Multiply and divide rational numbers: word problems

Key notes:

Understanding Rational Numbers

  • Rational numbers include integers, fractions, and decimals that can be written as a ratio of two integers (e.g., 3/4​, -5, 0.75).
  • They follow the same multiplication and division rules as whole numbers.

Multiplication of Rational Numbers

  • Multiply the numerators and denominators when dealing with fractions.
  • Apply integer sign rules:
    • Positive × Positive = Positive
    • Negative × Negative = Positive
    • Positive × Negative = Negative

Division of Rational Numbers

  • To divide fractions, multiply by the reciprocal (e.g., 2/3 ÷ 5/4 = 2/3 × 4/5​).
  • Follow the integer sign rules for division (same as multiplication).

Word Problems Strategy

  • Identify what is given and what is being asked.
  • Determine whether to multiply or divide based on the context.
  • Convert any decimals to fractions if needed for easier calculation.
  • Simplify your answer and check for reasonableness.

Common Word Problem Scenarios

  • Proportional reasoning: “A rope is 5.5 meters long. Each piece is 3/4​ meters. How many pieces can be cut?” → Use division.
  • Rate problems: “A car travels 3/2​ miles per minute. How far does it travel in 6 minutes?” → Use multiplication.
  • Scaling recipes: “A recipe calls for 2/3​ cup of sugar per cake. How much sugar is needed for 4 cakes?” → Use multiplication.

Real-Life Applications

  • Finance (calculating discounts, tax, or interest).
  • Science (density, speed, and conversion problems).
  • Construction (measuring and cutting materials).

Learn with an example

Carly collected 1/2 of a bin of glass bottles to recycle. Erin collected 9 times as many bins as Carly. How many bins of bottles did Erin collect?

Write your answer as a fraction or as a whole or mixed number.

_____ bins

Erin collected 9 times 1/2 of a bin of bottles.

Multiply:

9 x 1/2 = ?

Write 9 as an improper fraction.

9 = 1/9

Multiply the numerators and multiply the denominators.

9/1 x 1/2 = 9 x 1 / 1 x 2 = 9/2

Simplify the product.

9/2 = 4 1/2

Erin collected 4 1/2 bins of bottles.

Warren bought 5 rolls of paper towels. He got 86 meters of paper towels in all. How many meters of paper towels were on each roll?

Write your answer as a decimal.

____ metres

Add zeroes after the decimal point in the number you are dividing. Divide until there is no remainder.

There were 17.2 meters of paper towels on each roll.

A construction company transported 3 stones that weighed a total of 7.2 kilograms. How much did each stone weigh? _______kilograms

Divide until there is no remainder.

Each stone weighed 2.4 kilograms.

Let’s practice!🖊️