Multiply and divide rational numbers

key notes:

Multiplying Rational Numbers

  • Definition:
    • To multiply two rational numbers, multiply the numerators together and the denominators together.
  • Formula
  • Steps:
    • Multiply the numerators: a×c
    • Multiply the denominators: b×d
    • Simplify the resulting fraction, if possible.
  • Example:
  • Multiplying by a Whole Number:
    • When multiplying a rational number by a whole number, treat the whole number as a fraction with denominator 1.

Dividing Rational Numbers

  • Definition:
    • To divide two rational numbers, multiply the first rational number by the reciprocal of the second rational number.
  • Formula:
  • Steps:
    • Find the reciprocal of the divisor (flip the numerator and the denominator of the second fraction).
    • Multiply the first fraction by this reciprocal.
    • Simplify the resulting fraction, if possible.
  • Example:
  • Dividing by a Whole Number:
    • When dividing a rational number by a whole number, treat the whole number as a fraction with denominator 1 and then find its reciprocal.

Simplifying Rational Numbers

  • After multiplying or dividing rational numbers, always simplify the resulting fraction by finding the greatest common divisor (GCD) of the numerator and the denominator and dividing both by this number.Example:

Practice Problems

  • Multiplying Rational Numbers:
  • Dividing Rational Numbers:
  • Mixed Practice:

Tips for Success

  1. Reciprocals:
    • Remember to find the reciprocal when dividing by a fraction.
  2. Simplifying:
    • Always simplify your answer to its lowest terms.
  3. Multiplication and Division Rules:
    • Use the multiplication rule for both operations: multiply the numerators together and the denominators together for multiplication, and multiply by the reciprocal for division.
  4. Check Your Work:
    • Double-check your multiplication and division to avoid small mistakes.

Learn with an example

🔮 Divide.

1/10 ÷ 1/2 =

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

1/10 ÷ 1/2 = 1/10 x 2/1

Cancel common factors, then multiply.

1/10 x 2/1 = 1/5 x 1/1

= 1 x 1 / 5 x 1

= 1/5

🔮 Divide.

1/4 ÷ 1/2 =

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

1/4 ÷ 1/2 = 1/4 x 2/1

Cancel common factors, then multiply.

1/4 x 2/1 = 1/2 x 1/1

= 1 x 1 / 2 x 1

= 1/2

🔮 Divide.

1/8 ÷ 1/6 =

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

1/8 ÷ 1/6 = 1/8 x 6/1

Cancel common factors, then multiply.

1/8 x 6/1 = 1/4 x 3/1

= 1 x 3 / 4 x 1

= 3/4

Let’s Practice!