Question

Divide each of the following and express each answer in simplest form. $$a \cdot \frac{1}{5} \div \frac{1}{10}$$ $$\text { b. } \frac{-5}{9} \div \frac{25}{-21}$$ $$\text { c. }-8 \div \frac{8}{10}$$ $$d .-\frac{12}{26} \div \frac{8}{15}$$ $$\text { e. } \frac{\frac{-3}{10}}{\frac{-5}{9}}$$ $$\frac{\frac{23}{19}}{-46}$$

Short answer

Question: Simplify the following fractions and mixed numbers: a. $$a \cdot \frac{1}{5} \div \frac{1}{10}$$ Answer: $$2a$$ b. $$\frac{-5}{9} \div \frac{25}{-21}$$ Answer: $$\frac{7}{15}$$ c. $$-8 \div \frac{8}{10}$$ Answer: $$-10$$ d. $$-\frac{12}{26} \div \frac{8}{15}$$ Answer: $$\frac{-15}{4}$$ e. $$\frac{\frac{-3}{10}}{\frac{-5}{9}}$$ Answer: $$\frac{27}{50}$$ f. $$ \frac{\frac{23}{19}}{-46}$$ Answer: $$\frac{23}{-874}$$

Step-by-step solution

a. $$a \cdot \frac{1}{5} \div \frac{1}{10}$$

First, find the reciprocal of the second fraction: $$\frac{10}{1}$$ Now, convert the division to multiplication: $$a \cdot \frac{1}{5} \cdot \frac{10}{1}$$ Multiply the numerators: $$a \cdot 1 \cdot 10 = 10a$$ Multiply the denominators: $$5 \cdot 1 = 5$$ So, the final fraction is: $$\frac{10a}{5}$$ Now, simplify the fraction by dividing both numerator and denominator by the greatest common divisor (GCD) of 5. The final answer is: $$2a$$

b. $$\frac{-5}{9} \div \frac{25}{-21}$$

First, find the reciprocal of the second fraction: $$\frac{-21}{25}$$ Now, convert the division to multiplication: $$\frac{-5}{9} \cdot \frac{-21}{25}$$ Multiply the numerators: $$(-5) \cdot (-21) = 105$$ Multiply the denominators: $$9 \cdot 25 = 225$$ So, the final fraction is: $$\frac{105}{225}$$ Now, simplify the fraction by dividing both numerator and denominator by the greatest common divisor (GCD) of 15. The final answer is: $$\frac{7}{15}$$

c. $$-8 \div \frac{8}{10}$$

First, find the reciprocal of the second fraction: $$\frac{10}{8}$$ Now, convert the division to multiplication: $$-8 \cdot \frac{10}{8}$$ Multiply the numerators: $$-8 \cdot 10 = -80$$ Multiply the denominators: $$1 \cdot 8 = 8$$ So, the final fraction is: $$\frac{-80}{8}$$ Now, simplify the fraction by dividing both numerator and denominator by the greatest common divisor (GCD) of 8. The final answer is: $$-10$$

d. $$-\frac{12}{26} \div \frac{8}{15}$$

First, find the reciprocal of the second fraction: $$\frac{15}{8}$$ Now, convert the division to multiplication: $$-\frac{12}{26} \cdot \frac{15}{8}$$ Multiply the numerators: $$(-12) \cdot 15 = -180$$ Multiply the denominators: $$26 \cdot 8 = 208$$ So, the final fraction is: $$\frac{-180}{208}$$ Now, simplify the fraction by dividing both numerator and denominator by the greatest common divisor (GCD) of 52. The final answer is: $$\frac{-15}{4}$$

e. $$\frac{\frac{-3}{10}}{\frac{-5}{9}}$$

First, find the reciprocal of the second fraction: $$\frac{-9}{5}$$ Now, convert the division to multiplication: $$\frac{-3}{10} \cdot \frac{-9}{5}$$ Multiply the numerators: $$(-3) \cdot (-9) = 27$$ Multiply the denominators: $$10 \cdot 5 = 50$$ So, the final fraction is: $$\frac{27}{50}$$

f.$$ \frac{\frac{23}{19}}{-46}$$

First, find the reciprocal of the second fraction: $$\frac{1}{-46}$$ Now, convert the division to multiplication: $$\frac{23}{19} \cdot \frac{1}{-46}$$ Multiply the numerators: $$23 \cdot 1 = 23$$ Multiply the denominators: $$19 \cdot (-46) = -874$$ So, the final fraction is: $$\frac{23}{-874}$$