$.
• Divide $\left(-\frac{3}{4}\right)$ by $\frac{2}{3}$.
• Divide $\frac{2}{3}$ by $\frac{4}{5}$.

Step-by-Step Solutions

• Divide $\frac{1}{2}$ by $\frac{3}{4}$.
  1. Invert the second fraction: $\frac{3}{4} \rightarrow \frac{4}{3}$
  2. Multiply the fractions: $\frac{1}{2} \times \frac{4}{3} = \frac{4}{6}$
  3. Simplify the fraction: $\frac{4}{6} \rightarrow \frac{2}{3}$
• Divide $\left(-\frac{3}{4}\right)$ by $\frac{2}{3}$.
  1. Invert the second fraction: $\frac{2}{3} \rightarrow \frac{3}{2}$
  2. Multiply the fractions: $\left(-\frac{3}{4}\right) \times \frac{3}{2} = -\frac{9}{8}$
• Divide $\frac{2}{3}$ by $\frac{4}{5}$.
  1. Invert the second fraction: $\frac{4}{5} \rightarrow \frac{5}{4}$
  2. Multiply the fractions: $\frac{2}{3} \times \frac{5}{4} = \frac{10}{12}$
  3. Simplify the fraction: $\frac{10}{12} \rightarrow \frac{5}{6}$
     

Q: What is the rule for dividing fractions?

A: To divide fractions, we need to invert the second fraction and then multiply.

Q: How do we invert a fraction?

A: To invert a fraction, we flip the numerator and denominator. For example, if we have the fraction $\frac{a}{b}$, the inverted fraction is $\frac{b}{a}$.

Q: What is the result of dividing $\left(-\frac{5}{14}\right)$ by $\frac{15}{7}$?

A: The result of dividing $\left(-\frac{5}{14}\right)$ by $\frac{15}{7}$ is $-\frac{1}{6}$.

Q: Can we divide a fraction by a whole number?

A: Yes, we can divide a fraction by a whole number. To do this, we can multiply the fraction by the reciprocal of the whole number. For example, if we have the fraction $\frac{a}{b}$ and the whole number $c$, we can divide the fraction by the whole number as follows:

$\frac{a}{b} \div c = \frac{a}{b} \times \frac{1}{c}$

Q: Can we divide a whole number by a fraction?

A: Yes, we can divide a whole number by a fraction. To do this, we can multiply the whole number by the reciprocal of the fraction. For example, if we have the whole number $a$ and the fraction $\frac{b}{c}$, we can divide the whole number by the fraction as follows:

$a \div \frac{b}{c} = a \times \frac{c}{b}$

Q: What is the result of dividing $\frac{3}{4}$ by $2$?

A: To divide $\frac{3}{4}$ by $2$, we can multiply the fraction by the reciprocal of $2$, which is $\frac{1}{2}$. The result of the division is:

$\frac{3}{4} \div 2 = \frac{3}{4} \times \frac{1}{2} = \frac{3}{8}$

Q: What is the result of dividing $3$ by $\frac{4}{5}$?

A: To divide $3$ by $\frac{4}{5}$, we can multiply the whole number by the reciprocal of the fraction. The result of the division is:

$3 \div \frac{4}{5} = 3 \times \frac{5}{4} = \frac{15}{4}$

Q: Can we divide a negative fraction by a positive fraction?

A: Yes, we can divide a negative fraction by a positive fraction. To do this, we can follow the same steps as dividing two positive fractions. For example, if we have the negative fraction $\left(-\frac{a}{b}\right)$ and the positive fraction $\frac{c}{d}$, we can divide the negative fraction by the positive fraction as follows:

$\left(-\frac{a}{b}\right) \div \frac{c}{d} = \left(-\frac{a}{b}\right) \times \frac{d}{c}$

Q: Can we divide a positive fraction by a negative fraction?

A: Yes, we can divide a positive fraction by a negative fraction. To do this, we can follow the same steps as dividing two negative fractions. For example, if we have the positive fraction $\frac{a}{b}$ and the negative fraction $\left(-\frac{c}{d}\right)$, we can divide the positive fraction by the negative fraction as follows:

$\frac{a}{b} \div \left(-\frac{c}{d}\right) = \frac{a}{b} \times \left(-\frac{d}{c}\right)$

Q: What is the result of dividing $\left(-\frac{2}{3}\right)$ by $\frac{3}{4}$?

A: To divide $\left(-\frac{2}{3}\right)$ by $\frac{3}{4}$, we can follow the same steps as dividing two positive fractions. The result of the division is:

$\left(-\frac{2}{3}\right) \div \frac{3}{4} = \left(-\frac{2}{3}\right) \times \frac{4}{3} = -\frac{8}{9}$

Q: What is the result of dividing $\frac{2}{3}$ by $\left(-\frac{3}{4}\right)$?

A: To divide $\frac{2}{3}$ by $\left(-\frac{3}{4}\right)$, we can follow the same steps as dividing two negative fractions. The result of the division is:

$\frac{2}{3} \div \left(-\frac{3}{4}\right) = \frac{2}{3} \times \left(-\frac{4}{3}\right) = -\frac{8}{9}$