GED Mathematical Reasoning: Number Line And Signed Numbers III – Multiplying & Dividing

 

  • In algebra, it is common to use the letter “x” to represent numbers we don’t know. As you can imagine, the traditional multiplication sign would be easily confused with the letter “x.” So from this point forward, we will usually denote multiplication using the multiplication dot or parenthesis. Keep in mind that all the following statements mean the same thing:  “negative 2 multiplied by 3”

-2 \cdot 3
(-2)3
-2(3)
(-2)(3)

  • The division sign \div is used to denote division. Recall that the fraction bar can also be used to denote division. For instance, both these statements mean the same thing: “-9 divided by 3”

(-9) \div 3
\frac{-9}{3}

 

Multiplying or dividing two signed numbers

  • First, multiply or divide the number parts together.
  • Second, determine the sign of the final answer. If the signs of the two numbers being multiplied are the same, the answer is positive. If the signs are different, the answer is negative.

 

Multiplying more than two signed numbers together

  • First, multiply the number parts together.
  • Then, determine the sign of the final answer. If there are no negative signs or an even number of negative signs among the numbers being multiplied, the final answer is positive. This is because each pair of negative numbers yields a positive result.
  • If there are an odd number of negative signs, the final answer is negative.

 

Solving problems containing multiple operations

  • Always multiply or divide (working from left to right) before adding or subtracting (working from left to right).
  • And if the problem contains a division bar, as in example 6, simplify the numerator and denominator separately before dividing.

 

Example 1

Multiply:  3(-10)

 

We will first multiply the number parts together. And then, determine the sign of the final answer. If the signs of the two numbers being multiplied are the same, the answer is positive. If the signs are different, the answer is negative.

So to multiply 3 and -10, we’ll first multiply 3 times 10, which is equal to 30. And since our two numbers have different signs, the final answer is negative. So 3 times -10 equals -30.

 

Example 2

Multiply:  (-11)(-4)

 

Here we are multiplying -11 by -4. Both numbers are negative, meaning that they have the same sign.

We’ll first multiply the number parts:  11 times 4, which is equal to 44. And since the two numbers being multiplied have the same sign (both are negative), the final answer is positive. So -11 times -4 equals +44.

 

Example 3

Divide: 40 \div (-5)

 

To divide 40 by -5, we’ll first divide the number parts:  40 divided by 5 equals 8. And since the two numbers being divided have different signs, the final answer is negative. So 40 divided by -5 equals -8.

 

Example 4

Divide:  \frac{-36}{-6}

 

First, we’ll divide the number parts:  36 divided by 6 equals 6. And since the two numbers being divided have the same sign, the final answer is positive. So -36 divided by -6 equals +6.

 

Example 5

Multiply:  (-1)(2)(4)(-3)(-2)

 

When multiplying more than two signed numbers together,  the steps are as follows:

First, multiply the number parts together.

Then, determine the sign of the final answer. If there are no negative signs or an even number of negative signs among the numbers being multiplied, the final answer is positive. This is because each pair of negative numbers yields a positive result. If there are an odd number of negative signs, the final answer is negative.

So to multiply -1 , 2 , 4 , -3 and -2, first multiply their number parts: 1, 2, 4, 3, and 2. The answer is: 48

There are three negative signs among the numbers being multiplied, which is an ODD number of negative signs. This means our final answer is negative. So the final answer is:  -48

 

Example 6

Simplify:  \frac{(-2)(-9)}{-6}

 

Notice that example 6 involves both multiplication AND division. And we are being asked to “simplify,” which means to reduce down to the simplest form.

We will be talking a lot more in future lessons about the accepted order in which to solve problems containing multiple operations. For now, follow these simple rules:

Always multiply or divide (working from left to right) before adding or subtracting (working from left to right).

And if the problem contains a division bar, as in example 6, simplify the numerator and denominator separately before dividing.

Applying these guidelines to example six, we will first multiply the numerator before dividing.

(-2)(-9) = +18 or just 18

Then, we will divide 18 by -6, which is equal to -3. The final answer is: -3.

 \frac{18}{-6} = -3

As a final note, don’t let the use of parenthesis confuse you. They are there to help!

At this point, we’ve talked about how parentheses might be used around negative numbers to emphasize the negative sign in addition or subtraction. We’ve also talked about how parentheses are used to denote multiplication. When a problem contains parenthesis, look carefully at the other symbols in the problem and their placement so you know the purpose of the parenthesis and can work the problem accordingly. For example, here the parentheses are used to help us distinguish negative numbers in a subtraction problem:

Subtraction: (-2) – (-4)
Note: there is a subtraction sign between the numbers in parenthesis. **

Whereas here, the parentheses denote multiplication since they are placed right next to each other with no operation or symbol in between:

Multiplication: (-2)(-4)
Note: the parentheses are placed right next to each other with no operation or symbol in between.**

You have seen 1 out of 15 free pages this month.
Get unlimited access, over 1000 practice questions for just $29.99. Enroll Now