5.3.2: Multiplying Rational Numbers

Lesson

Let's multiply signed numbers.

Exercise \(\PageIndex{1}\): Before and After

Figure \(\PageIndex{1}\)

Where was the girl:

1. 5 seconds after this picture was taken? Mark her approximate location on the picture.
2. 5 seconds before this picture was taken? Mark her approximate location on the picture.

Exercise \(\PageIndex{2}\): Backwards in Time

A traffic safety engineer was studying travel patterns along a highway. She set up a camera and recorded the speed and direction of cars and trucks that passed by the camera. Positions to the east of the camera are positive, and to the west are negative.

1. Here are some positions and times for one car:

   position (feet)	\(-180\)	\(-120\)	\(-60\)	\(0\)	\(60\)	\(120\)
   time (seconds)	\(-3\)	\(-2\)	\(-1\)	\(0\)	\(1\)	\(2\)

   Table \(\PageIndex{1}\)
   1. In what direction is this car traveling?
   2. What is its velocity?
2. 1. What does it mean when the time is zero?
   2. What could it mean to have a negative time?
3. Here are the positions and times for a different car whose velocity is -50 feet per second:

   position (feet)				\(0\)	\(-50\)	\(-100\)
   time (seconds)	\(-3\)	\(-2\)	\(-1\)	\(0\)	\(1\)	\(2\)

   Table \(\PageIndex{2}\)
   1. Complete the table with the rest of the positions.
   2. In what direction is this car traveling? Explain how you know.
4. Complete the table for several different cars passing the camera.

   Table \(\PageIndex{3}\)

   	velocity (meters per second)	time after passing the camera (seconds)	ending position (meters)	equation
   car C	\(+25\)	\(+10\)	\(+250\)	\(25\cdot 10=250\)
   car D	\(-20\)	\(+30\)
   car E	\(+32\)	\(-40\)
   car F	\(-35\)	\(-20\)
   car G	\(-15\)	\(-8\)

5. 1. If a car is traveling east when it passes the camera, will its position be positive or negative 60 seconds before it passes the camera?
   2. If we multiply a positive number and a negative number, is the result positive or negative?
6. 1. If a car is traveling west when it passes the camera, will its position be positive or negative 60 seconds before it passes the camera?
   2. If we multiply two negative numbers, is the result positive or negative?

Exercise \(\PageIndex{3}\): Cruising

Around noon, a car was traveling -32 meters per second down a highway. At exactly noon (when time was 0), the position of the car was 0 meters.

1. Complete the table.

   Table \(\PageIndex{4}\)

   time (s)	\(-10\)	\(-7\)	\(-4\)	\(-1\)	\(2\)	\(5\)	\(8\)	\(11\)
   position (m)

2. Graph the relationship between the time and the car's position.
   1. What was the position of the car at -3 seconds?
   2. What was the position of the car at 6.5 seconds?

Are you ready for more?

Find the value of these expressions without using a calculator.

\((-1)^{2}\qquad (-1)^{3}\qquad (-1)^{4}\qquad (-1)^{99}\)

Exercise \(\PageIndex{4}\): Rational Numbers Multiplication Grid

Look at the patterns along the rows and columns and continue those patterns to complete the table. When you have filled in all the boxes you can see, click on the "More Boxes" button.

What does this tell you about multiplication by a negative?

Summary

We can use signed numbers to represent time relative to a chosen point in time. We can think of this as starting a stopwatch. The positive times are after the watch starts, and negative times are times before the watch starts.

Figure \(\PageIndex{2}\): Three points are labeled on a number line. The numbers negative 10 through 10, in increments of 1, are indicated. A dot at negative 4 is labeled 4 seconds before the start time. A dot at 0 is labeled start time. A dot at 7 is labeled 7 seconds after the start time.

If a car is at position 0 and is moving in a positive direction, then for times after that (positive times), it will have a positive position. A positive times a positive is positive.

Figure \(\PageIndex{3}\): Number line. 7 evenly spaced tick marks. Scale negative 15 to 15, by 5s. Three equal sized arrows pointing to the right, one from 0 to 5, 5 to 10, and 10 to 15.

If a car is at position 0 and is moving in a negative direction, then for times after that (positive times), it will have a negative position. A negative times a positive is negative.

Figure \(\PageIndex{4}\): A blank horizontal number line from negative 15 to 15 by 5s. Above the number line with three arrows pointing left and a dot are plotted. The first arrow points from 0 to negative 5. The second arrow points from negative 5 to negative 10. The third arrow points from negative 10 to negative 15. A dot is above 0.

If a car is at position 0 and is moving in a positive direction, then for times before that (negative times), it must have had a negative position. A positive times a negative is negative.

Figure \(\PageIndex{5}\): A blank horizontal number line from negative 15 to 15 by 5s. Above the number line with three arrows pointing right and a dot are plotted. The first arrow points from negative 15 to negative 10. The second arrow points from negative 10 to negative 5. The third arrow points from negative 5 to 0. A dot is above 0.

If a car is at position 0 and is moving in a negative direction, then for times before that (negative times), it must have had a positive position. A negative times a negative is positive.

Figure \(\PageIndex{6}\): A blank horizontal number line from negative 15 to 15 by 5s. Above the number line with three arrows pointing left and a dot are plotted. The first arrow points from 15 to 10. The second arrow points from 10 to 5. The third arrow points from 5 to 0. A dot is above 0.

Here is another way of seeing this:

We can think of \(3\cdot 5\) as \(5+5+5\), which has a value of 15.

We can think of \(3\cdot (-5)\) as \(-5+-5+-5\), which has a value of -15.

We know we can multiply positive numbers in any order: \(3\cdot 5=5\cdot 3\)

If we can multiply signed numbers in any order, then \((-5)\cdot 3\) would also equal -15.

Now let’s think about multiplying two negatives.

We can find \(-5\cdot (3+-3)\) in two ways:

• Applying the distributive property:
  \(-5\cdot 3+-5\cdot (-3)\)
• Adding the numbers in parentheses:
  \(-5\cdot (0)=0\)

This means that these expressions must be equal.

\(-5\cdot 3+-5\cdot (-3)=0\)

Multiplying the first two numbers gives

\(-15+-5\cdot (-3)=0\)

Which means that

\(-5\cdot (-3)=15\)

There was nothing special about these particular numbers. This always works!

• A positive times a positive is always positive.
• A negative times a positive or a positive times a negative is always negative.
• A negative times a negative is always positive.

Practice

Exercise \(\PageIndex{5}\)

Fill in the missing numbers in these equations

1. \(-2\cdot (-4.5)=?\)
2. \((-8.7)\cdot (-10)=?\)
3. \((-7)\cdot ?=14\)
4. \(?\cdot (-10)=90\)

Exercise \(\PageIndex{6}\)

A weather station on the top of a mountain reports that the temperature is currently \(0^{\circ}\text{C}\) and has been falling at a constant rate of \(3^{\circ}\text{C}\) per hour. If it continues to fall at this rate, find each indicated temperature. Explain or show your reasoning.

1. What will the temperature be in 2 hours?
2. What will the temperature be in 5 hours?
3. What will the temperature be in half an hour?
4. What was the temperature 1 hour ago?
5. What was the temperature 3 hours ago?
6. What was the temperature 4.5 hours ago?

Exercise \(\PageIndex{7}\)

Find the value of each expression.

1. \(\frac{1}{4}\cdot (-12)\)
2. \(-\frac{1}{3}\cdot 39\)
3. \((-\frac{4}{5})\cdot (-75)\)
4. \(-\frac{2}{5}\cdot (-\frac{3}{4})\)
5. \(\frac{8}{3}\cdot -42\)

Exercise \(\PageIndex{8}\)

To make a specific hair dye, a hair stylist uses a ratio of \(1\frac{1}{8}\) oz of red tone, \(\frac{3}{4}\) oz of gray tone, and \(\frac{5}{8}\) oz of brown tone.

1. If the stylist needs to make 20 oz of dye, how much of each dye color is needed?
2. If the stylist needs to make 100 oz of dye, how much of each dye color is needed?

(From Unit 4.1.2)

Exercise \(\PageIndex{9}\)

1. Here are the vertices of rectangle \(FROG\): \((-2,5),\: (-2,1),\: (6,5),\: (6,1)\).
   Find the perimeter of this rectangle. If you get stuck, try plotting the points on a coordinate plane.
2. Find the area of the rectangle \(FROG\).
3. Here are the coordinates of rectangle \(PLAY\): \((-11,20),\: (-11,-3),\: (-1,20),\: (-1,-3)\). Find the perimeter and area of this rectangle. See if you can figure out its side lengths without plotting the points.

(From Unit 5.2.6)