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Division of Rational Numbers

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Division of Rational Numbers

Suppose a box of cereal is \frac{4}{5} full, and you want to divide the remaining cereal into portions so that each portion is \frac{1}{5} of the full box. In this case, you would have to divide a fraction by a fraction to come up with the number of portions you could make. After completing this Concept, you'll be able to use reciprocals to perform division problems such as these.

Guidance

Division of Rational Numbers

Previously, you have added, subtracted, and multiplied rational numbers. It now makes sense to learn how to divide rational numbers. We will begin with a definition of inverse operations.

Inverse operations "undo" each other.

For example, addition and subtraction are inverse operations because addition cancels subtraction and vice versa. The additive identity results in a sum of zero. In the same sense, multiplication and division are inverse operations. This leads into the next property: The Inverse Property of Multiplication.

The Inverse Property of Multiplication: For every nonzero number a , there is a multiplicative inverse \frac{1}{a} such that a \left ( \frac{1}{a} \right ) = 1 .

This means that the multiplicative inverse of a is \frac{1}{a} . The values of a and \frac{1}{a} are called also called reciprocals. In general, two nonzero numbers whose product is 1 are multiplicative inverses or reciprocals.

Reciprocal: The reciprocal of a nonzero rational number \frac{a}{b} is \frac{b}{a} .

Note: The number zero does not have a reciprocal.

Using Reciprocals to Divide Rational Numbers

When dividing rational numbers, use the following rule:

“When dividing rational numbers, multiply by the ‘right’ reciprocal.”

In this case, the “right” reciprocal means to take the reciprocal of the fraction on the right-hand side of the division operator.

Example A

Simplify \frac{2}{9} \div \frac{3}{7} .

Solution:

Begin by multiplying by the “right” reciprocal.

\frac{2}{9} \times \frac{7}{3} = \frac{14}{27}

Example B

Simplify \frac{7}{3} \div \frac{2}{3} .

Solution:

Begin by multiplying by the “right” reciprocal.

\frac{7}{3} \div \frac{2}{3} = \frac{7}{3} \times \frac{3}{2} = \frac{7 \cdot 3} {2 \cdot 3} = \frac{7}{2}

Instead of the division symbol \div , you may see a large fraction bar. This is seen in the next example.

Example C

Simplify \frac{\frac{2}{3}}{\frac{7}{8}} .

Solution:

The fraction bar separating \frac{2}{3} and \frac{7}{8} indicates division.

\frac{2}{3} \div \frac{7}{8}

Simplify as in Example B:

\frac{2}{3} \times \frac{8}{7} = \frac{16}{21}

Video Review

Guided Practice

1. Find the multiplicative inverse of \frac{5}{7} .

2. Simplify  5\div \frac{3}{2} .

Solutions:

1. The multiplicative inverse of \frac{5}{7} is \frac{7}{5}. We can see that by multiplying them together:

\frac{5}{7}\times \frac{7}{5}= \frac{5\times 7}{7\times 5}=\frac{35}{35}=1.

2. When we are asked to divide by a fraction, we know we can rewrite the problem as multiplying by the reciprocal:

 5\div \frac{3}{2}=5 \times \frac{2}{3}=\frac{5\times 2}{3}=\frac{10}{3}

Explore More

  1. Define inverse.
  2. What is a multiplicative inverse? How is this different from an additive inverse?

In 3 – 11, find the multiplicative inverse of each expression.

  1. 100
  2. \frac{2}{8}
  3. -\frac{19}{21}
  4. 7
  5. - \frac{z^3}{2xy^2}
  6. 0
  7. \frac{1}{3}
  8. \frac{-19}{18}
  9. \frac{3xy}{8z}

In 12 – 20, divide the rational numbers. Be sure that your answer is in the simplest form.

  1. \frac{5}{2} \div \frac{1}{4}
  2. \frac{1}{2} \div \frac{7}{9}
  3. \frac{5}{11} \div \frac{6}{7}
  4. \frac{1}{2} \div \frac{1}{2}
  5. - \frac{x}{2} \div \frac{5}{7}
  6. \frac{1}{2} \div \frac{x}{4y}
  7. \left ( - \frac{1}{3} \right ) \div \left ( - \frac{3}{5} \right )
  8. \frac{7}{2} \div \frac{7}{4}
  9. 11 \div \left ( - \frac{x}{4} \right )

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