When multiplying and dividing signed numbers, we start by applying the operation to the absolute value of both numbers. If the two numbers have the same sign, the answer will be positive, if the two numbers have different signs, the answer will be negative.
The following rules show that when the signs are the same the answer will always be positive, and when the signs are different the answer will always be negative when multiplying and dividing.
A negative times a positive is a negative: \((–)(+)=(–)\)
A negative times a negative is a positive: \((–)(–)=(+)\)
A positive times a positive is a positive: \((+)(+)=(+)\)
A positive divided by a negative is a negative: \((+)÷(–)=(–)\)
A negative divided by a positive is a negative: \((–)÷(+)=(–)\)
A negative divided by a negative is a positive: \((–)÷(–)=(+)\)
A positive divided by a positive is a positive: \((+)÷(+)=(+)\)
Example 1
What is the product of \((-5)\) and \((-12)\)?
We will start by applying the operation to the absolute value of the two numbers, which is, \(5·12=60\). Since both numbers are negative and a negative times a negative equals a positive, the answer will be positive \(60\).
Example 2
What is the product of \(17\) and \((-34)\)?
We will start by multiplying \(17\) and \(34\), which is \(578\). Since the two numbers have opposite signs and a positive times a negative is a negative, the sign of the answer will be negative. Therefore, the product of \(17\) and \((-34)\) is \(-578\).
Example 3
What is \(-700\) divided by \(-14\)?
First, we apply the operation to the absolute value of both terms, which is \(700÷14=50\). Since both numbers are negative and a negative divided by a negative is a positive, the sign of the answer is positive. Therefore, \(-700\) divided by \(-14\) is positive \(50\).
Example 4
Find the quotient: \(480÷(-30)\).
To find the quotient of \(480\) and \((-30)\), we will start by dividing the absolute value of the two numbers, which is \(480÷30=16\). Since the two numbers have different signs and a positive divided by a negative is a negative, the quotient of \(480(-30)=-16\).
Example 5
Evaluate the expression \([15÷(-3)](-7)\).
To evaluate the expression, we start by evaluating the expression inside the brackets based on the rules for the order of operations. The value of \(15\) divided by \(-3\) can be calculated by applying the operation to the absolute value of the two numbers, which is \(15÷3=5\). Since the two numbers have different signs and a positive divided by a negative is a negative, the quotient of \(15\) and \(-3\) is \(-5\).
Now we need to evaluate \((-5)(-7)\). We start by applying the operation to the absolute value of the two numbers, which is \(57=35\). Since the two numbers have the same sign and a negative times a negative is a positive, the product of \(-5\) and \(-7\) is positive \(35\).
Example 6
Evaluate the expression \(108÷[(-3)(6)]\).
To evaluate the expression, we will follow the order of operation rules and start by evaluating the expression inside the brackets. The product of \(-3\) and \(6\) can be evaluated by multiplying the absolute value of the two numbers, which is \(36=18\). Since the two numbers have different signs and a negative times a positive is a negative, the product is \(-18\).
Now we will evaluate \(108÷(-18)\). To find the quotient of \(108\) and \(-18\), we start by applying the operation to the absolute value of the two numbers, which is \(108÷18\), which equals \(6\). Since the two numbers have different signs and a positive divided by a negative is a negative, the answer will be \(-6\).
Multiplying and Dividing Signed Numbers Sample Questions
Here are a few sample questions going over multiplying and dividing signed numbers.
What is the product of \(19\) and \((-23)\)?
To find the product of \(19\) and \(-23\), we start by applying the operation to the absolute value of the two numbers, which is \(19·23=437\). Next, we determine what the sign of the answer is by following the rules for multiplying signed numbers. Since the numbers have different signs and a positive times a negative is a negative, the product of \(19\) and \(-23\) is \(-437\).
Find the product: \((-25)(-16)\).
To find the product of the \(-25\) and \(-16\), we will start by multiplying the absolute value of the two terms, which is \(25·16=400\). Since the two terms are negative and a negative times a negative is a positive, the product of \(-25\) and \(-16\) is positive \(400\).
Find the quotient of \(-420\) and \(21\).
To find the quotient of \(-420\) and \(21\), we will start by finding the quotient of the absolute value of the two terms, which is \(420÷21=20\). Since the two numbers have different signs and a negative divided by a positive is a negative, the answer will be negative. Therefore, the quotient of \(-420\) and \(21\) is \(-20\).
What is \((-655)÷(-5)\)?
To find the value of \(-65\div-5\), we start by applying the operation to the absolute value of the two numbers, which is \(655÷5=131\). Since the two numbers have the same sign and a negative divided by a negative is a positive, the quotient of \(-655\) and \(-5\) is positive \(131\).
Evaluate: \([4·(-3)]÷(-2)\).
To evaluate the expression, we use the order of operation rules and start by evaluating the expression inside the brackets. To find the product of \(4\) and \(-3\), we start by multiplying the absolute value of the two numbers which is \(4\times3=12\). Since the two numbers have different signs and a positive times a negative is a negative, the answer will be \(-12\). Now we must evaluate \(-12÷(-2)\). To find the quotient, we will start by applying the operation to the absolute value of the two numbers, which is \(12÷2\), which equals \(6\). Since the two numbers have the same sign and a negative divided by a negative is a positive, the answer will be positive \(6\).
