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:

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

\(\frac{35}{27}\qquad\qquad -\frac{35}{27}\)

Exercise \(\PageIndex{2}\): Multiplication and Division

1. Find the missing values in the equations
   1. \(-3\cdot 4=?\)
   2. \(-3\cdot ?=12\)
   3. \(3\cdot ?=12\)
   4. \(?\cdot -4=12\)
   5. \(?\cdot 4=-12\)
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.
   • Where is each person 10 seconds before they meet up?
   • When is each person at the position -10 feet from the meeting place?
     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?

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.

Table \(\PageIndex{1}\)

\(\otimes\)	\(0\)	\(1\)	\(2\)	\(3\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)
\(1\)		\(1\)	\(2\)	\(3\)
\(2\)			\(0\)	\(2\)
\(3\)

1. In this system, \(1\otimes 3=3\) and \(2\otimes 3=2\). How can you see that in the table?
2. What do you think \(2\otimes 1\) is?
3. What about \(3\otimes 3\)?
4. What do you think the solution to \(3\otimes n=2\) is?
5. 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.

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?

Figure \(\PageIndex{1}\): “US Navy 090226-N-9584H-018 Construction Electrician Construction man Greg Langdon, assigned to Naval Mobile Construction Battalion (NMCB) 1, installs a new section of drill steel during a water well drilling operation”, by U.S. Navy photo by Mass Communication Specialist Seaman Ernesto Hernandez Fonte. Public Domain. Source.

1. Use the coordinate grid to show the drill's progress.
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 \(2\cdot 3=6\)
• \(6\div -2=-3\) 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:

1. \(24\div -6\)
2. \(-15\div 0.3\)
3. \(-4\div -20\)

Exercise \(\PageIndex{5}\)

Find the quotients.

1. \(\frac{2}{5}\div\frac{3}{4}\)
2. \(\frac{9}{4}\div\frac{-3}{4}\)
3. \(\frac{-5}{7}\div\frac{-1}{3}\)
4. \(\frac{-5}{3}\div\frac{1}{6}\)

Exercise \(\PageIndex{6}\)

Is the solution positve or negative?

1. \(2\cdot x=6\)
2. \(-2\cdot x=6.1\)
3. \(2.9\cdot x=-6.04\)
4. \(-2.473\cdot x=-6.859\)

Exercise \(\PageIndex{7}\)

Find the solution mentally.

1. \(3\cdot -4=a\)
2. \(b=\cdot (-3)=-12\)
3. \(-12\cdot c=12\)
4. \(d\cdot 24=-12\)

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.

Table \(\PageIndex{2}\)

	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	\(4,892\)	\(-50\)

1. Which continent has the largest difference in elevation? The smallest?
2. 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)