Lesson 11: Dividing Rational Numbers

Let's divide signed numbers.

11.1: Tell Me Your Sign

Consider the equation:   $\text- 27x = \text- 35$

Without computing:

  1. Is the solution to this equation positive or negative?
  2. Are either of these two numbers solutions to the equation?

    $$\frac{35}{27}$$

    $$\text-\frac{35 }{ 27}$$

11.2: Multiplication and Division

  1. Find the missing values in the equations

    1. $\text-3 \boldcdot 4 = \text{?}$

    2. $\text-3 \boldcdot \text{?} = 12$

    3. $3 \boldcdot \text{?} = 12$

    4. $\text{?} \boldcdot \text-4 = 12$

    5. $\text{?} \boldcdot 4 = \text-12$

  2. Rewrite the unknown factor problems as division problems.

  3. Complete the sentences. Be prepared to explain your reasoning.

    1. The sign of a positive number divided by a positive number is always:

    2. The sign of a positive number divided by a negative number is always:

    3. The sign of a negative number divided by a positive number is always:

    4. The sign of a negative number divided by a negative number is always:

  4. Han and Clare walk towards each other at a constant rate, meet up, and then continue past each other in opposite directions. We will call the position where they meet up 0 feet and the time when they meet up 0 seconds.

    • Han's velocity is 4 feet per second.

    • Clare's velocity is -5 feet per second.

    1. Where is each person 10 seconds before they meet up?

    2. When is each person at the position -10 feet from the meeting place?

11.3: Drilling Down

A water well drilling rig has dug to a height of -60 feet after one full day of continuous use.

  1. Assuming the rig drilled at a constant rate, what was the height of the drill after 15 hours?
  2. If the rig has been running constantly and is currently at a height of -147.5 feet, for how long has the rig been running?

US Navy 090226-N-9584H-018 Construction Electrician Constructionman Greg Langdon, assigned to Naval Mobile Construction Battalion (NMCB) 1, installs a new section of drill steel during a water well drilling operation Copyright Owner: Mass Communication Specialist Seaman Ernesto Hernandez Fonte License: Public Domain Via: Wikimedia Commons

  1. Use the coordinate grid to show the drill’s progress.

    GeoGebra Applet bz7v4QgM

  2. At this rate, how many hours will it take until the drill reaches -250 feet?

Summary

Any division problem is actually a multiplication problem:

  • $6 \div 2 = 3$ because $2 \boldcdot 3 = 6$
  • $6 \div \text- 2 = \text-3$ because $\text-2 \boldcdot \text-3 = 6$
  • $\text-6 \div 2 = \text-3$ because $2 \boldcdot \text-3 = \text-6$
  • $\text-6 \div \text-2 = 3$ because $\text-2 \boldcdot 3 = \text-6$

Because we know how to multiply signed numbers, that means we know how to divide them.

  • The sign of a positive number divided by a negative number is always negative.
  • The sign of a negative number divided by a positive number is always negative.
  • The sign of a negative number divided by a negative number is always positive.

Practice Problems ▶