Multiplying and Dividing Integers

Multiplying and Dividing Integers exercise

• Determine the correct distance and direction of the blue car by multiplying integers.

  Hints

  Think about in which direction the car is traveling.

  The distance the car travels can be expressed by: $distance = rate\times time$.

  Solution

  Let's see what information we have.

  • The car is traveling at a constant $50$ miles per hour.
  • The car drives for one hour.

  We can rewrite this as a mathematical expression:

  • $50\times1$

  Since both of the numbers are positive, we don't need to worry about the sign changing. You should also remember that no number changes when it is multiplied by one!

  • $50\times1 = 50$

  Since the car you're observing starts at position zero, and travels east (the positive direction), our final answer is $50$ miles east!

• Explain how multiplying integers gives you the distance and direction of each car.

  Hints

  A positive number times a negative number results in a negative number.

  A negative number times a negative number results in a positive number.

  If we turn back time, it is equivalent to "negative" time.

  Solution

  Let's look at our word problems again.

  1. When our car travels at $50$ miles per hour for $2$ hours in a positive direction, we can write $50\times2$. Therefore, since we have two positive numbers, the car will travel $100$ miles in the positive direction which corresponds to an easterly direction.
  2. However, if our car is traveling in the negative direction at $50$ miles an hour for $2$ hours, we should write $(-50)\times2$. Now we are multiplying one positive number and one negative number. A positive number times a negative number always results in a negative product. In this case the car traveled $-100$ miles which means we travelled $100$ miles west.
  3. For "negative" time, we have to write the hours as a negative number. We can rewrite the third sentence mathematically as $50\times(-2)$. As you saw before, a positive number times a negative number always results in a negative product. So the car traveled $-100$ miles which is $100$ miles in an easterly direction.
  4. When the car drives $50$ miles per hour for $2$ hours in a negative direction, and then we turn back time $2$ hours, we should write this mathematically as $(-50)\times(-2)$. Remember, if we are multiplying like signs, the product is positive. Therefore, the car will travel $100$ miles in the positive direction which means it is traveling east.

• Calculate the correct distance and direction using multiplication.

  Hints

  Remember to multiply the car's speed by how many hours it is driven. Don't forget the signs!

  Solution

  In this example, the car is always driving the speed limit ($65$ miles per hour).

  1. The car drives for $3$ hours in an easterly direction. We can write this sentence mathematically as $65\times3 = 195$.
  2. The car drives west for $2$ hours. Since west is our negative direction it makes $(-65)\times2 = -130$ miles.
  3. The car drives in a westerly direction for $4$ hours. Remember, west is our negative direction! We calculate $(-65)\times4 = -260$.
  4. The car drives east for $5$ hours. We have to evaluate $65\times5 = 325$,

  Grandma lives FAR AWAY!

• Calculate the amount of money you have after going to the movies with your friends.

  Hints

  The money you receive is positive money, while money you spend is negative money.

  Remember, you need two digits after the decimal!

  Solution

  We can write $2\times\$20$ for the money you receive from your parents, $\$20$ from your mom and $\$20$ from your dad.

  And we can write $(-1)\times((\$8\times3) + (\$5) + (\$1.50\times3))$ to express the amount of money you spent on movie tickets, popcorn and drinks, respectively. So our final expression would be:

  • $2\times\$20 + (-1)\times(\$8\times3 + \$5 + \$1.50\times3)$

  Using PEMDAS and working from left to right we first have to solve the parentheses. Doing this gives us:

  • $2\times\$20 + (-1)\times(\$24 + \$5) + \$4.50)$

  Next, we distribute the $(-1)$ to all terms inside the final set of parentheses.

  • $2\times\$20 + (-1)\times\$24 + (-1)\times\$5 + (-1)\times\$4.50$

  Simplifying then gives us:

  • $(\$40) + (-\$24) + (-\$5) + (-\$4.50)$

  Next, we should simplify the signs. Doing this leaves us with:

  • $\$40 - \$24 - \$5 - \$4.50$

  Now all we have left is addition and subtraction, so we work from left to right.

  $\begin{array}{rcl} \$40 - \$24 - \$5 - \$4.50 & = & \$16 - \$5 - \$4.50\\ & = & \$11 - \$4.50\\ & = & \$ 6.50 \end{array}$

  After treating your friends to movies and snacks you have $\$6.50$ left.

• Determine the correct sign for the product of the pairs of signs below.

  Hints

  See if you can find a pattern:

  • $5 \times 1 = 5$
  • $5 \times 0 = 0$
  • $(-5) \times (-1) = (5)$
  • $5 \times (-2) = (-10)$

  Solution

  You should have no problem remembering that a positive times a positive yields a positive answer.

  • $(+) \times (+) = (+)$

  When multiplying terms, if there is an odd number of negative terms, then the resulting answer will be negative, as in the following two examples:

  • $(-) \times (+) = (-)$
  • $(+) \times (-) = (-)$

  However, when if you are multiplying terms in which an even number of terms is negative, then the resulting answer will be positive, like in the example below:

  • $(-) \times (-) = (+)$

• Evaluate each of the expressions using multiplication and division.

  Hints

  The rule for signs when dividing two numbers is the same as for multiplication.

  Zero is less than positive numbers and greater than negative numbers.

  Solution

  • A positive number times another positive number equals a positive number: $(+)\times(+) = (+)$. As an example we have $5\times3 = 15$.
  • A negative number times another negative number equals a positive number, too: $(-)\times(-) = (+)$. Thus we have $(-3)\times(-4) = 12$.
  • Any number, positive or negative, times zero equals zero: $(+/-)\times(0) = (0)$. That leaves us with $10\times0 = 0$.
  • The rules for signs when dividing numbers are the same as the rules when multiplying. Additionally, the location of the positive and negative numbers does not matter: $\frac{+}{-} = (-)$ or $\frac{-}{+} = (-)$. So $\frac{35}{-7} = -5$ equals $\frac{-35}{7} = -5$.
  • A negative number times a positive number equals a negative number: $(-)\times(+) = (-)$. As an example we have $(-4)\times4 = -16$.

  So, the right order beginning on the left side of the number line is:

  $\begin{array}{rcl} -4 \times 4 & = & -16 \\ \frac{35}{-7} & = & -~~5\\ 10 \times 0 & = & ~~~~~0\\ -3 \times -4 & = & ~~~12\\ 5 \times 3 & = & ~~~15 \end{array}$