Multiply and divide rational numbers

Key notes:

Understanding Rational Numbers

  • A rational number is any number that can be expressed as a fraction a/b​, where a and b are integers, and b ≠ 0.
  • Rational numbers include integers, fractions, and terminating or repeating decimals.

Multiplying Rational Numbers

Multiply the numerators together and multiply the denominators together: a/b × c/d = a×c / b×d

If multiplying decimals, count the total number of decimal places and place the decimal correctly in the product.

Sign rules:

  • Positive × Positive = Positive
  • Positive × Negative = Negative
  • Negative × Negative = Positive

Dividing Rational Numbers

To divide fractions, multiply by the reciprocal (flip the second fraction): a/b ÷ c/d = a/b × d/c

When dividing decimals, convert them into fractions or move the decimal point appropriately.

Sign rules:

  • Positive ÷ Positive = Positive
  • Positive ÷ Negative = Negative
  • Negative ÷ Negative = Positive

Simplifying Before Multiplying or Dividing

  • Reduce fractions by canceling out common factors before multiplying.
  • Cross-cancel when multiplying fractions to simplify calculations.

Multiplying and Dividing Mixed Numbers

  • Convert mixed numbers into improper fractions before multiplying or dividing.

Word Problems and Real-Life Applications

  • Understand the context of the problem before applying operations.
  • Use multiplication and division of rational numbers in scenarios like rates, ratios, and proportions.

positive × positive = positive

positive × negative = negative

negative × positive = negative

negative × negative = positive

To multiply two fractions, multiply the numerators and multiply the denominators.

ex: 2/3 × 1/5 = 2/15

positive ÷ positive = positive

positive ÷ negative = negative

negative ÷ positive = negative

negative ÷ negative = positive

Dividing by a fraction is the same as multiplying by its reciprocal.

Ex: 2/3 ÷ 1/3 = 2/3 x 3/1 =6/3 =2

Learn with an example

🎯 Divide.

3/10 ÷ 1/3 = _____

Turn this from a division problem into a multiplication problem by multiplying by the reciprocal.

3/10 ÷ 1/3 = 3/10 x 3/1

Now multiply.

3/10 x 3/1 = 3 x 3 / 10 x 1

= 9/10

🎯 Divide.

-1/5 ÷ -1/2 = ____

Turn this from a division problem into a multiplication problem by multiplying by the reciprocal.

-1/5 ÷ -1/2 = -1/5 x -2/1

Now multiply.

-1/5 x -2/1 = -1 x -2 / 5 x 1

= 2/5

Divide.

1/4 ÷ -2/3 =

Turn this from a division problem into a multiplication problem by multiplying by the reciprocal.

1/4 ÷ -2/3 = 1/4 x -3/2

Now multiply.

1/4 x -3/2 = 1 × –3 / 4 × 2

= -3/8

Let’s practice!🖊️