Pre-Algebra Workbook 6: Fractions III – Multiplying, Dividing & Mixed Numbers

To multiply fractions, multiply the numerators together and the denominators together, then simplify the answer if possible.

Multiply the numerators and denominators:

$$\dfrac{2}{3} \cdot \dfrac{3}{5} = \dfrac{2 \cdot 3}{3 \cdot 5} = \dfrac{6}{15}$$

Simplify $\dfrac{6}{15}$ by dividing numerator and denominator by their GCF $3$

$$\dfrac{6}{15} = \dfrac{6 \div 3}{15 \div 3} = \dfrac{2}{5}$$

So the product is $\dfrac{2}{5}$

The correct choice is B. $\dfrac{2}{5}$.

Think of the whole number $4$ as $\dfrac{4}{1}$ then multiply the fractions and simplify.

Write $4$ as $\dfrac{4}{1}$

$$4 \cdot \dfrac{2}{5} = \dfrac{4}{1} \cdot \dfrac{2}{5} = \dfrac{4 \cdot 2}{1 \cdot 5} = \dfrac{8}{5}$$

Convert $\dfrac{8}{5}$ to a mixed number if desired:

$$8 \div 5 = 1 \text{ remainder } 3 \quad \dfrac{8}{5} = 1 \dfrac{3}{5}$$

So the product is $1 \dfrac{3}{5}$

The correct choice is C. $1 \dfrac{3}{5}$.

Convert the mixed number $1 \dfrac{1}{2}$ to an improper fraction, then multiply and simplify the result.

Convert $1 \dfrac{1}{2}$ to an improper fraction:

$$1 \dfrac{1}{2} = \dfrac{1 \cdot 2 + 1}{2} = \dfrac{3}{2}$$

Now multiply:

$$\dfrac{3}{2} \cdot \dfrac{2}{3} = \dfrac{3 \cdot 2}{2 \cdot 3} = \dfrac{6}{6} = 1$$

So the product is $1$

The correct choice is A. $1$.

To divide by a fraction, multiply by its reciprocal. Flip $\dfrac{2}{5}$ to get $\dfrac{5}{2}$ and then multiply.

Rewrite the division as multiplication by the reciprocal:

$$\dfrac{3}{4} \div \dfrac{2}{5} = \dfrac{3}{4} \cdot \dfrac{5}{2}$$

Multiply the fractions:

$$\dfrac{3}{4} \cdot \dfrac{5}{2} = \dfrac{3 \cdot 5}{4 \cdot 2} = \dfrac{15}{8}$$

Convert $\dfrac{15}{8}$ to a mixed number:

$$15 \div 8 = 1 \text{ remainder } 7 \quad \dfrac{15}{8} = 1 \dfrac{7}{8}$$

So the quotient is $1 \dfrac{7}{8}$

The correct choice is D. $1 \dfrac{7}{8}$.

Convert $2 \dfrac{1}{3}$ to an improper fraction, then multiply by the reciprocal of $\dfrac{1}{6}$

Convert $2 \dfrac{1}{3}$ to an improper fraction:

$$2 \dfrac{1}{3} = \dfrac{2 \cdot 3 + 1}{3} = \dfrac{7}{3}$$

Divide by $\dfrac{1}{6}$ by multiplying by its reciprocal $\dfrac{6}{1}$

$$\dfrac{7}{3} \div \dfrac{1}{6} = \dfrac{7}{3} \cdot \dfrac{6}{1} = \dfrac{7 \cdot 6}{3 \cdot 1} = \dfrac{42}{3} = 14$$

So the quotient is $14$

The correct choice is B. $14$.

Multiply the fractions as usual, then determine the sign of the result. A negative times a positive is negative.

Multiply the numerators and denominators:

$$-\dfrac{3}{5} \cdot \dfrac{4}{7} = -\dfrac{3 \cdot 4}{5 \cdot 7} = -\dfrac{12}{35}$$

The fraction $\dfrac{12}{35}$ cannot be simplified further.

So the product is $-\dfrac{12}{35}$

The correct choice is C. $-\dfrac{12}{35}$.

Divide by multiplying by the reciprocal. Pay attention to the signs: a negative divided by a negative is positive.

Rewrite the division as multiplication by the reciprocal of $-\dfrac{1}{3}$ which is $-\dfrac{3}{1}$

$$\dfrac{-5}{6} \div \left(-\dfrac{1}{3}\right) = \dfrac{-5}{6} \cdot \left(-\dfrac{3}{1}\right)$$

Multiply the fractions:

$$\dfrac{-5}{6} \cdot \left(-\dfrac{3}{1}\right) = \dfrac{(-5) \cdot (-3)}{6 \cdot 1} = \dfrac{15}{6}$$

Since negative times negative is positive, the result is positive.

Simplify $\dfrac{15}{6}$ by dividing numerator and denominator by $3$

$$\dfrac{15}{6} = \dfrac{5}{2} = 2 \dfrac{1}{2}$$

So the quotient is $2 \dfrac{1}{2}$

The correct choice is D. $2 \dfrac{1}{2}$.

The phrase “$\dfrac{2}{3}$ of $18$” means $\dfrac{2}{3} \cdot 18$ Multiply and simplify.

Translate the phrase to multiplication:

$$\dfrac{2}{3} \text{ of } 18 = \dfrac{2}{3} \cdot 18$$

Write $18$ as $\dfrac{18}{1}$ and multiply:

$$\dfrac{2}{3} \cdot \dfrac{18}{1} = \dfrac{2 \cdot 18}{3 \cdot 1} = \dfrac{36}{3} = 12$$

So $\dfrac{2}{3}$ of $18$ is $12$

The correct choice is B. $12$.

“$\dfrac{3}{4}$ of $32$” means $\dfrac{3}{4} \cdot 32$ Multiply and simplify.

Compute $\dfrac{3}{4}$ of $32$

$$\dfrac{3}{4} \cdot 32 = \dfrac{3}{4} \cdot \dfrac{32}{1} = \dfrac{3 \cdot 32}{4 \cdot 1} = \dfrac{96}{4} = 24$$

So there are $24$ ounces of water in the bottle.

The correct choice is C. $24$ ounces.

Multiply the two fractions $\dfrac{3}{4}$ and $\dfrac{2}{5}$ to find the area. Then simplify the result if possible.

Multiply the length and width:

$$\text{Area} = \dfrac{3}{4} \cdot \dfrac{2}{5} = \dfrac{3 \cdot 2}{4 \cdot 5} = \dfrac{6}{20}$$

Simplify $\dfrac{6}{20}$ by dividing numerator and denominator by $2$

$$\dfrac{6}{20} = \dfrac{3}{10}$$

So the area is $\dfrac{3}{10}$ square meters.

The correct choice is A. $\dfrac{3}{10}$ square meters.