Multiplication and Division of Fractions

\(\dfrac{2}{3}, \dfrac{1}{2}, \dfrac{7}{8}, \cdots\) are examples of fractions.

Multiply the fractions \(\dfrac{3}{7}\) and \(\dfrac{5}{11}\).

Solution

Step 1. Multiplying the numerators of the fractions together, we get \[3\times 5=15.\]

Multiplying the denominators of the fractions together, we get

\[7\times 11=77.\]

Step 2. Dividing the resultant numbers gives the new fraction \(\dfrac{15}{77}.\)

Since the numerator and denominator of the new fraction do not have any common factors, this is the simplest form.

Multiply \(\dfrac{2}{5}\) and \(\dfrac{7}{9}\).

Solution

Multiplying the numerators and denominators, we get

\[\dfrac{2}{5}\times \dfrac{7}{9}=\dfrac{2\times 7}{5 \times 9}=\dfrac{14}{45}.\]

Multiply \(\dfrac{5}{8}\) and \(\dfrac{2}{3}.\)

Solution

Step 1. Multiplying the numerators of the two fractions together, we get

\(5 \times 2=10.\) Similarly, doing the same with the denominators gives \(8\times 3=24.\)

Step 2. Dividing the resultant numbers gives us the new fraction \(\dfrac{10}{24}.\)

We notice that the numerator and denominator of the new fraction have a common factor of 2.

Step 3. We get the simplest form of this fraction by dividing out the common factor 2 from the numerator 10 and the denominator 24. This gives us, \(10 \divsymbol 2=5\) and \(24\divsymbol 2=12\).

The simplest fraction is therefore \(\dfrac{5}{12}.\)

Divide \(\dfrac{5}{8}\) by \(\dfrac{2}{3}.\)

Solution

Step 1. Inverting the divisor, we get \(\dfrac{3}{2}\).

Step 2. Now we perform the multiplication of the obtained fractions,

\(\dfrac{5}{8}\) and \(\dfrac{3}{2}\) to get,

\[\dfrac{5}{8}\times \dfrac{3}{2}=\dfrac{5\times 3}{8\times 2}=\dfrac{15}{16}.\]

Since the numerator and denominator have no common factors, this is the simplest form.

Find \(\dfrac{2}{5}\divsymbol \dfrac{3}{8}\).

Solution

Here \(\dfrac{2}{5}\)is the dividend fraction and \(\dfrac{3}{8}\)is the divisor fraction.

Step 1. Invert the divisor, we get \(\dfrac{8}{3}.\)

Step 2. Now multiply the fractions we get,

\[\frac{2}{5}\divsymbol\frac{3}{8}=\frac{2}{5}\times \frac{8}{3}=\frac{2\times 8}{3\times 5} =\frac{16}{15}.\]

Since the numerator and denominator has no common factors, this is the simplest form.

Find \(\dfrac{\dfrac{2}{5}}{3}.\)

Solution

Here \(\dfrac{2}{5}\)is the dividend fraction and \(3=\dfrac{3}{1}\) is the divisor fraction.

Step 1. Invert the divisor, we get \(\dfrac{1}{3}\).

Step 2. Now multiply the fractions to get,

\[\dfrac{2}{5}\times \dfrac{1}{3}=\dfrac{2\times 1}{5\times 3}=\dfrac{2}{15}.\]

Since the numerator and denominator have no common factors, this is the simplest form.

Simplify \(\dfrac{4}{\dfrac{7}{9}}\).

Solution

Here \(4=\dfrac{4}{1}\)is the dividend fraction and \(\dfrac{7}{9}\)is the divisor fraction.

Solution

Step 1. Invert the divisor, we get \(\dfrac{9}{7}\).

Step 2. Now multiply the fractions together to get,

\[\dfrac{4}{\dfrac{7}{9}}=\dfrac{4}{1}\times \dfrac{9}{7}=\dfrac{4\times 9}{1\times 7}=\dfrac{36}{7}.\]

Since the numerator and denominator have no common factors, this is the simplest form.

Simplify \(\dfrac{5}{9}\times\dfrac{18}{13}\times\dfrac{21}{20}\)

Solution

Here we have three fractions under multiplication. The first step is to multiply the numerators of the fractions together \(5\times 18\times 21\) and the denominators together \(9\times 13\times 20.\)

We see here that we end up with a multiplication of huge numbers. To avoid this, we are going first to cancel the common factors, where possible.

Step 1 . The numerators are 5,18,21 and the denominators are 9,13,20. We see 9 and 18 has 9 as common factor and 5 and 20 has 5 as a factor, thus we have

\[\frac{5}{9}\times\dfrac{18}{13}\times\dfrac{21}{20}=\dfrac{1}{1}\times\dfrac{2}{13}\times\dfrac{21}{4}.\]

Further, we can simplify 2 and 4 by dividing by 2, to get

\[\dfrac{5}{9}\times\dfrac{18}{13}\times\dfrac{21}{20}=\dfrac{1}{13} \times\dfrac{21}{2}.\]

Step 2. And the final answer is,

\[\dfrac{5}{9}\times\dfrac{18}{13}\times\dfrac{21}{20}=\dfrac{21}{13\times 2}=\dfrac{21}{26}.\]

Simplify

\[\dfrac{14}{39}\times\dfrac{12}{35}\divsymbol\dfrac{8}{13}\times\dfrac{2}{9}\]

Solution

Step 1. Invert the divisor fraction to get,

\[\dfrac{14}{39}\times\dfrac{12}{35}\divsymbol\dfrac{8}{13}\times\dfrac{2}{9}=\dfrac{14}{39}\times\dfrac{12}{35}\times\dfrac{13}{8}\times\dfrac{2}{9}\]

Step 2. Now we try to bring the terms to the simplest form. Dividing 14 and 35 by 7, 13 and 39 by 13, 12 and 9 by 3, 2 and 8 by 2 we get,

\[\dfrac{14}{39}\times\frac{12}{35}\times\dfrac{13}{8}\times\dfrac{2}{9}=\dfrac{2}{3}\times\dfrac{4}{5}\times\dfrac{1}{4}\times\dfrac{1}{3}\]

Step 3. Cancel out the 4, we get\[\dfrac{2}{3}\times\dfrac{4}{5}\times\dfrac{1}{4}\times\dfrac{1}{3}=\dfrac{2}{5}\times\dfrac{1}{5}\times \dfrac{1}{3}=\dfrac{2}{45}.\]

Simplify

\[4\dfrac{2}{7}\times 2\dfrac{1}{3}\div \dfrac{3}{5}.\]

Solution

Converting the mixed fractions into improper fractions, we get,

\[4\dfrac{2}{7}\times 2\dfrac{1}{3}\div \frac{3}{5} = \dfrac{30}{7}\times \dfrac{7}{3} \div \dfrac{3}{5}.\]

Inverting the divisor, we get,

\[\dfrac{30}{7}\times\dfrac{7}{3}\div\dfrac{3}{5}= \dfrac{30}{7} \times \dfrac{7}{3} \times \dfrac{5}{3}\]

Dividing 30 and 3 by 3, cancelling the 7 in the numerator and denominator, we have

\[\dfrac{30}{7}\times\dfrac{7}{3}\times \dfrac{5}{3}= \dfrac{10}{1} \times \dfrac{1}{1} \times \dfrac{5}{3}.\]

Multiplying the above fractions gives,

\[\dfrac{10}{1}\times\dfrac{5}{3}= \dfrac{50}{3} = 16\dfrac{2}{3}.\]

Simplify \(\dfrac{4xy}{5} \times \dfrac{2y}{x^3}\div \dfrac{y}{x}\).

Solution

Inverting the divisor, we get

\[\dfrac{4xy}{5} \times \dfrac{2y}{x^3} \div \dfrac{y}{x} = \dfrac{4xy}{5} \times \dfrac{2y}{x^3} \times \dfrac{x}{y}.\]

Dividing \(4xy\) and \(x^{3}\) by \(x\) and \(2y\) and \(y\) by \(y\), we get

\[ \dfrac{4xy}{5}\times\dfrac{2y}{x^3}\times\dfrac{x}{y}= \dfrac{4y}{5} \times \dfrac{2}{x^2} \times \dfrac{x}{1}.\]

Dividing \(x^2\) and \(x\) by \(x\) we get,

\[ \dfrac{4y}{5}\times\dfrac{2}{x^2}\dfrac{x}{1}= \dfrac{4y}{5} \times \dfrac{2}{x} \times \dfrac{1}{1}\]

Multiplying the above fractions gives,

\[ \dfrac{4y}{5}\times\dfrac{2}{x}\times\dfrac{1}{1}= \dfrac{8y}{5x}.\]

Multiply \( 2y^3 + 3xy + 5x^2 + 7\) by \(4x^2\).

Solution

\[\begin{align} &(2y^3 + 3xy + 5x^2 + 7) \times 4x^2 \\ &= (2y^3 \times 4x^2) + (3xy\times 4x^2) + (5x^2\times 4x^2) + (7\times 4x^2)\\ &= 8x^2y^3 + 12x^3 y + 20x^4 + 28x^2.\end{align}\]

Simplify \(\dfrac{2x^2 y^3}{7} \times \dfrac{14}{xy} \times \dfrac{y}{x^3}\).

Solution

Dividing \(2x^2y^3\) and \(xy\) by \(xy\), and 7 and 14 by 7, we get

\[ \frac{2x^2y^3}{7} \times \frac{14}{xy} \times \frac{y}{x^3} = \frac{2xy^2}{1} \times \frac{2}{1} \times \frac{y}{x^3} .\]

Dividing \(2xy^2\) and \(x^3\) by \(x\), we get,

\[\frac{2xy^{2}}{1}\times\frac{2}{1}\times\frac{y}{x^3}= \frac{2y^2}{1} \times \frac{2}{1} \times \frac{y}{x^2} .\]

Multiplying the above fractions, we get

\[ \frac{2y^2}{1} \times \frac{2}{1} \times \frac{y}{x^2} = \frac{4y^3}{x^2}. \]