5.3.4: Dividing Rational Numbers
Lesson
Let's divide signed numbers.
Exercise \(\PageIndex{1}\): Tell Me Your Sign
Consider the equation: \(-27x=-35\)
Without computing:
- Is the solution to this equation positive or negative?
- Are either of these two number solutions to the equation?
\(\frac{35}{27}\qquad\qquad -\frac{35}{27}\)
Exercise \(\PageIndex{2}\): Multiplication and Division
-
Find the missing values in the equations
- \(-3\cdot 4=?\)
- \(-3\cdot ?=12\)
- \(3\cdot ?=12\)
- \(?\cdot -4=12\)
- \(?\cdot 4=-12\)
- Rewrite the unknown factor problems as division problems.
-
Complete the sentences. Be prepared to explain your reasoning.
- The sign of a positive number divided by a positive number is always:
- The sign of a positive number divided by a negative number is always:
- The sign of a negative number divided by a positive number is always:
- The sign of a negative number divided by a negative number is always:
-
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.
- Where is each person 10 seconds before they meet up?
-
When is each person at the position -10 feet from the meeting place?
- Where is each person 10 seconds before they meet up?
- When is each person at the position -10 feet from the meeting place?
Are you ready for more?
It is possible to make a new number system using only the numbers 0, 1, 2, and 3. We will write the symbols for multiplying in this system like this: \(1\otimes 2=2\). The table shows some of the products.
| \(\otimes\) | \(0\) | \(1\) | \(2\) | \(3\) |
|---|---|---|---|---|
| \(0\) | \(0\) | \(0\) | \(0\) | \(0\) |
| \(1\) | \(1\) | \(2\) | \(3\) | |
| \(2\) | \(0\) | \(2\) | ||
| \(3\) |
- In this system, \(1\otimes 3=3\) and \(2\otimes 3=2\). How can you see that in the table?
- What do you think \(2\otimes 1\) is?
- What about \(3\otimes 3\)?
- What do you think the solution to \(3\otimes n=2\) is?
- What about \(2\otimes n=3\)?
Exercise \(\PageIndex{3}\): Drilling Down
A water well drilling rig has dug to a height of -60 feet after one full day of continuous use.
- Assuming the rig drilled at a constant rate, what was the height of the drill after 15 hours?
- 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?
- Use the coordinate grid to show the drill's progress.
- 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 \(2\cdot 3=6\)
- \(6\div -2=-3\) because \(-2\cdot -3=6\)
- \(-6\div 2=-3\) because \(2\cdot -3=-6\)
- \(-6\div 2=3\) because \(-2\cdot 3=-6\)
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.
A number that can be used in place of the variable that makes the equation true is called a solution to the equation. For example, for the equation \(x\div -2=5\), the solution is -10, because it is true that \(-10\div -2=5\).
Glossary Entries
Definition: Solution to an Equation
A solution to an equation is a number that can be used in place of the variable to make the equation true.
For example, 7 is the solution to the equation \(m+1=8\), because it is true that \(7+1=8\). The solution to \(m+1=8\) is not 9, because \(9+1\neq 8\).
Practice
Exercise \(\PageIndex{4}\)
Find the quotients:
- \(24\div -6\)
- \(-15\div 0.3\)
- \(-4\div -20\)
Exercise \(\PageIndex{5}\)
Find the quotients.
- \(\frac{2}{5}\div\frac{3}{4}\)
- \(\frac{9}{4}\div\frac{-3}{4}\)
- \(\frac{-5}{7}\div\frac{-1}{3}\)
- \(\frac{-5}{3}\div\frac{1}{6}\)
Exercise \(\PageIndex{6}\)
Is the solution positve or negative?
- \(2\cdot x=6\)
- \(-2\cdot x=6.1\)
- \(2.9\cdot x=-6.04\)
- \(-2.473\cdot x=-6.859\)
Exercise \(\PageIndex{7}\)
Find the solution mentally.
- \(3\cdot -4=a\)
- \(b=\cdot (-3)=-12\)
- \(-12\cdot c=12\)
- \(d\cdot 24=-12\)
Exercise \(\PageIndex{8}\)
In order to make a specific shade of green paint, a painter mixes \(1\frac{1}{2}\) quarts of blue paint, 2 cups of green paint, and \(\frac{1}{2}\) gallon of white paint. How much of each color is needed to make 100 cups of this shade of green paint?
(From Unit 4.1.2)
Exercise \(\PageIndex{9}\)
Here is a list of the highest and lowest elevation on each continent.
| highest point (m) | lowest point (m) | |
|---|---|---|
| Europe | \(4,810\) | \(-28\) |
| Asia | \(8,848\) | \(-427\) |
| Africa | \(5,895\) | \(-155\) |
| Australia | \(4,884\) | \(-15\) |
| North America | \(6,198\) | \(-86\) |
| South America | \(6,960\) | \(-105\) |
| Antartica | \(4,892\) | \(-50\) |
- Which continent has the largest difference in elevation? The smallest?
- Make a display (dot plot, box plot, or histogram) of the data set and explain why you chose that type of display to represent this data set.
(From Unit 5.2.2)