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Mathematics LibreTexts

1.5: Multiply and Divide Integers

  • Page ID
    18924
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    Learning Objectives

    By the end of this section, you will be able to:

    • Multiply integers
    • Divide integers
    • Simplify expressions with integers
    • Evaluate variable expressions with integers
    • Translate English phrases to algebraic expressions
    • Use integers in applications

    A more thorough introduction to the topics covered in this section can be found in the Prealgebra chapter, Integers.

    Multiply Integers

    Since multiplication is mathematical shorthand for repeated addition, our model can easily be applied to show multiplication of integers. Let’s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction. Here, we will use the model just to help us discover the pattern.

    We remember that \(a\cdot b\) means add \(a,\, b\) times. Here, we are using the model just to help us discover the pattern.

    Two images are shown side-by-side. The image on the left has the equation five times three at the top. Below this it reads “add 5, 3 times.” Below this depicts three rows of blue counters, with five counters in each row. Under this, it says “15 positives.” Under thisis the equation“5 times 3 equals 15.” The image on the right reads “negative 5 times three. The three is in parentheses. Below this it reads, “add negative five, three times.” Under this are fifteen red counters in three rows of five. Below this it reads” “15 negatives”. Below this is the equation negative five times 3 equals negative 15.”
    Figure \(\PageIndex{1}\)

    The next two examples are more interesting.

    What does it mean to multiply \(5\) by \(−3\)? It means subtract \(5, 3\) times. Looking at subtraction as “taking away,” it means to take away \(5, 3\) times. But there is nothing to take away, so we start by adding neutral pairs on the workspace. Then we take away \(5\) three times.

    This figure has two columns. In the top row, the left column contains the expression 5 times negative 3. This means take away 5, three times. Below this, there are three groups of five red negative counters, and below each group of red counters is an identical group of five blue positive counters. What are left are fifteen negatives, represented by 15 red counters. Underneath the counters is the equation 5 times negative 3 equals negative 15. In the top row, the right column contains the expression negative 5 times negative 3. This means take away negative 5, three times. Below this, there are three groups of five blue positive counters, and below each group of blue counters is an identical group of five red negative counters. What are left are fifteen positives, represented by 15 blue counters. Underneath the blue counters is the equation negative 5 times negative 3 equals 15.
    Figure \(\PageIndex{2}\)

    In summary:

    \[\begin{array} {ll} {5 \cdot 3 = 15} &{-5(3) = -15} \\ {5(-3) = -15} &{(-5)(-3) = 15} \end{array}\]

    Notice that for multiplication of two signed numbers, when the:

    • signs are the same, the product is positive.
    • signs are different, the product is negative.

    We’ll put this all together in the chart below.

    MULTIPLICATION OF SIGNED NUMBERS

    For multiplication of two signed numbers:

    Same signs Product Example
    Two positives Positive \(7\cdot 4 = 28\)
    Two negatives Positive \(-8(-6) = 48\)

    Table \(\PageIndex{1}\)

    Different signs Product Example
    Positives \(\cdot\) negative Negative \(7(-9) = -63\)
    Negative \(\cdot\) positives Negative \(-5\cdot 10= -50\)

    Table \(\PageIndex{2}\)

    Exercise \(\PageIndex{1}\)

    Multiply:

    1. \(-9\cdot 3\)
    2. \(-2(-5)\)
    3. \(4(-8)\)
    4. \(7\cdot 6\)
    Answer
    1. \[\begin{array} {ll} {} &{-9\cdot 3} \\ {\text{Multiply, noting that the signs are different, so the product is negative.}} &{-27} \end{array}\]
    2. \[\begin{array} {ll} {} &{-2(-5)} \\ {\text{Multiply, noting that the signs are same, so the product is positive.}} &{10} \end{array}\]
    3. \[\begin{array} {ll} {} &{4(-8)} \\ {\text{Multiply, with different signs.}} &{-32} \end{array}\]
    4. \[\begin{array} {ll} {} &{7\cdot 6} \\ {\text{Multiply, with different signs.}} &{42} \end{array}\]

    Exercise \(\PageIndex{2}\)

    Multiply:

    1. \(-6\cdot 8\)
    2. \(-4(-7)\)
    3. \(9(-7)\)
    4. \(5\cdot 12\)
    Answer
    1. \(-48\)
    2. \(28\)
    3. \(-63\)
    4. \(60\)

    Exercise \(\PageIndex{3}\)

    Multiply:

    1. \(-8\cdot 7\)
    2. \(-6(-9)\)
    3. \(7(-4)\)
    4. \(3\cdot 13\)
    Answer
    1. \(-56\)
    2. \(54\)
    3. \(-28\)
    4. \(39\)

    When we multiply a number by \(1\), the result is the same number. What happens when we multiply a number by \(−1\)? Let’s multiply a positive number and then a negative number by \(−1\) to see what we get.

    \[\begin{array} {lll} {} &{-1\cdot 4} &{-1(-3)}\\ {\text{Multiply.}} &{-4} &{3} \\ {} &{-4\text{ is the opposite of 4.}} &{3\text{ is the opposite of } -3} \end{array}\]
    Each time we multiply a number by \(−1\), we get its opposite!

     

    MULTIPLICATION BY −1

    \[−1a=−a\]

    Multiplying a number by \(−1\) gives its opposite.

    Exercise \(\PageIndex{4}\)

    Multiply:

    1. \(-1 \cdot 7\)
    2. \(-1(-11)\)
    Answer
    1. \[\begin{array} {ll} {} &{-1\cdot 7} \\ {\text{Multiply, noting that the signs are different}} &{-7} \\ {\text{so the product is negative.}} &{-7\text{ is the opposite of 7.}} \end{array}\]
    2. \[\begin{array} {ll} {} &{-1(-11)} \\ {\text{Multiply, noting that the signs are different}} &{11} \\ {\text{so the product is positive.}} &{11\text{ is the opposite of -11.}} \end{array}\]

    Exercise \(\PageIndex{5}\)

    Multiply:

    1. \(-1\cdot 9\)
    2. \(-1\cdot(-17)\)
    Answer
    1. \(-9\)
    2. \(17\)

    Exercise \(\PageIndex{6}\)

    Multiply:

    1. \(-1\cdot 8\)
    2. \(-1\cdot(-16)\)
    Answer
    1. \(-8\)
    2. \(16\)

    Divide Integers

    What about division? Division is the inverse operation of multiplication. So, \(15\div 3=5\) because \(15\cdot 3=5\). In words, this expression says that \(15\) can be divided into three groups of five each because adding five three times gives \(15\). Look at some examples of multiplying integers, to figure out the rules for dividing integers.

    \[\begin{array} {ll} {5\cdot 3 = 15\text{ so }15\div 3 = 5} &{-5(3) = -15\text{ so }-15\div 3 = -5} \\ {(-5)(-3) = 15\text{ so }15\div (-3) = -5} &{5(-3) = -15\text{ so }-15\div (-3) = 5} \end{array}\]

    Division follows the same rules as multiplication!

    For division of two signed numbers, when the:

    • signs are the same, the quotient is positive.
    • signs are different, the quotient is negative.

    And remember that we can always check the answer of a division problem by multiplying.

    MULTIPLICATION AND DIVISION OF SIGNED NUMBERS

    For multiplication and division of two signed numbers:

    • If the signs are the same, the result is positive.
    • If the signs are different, the result is negative.
    Same signs Result
    Two positives Positive
    Two negatives Positive
    If the signs are the same, the result is positive.

    Table \(\PageIndex{3}\)

    Different signs Result
    Positive and negative Negative
    Negative and positive Negative
    If the signs are different, the result is negative.

    Table \(\PageIndex{4}\)

    Exercise \(\PageIndex{7}\)

    1. \(-27\div 3\)
    2. \(-100\div (-4)\)
    Answer
    1. \[\begin{array} {ll} {} &{-27 \div 3} \\ {\text{Divide, with different signs, the quotient is}} &{-9} \\ {\text{negative.}} &{} \end{array}\]
    2. \[\begin{array} {ll} {} &{-100 \div (-4)} \\ {\text{Divide, with signs that are the same the}} &{25} \\ {\text{ quotient is negative.}} &{} \end{array}\]

    Exercise \(\PageIndex{8}\)

    Divide:

    1. \(-42\div 6\)
    2. \(-117\div (-3)\)
    Answer
    1. \(-7\)
    2. \(39\)

    Exercise \(\PageIndex{9}\)

    Divide:

    1. \(-63\div 7\)
    2. \(-115\div (-5)\)
    Answer
    1. \(-9\)
    2. \(23\)

    Simplify Expressions with Integers

    What happens when there are more than two numbers in an expression? The order of operations still applies when negatives are included. Remember My Dear Aunt Sally?

    Let’s try some examples. We’ll simplify expressions that use all four operations with integers—addition, subtraction, multiplication, and division. Remember to follow the order of operations.

    Exercise \(\PageIndex{10}\)

    Simplify:

    \(7(-2)+4(-7)-6\)

    Answer

    \[\begin{array} {ll} {} &{7(-2)+4(-7)-6} \\ {\text{Multiply first.}} &{-14+(-28)-6} \\ {\text{Add.}} &{-42-6} \\{\text{Subtract}} &{-48} \end{array}\]

    Exercise \(\PageIndex{11}\)

    Simplify:

    \(8(-3)+5(-7)-4\)

    Answer

    \(-63\)

    Exercise \(\PageIndex{12}\)

    Simplify:

    \(9(-3)+7(-8)-1\)

    Answer

    \(-84\)

    Exercise \(\PageIndex{13}\)

    Simplify:

    1. \((-2)^{4}\)
    2. \(-2^{4}\)
    Answer
    1. \[\begin{array} {ll} {} &{(-2)^{4}} \\ {\text{Write in expanded form.}} &{(-2)(-2)(-2)(-2)} \\ {\text{Multiply}} &{4(-2)(-2)} \\{\text{Multiply}} &{-8(-2)} \\{\text{Multiply}} &{16} \end{array}\]
    2. \[\begin{array} {ll} {} &{-2^{4}} \\ {\text{Write in expanded form. We are asked to find the opposite of }2^{4}.} &{-(2\cdot 2\cdot 2 \cdot 2)} \\ {\text{Multiply}} &{-(4\cdot 2\cdot 2)} \\{\text{Multiply}} &{-(8\cdot 2)} \\{\text{Multiply}} &{-16} \end{array}\]

    Notice the difference in parts (1) and (2). In part (1), the exponent means to raise what is in the parentheses, the \((−2)\) to the \(4^{th}\) power. In part (2), the exponent means to raise just the \(2\) to the \(4^{th}\) power and then take the opposite.

    Exercise \(\PageIndex{14}\)

    Simplify:

    1. \((-3)^{4}\)
    2. \(-3^{4}\)
    Answer
    1. \(81\)
    2. \(-81\)

    Exercise \(\PageIndex{15}\)

    Simplify:

    1. \((-7)^{2}\)
    2. \(-7^{2}\)
    Answer
    1. \(49\)
    2. \(-49\)

    The next example reminds us to simplify inside parentheses first.

    Exercise \(\PageIndex{16}\)

    Simplify:

    \(12-3(9 - 12)\)

    Answer

    \[\begin{array} {llll} {} &{12-3(9 - 12)} \\ {\text{Subtract parentheses first}} &{12-3(-3)} \\ {\text{Multiply.}} &{12-(-9)} \\{\text{Multiply}} &{-(8\cdot 2)} \\{\text{Subtract}} &{21} \end{array}\]

    Exercise \(\PageIndex{17}\)

    Simplify:

    \(17 - 4(8 - 11)\)

    Answer

    \(29\)

    Exercise \(\PageIndex{18}\)

    Simplify:

    \(16 - 6(7 - 13)\)

    Answer

    \(52\)

    Exercise \(\PageIndex{19}\)

    Simplify:

    \(8(-9)\div (-2)^{3}\)

    Answer

    \[\begin{array} {ll} {} &{8(-9)\div(-2)^{3}} \\ {\text{Exponents first}} &{8(-9)\div(-8)} \\ {\text{Multiply.}} &{-72\div (-8)} \\{\text{Divide}} &{9} \end{array}\]

    Exercise \(\PageIndex{20}\)

    Simplify:

    \(12(-9)\div (-3)^{3}\)

    Answer

    \(4\)

    Exercise \(\PageIndex{21}\)

    Simplify:

    \(18(-4)\div (-2)^{3}\)

    Answer

    \(9\)

    Exercise \(\PageIndex{22}\)

    Simplify:

    \(-30\div 2 + (-3)(-7)\)

    Answer

    \[\begin{array} {ll} {} &{-30\div 2 + (-3)(-7)} \\ {\text{Multiply and divide left to right, so divide first.}} &{-15+(-3)(-7)} \\ {\text{Multiply.}} &{-15+ 21} \\{\text{Add}} &{6} \end{array}\]

    Exercise \(\PageIndex{23}\)

    Simplify:

    \(-27\div 3 + (-5)(-6)\)

    Answer

    \(21\)

    Exercise \(\PageIndex{24}\)

    Simplify:

    \(-32\div 4 + (-2)(-7)\)

    Answer

    \(6\)

    Evaluate Variable Expressions with Integers

    Remember that to evaluate an expression means to substitute a number for the variable in the expression. Now we can use negative numbers as well as positive numbers.

    Exercise \(\PageIndex{25}\)

    When \(n=−5\), evaluate:

    1. \(n+1\)
    2. \(−n+1\).
    Answer
    1. \[\begin{array} {ll} {} &{n+ 1} \\ {\text{Substitute}-5\text{ for } n} &{-5+1} \\ {\text{Simplify.}} &{-4} \end{array}\]
    2. \[\begin{array} {ll} {} &{-n+ 1} \\ {\text{Substitute}-5\text{ for } n} &{-(-5)+1} \\ {\text{Simplify.}} &{-4} \\{\text{Add.}} &{6} \end{array}\]

    Exercise \(\PageIndex{26}\)

    When \(n=−8\), evaluate:

    1. \(n+2\)
    2. \(−n+2\).
    Answer
    1. \(-6\)
    2. \(10\)

    Exercise \(\PageIndex{27}\)

    When \(y=−9\), evaluate:

    1. \(y+8\)
    2. \(−y+8\).
    Answer
    1. \(-1\)
    2. \(17\)

    Exercise \(\PageIndex{28}\)

    Evaluate \((x+y)^{2}\) when \(x = -18\) and \(y = 24\).

    Answer

    \[\begin{array} {ll} {} &{(x+y)^{2}} \\ {\text{Substitute }-18\text{ for }x \text{ and } 24 \text{ for } y} &{(-18 + 24)^{2}} \\ {\text{Add inside parentheses}} &{(6)^{2}} \\{\text{Simplify.}} &{36} \end{array}\]

    Exercise \(\PageIndex{29}\)

    Evaluate \((x+y)^{2}\) when \(x = -15\) and \(y = 29\).

    Answer

    \(196\)

    Exercise \(\PageIndex{30}\)

    Evaluate \((x+y)^{3}\) when \(x = -8\) and \(y = 10\).

    Answer

    \(8\)

    Exercise \(\PageIndex{31}\)

    Evaluate \(20 -z \) when

    1. \(z = 12\)
    2. \(z = -12\)
    Answer
    1. \[\begin{array} {ll} {} &{20 - z} \\ {\text{Substitute }12\text{ for }z.} &{20 - 12} \\ {\text{Subtract}} &{8} \end{array}\]
    2. \[\begin{array} {ll} {} &{20 - z} \\ {\text{Substitute }-12\text{ for }z.} &{20 - (-12)} \\ {\text{Subtract}} &{32} \end{array}\]

    Exercise \(\PageIndex{32}\)

    Evaluate \(17 - k\) when

    1. \(k = 19\)
    2. \(k = -19\)
    Answer
    1. \(-2\)
    2. \(36\)

    Exercise \(\PageIndex{33}\)

    Evaluate \(-5 - b\) when

    1. \(b = 14\)
    2. \(b = -14\)
    Answer
    1. \(-19\)
    2. \(9\)

    Exercise \(\PageIndex{34}\)

    Evaluate:

    \(2x^{2} + 3x + 8\) when \(x = 4\).

    Answer

    Substitute \(4\) for \(x\). Use parentheses to show multiplication.

    \[\begin{array} {ll} {} &{2x^{2} + 3x + 8} \\ {\text{Substitute }} &{2(4)^{2} + 3(4) + 8} \\ {\text{Evaluate exponents.}} &{2(16) + 3(4) + 8} \\ {\text{Multiply.}} &{32 + 12 + 8} \\{\text{Add.}} &{52} \end{array}\]

    Exercise \(\PageIndex{35}\)

    Evaluate:

    \(3x^{2} - 2x + 6\) when \(x =-3\).

    Answer

    \(39\)

    Exercise \(\PageIndex{36}\)

    Evaluate:

    \(4x^{2} - x - 5\) when \(x = -2\).

    Answer

    \(13\)

    Translate Phrases to Expressions with Integers

    Our earlier work translating English to algebra also applies to phrases that include both positive and negative numbers.

    Exercise \(\PageIndex{37}\)

    Translate and simplify: the sum of \(8\) and \(−12\), increased by \(3\).

    Answer

    \[\begin{array} {ll} {} &{\text{the } \textbf{sum} \text{of 8 and -12, increased by 3}} \\ {\text{Translate.}} &{[8 + (-12)] + 3} \\ {\text{Simplify. Be careful not to confuse the}} &{(-4) + 3} \\{\text{brackets with an absolute value sign.}} \\{\text{Add.}} &{-1} \end{array}\]

    Exercise \(\PageIndex{38}\)

    Translate and simplify: the sum of \(9\) and \(−16\), increased by \(4\).

    Answer

    \((9 + (-16)) + 4 - 3\)

    Exercise \(\PageIndex{39}\)

    Translate and simplify: the sum of \(-8\) and \(−12\), increased by \(7\).

    Answer

    \((-8 + (-12)) + 7 - 13\)

    When we first introduced the operation symbols, we saw that the expression may be read in several ways. They are listed in the chart below.

    \(a−b\)
    \(a\) minus \(b\)
    the difference of \(a\) and \(b\)
    \(b\) subtracted from \(a\)
    \(b\) less than \(a\)

    Table \(\PageIndex{5}\)

    Be careful to get a and b in the right order!

    Exercise \(\PageIndex{40}\)

    Translate and then simplify

    1. the difference of \(13\) and \(−21\)
    2. subtract \(24\) from \(−19\).
    Answer
    1. \[\begin{array} {ll} {} &{\text{the } \textbf{difference } \text{of 13 and -21}} \\ {\text{Translate.}} &{13 - (-21)} \\ {\text{Simplify.}} &{34} \end{array}\]
    2. \[\begin{array} {ll} {} &\textbf{subtract }24 \textbf{ from }-19 \\ {\text{Translate.}} &{-19 - 24} \\ {\text{Remember, subtract b from a means }a - b} &{} \\{\text{Simplify.}} &{-43} \end{array}\]

    Exercise \(\PageIndex{41}\)

    Translate and simplify

    1. the difference of \(14\) and \(−23\)
    2. subtract \(21\) from \(−17\).
    Answer
    1. \(14 - (-23); 37\)
    2. \(-17 - 21; -38\)

    Exercise \(\PageIndex{42}\)

    Translate and simplify

    1. the difference of \(11\) and \(−19\)
    2. subtract \(18\) from \(−11\).
    Answer
    1. \(11 - (-19); 30\)
    2. \(-11 - 18; -29\)

    Once again, our prior work translating English to algebra transfers to phrases that include both multiplying and dividing integers. Remember that the key word for multiplication is “product” and for division is “quotient.”

    Exercise \(\PageIndex{43}\)

    Translate to an algebraic expression and simplify if possible: the product of \(−2\) and \(14\).

    Answer

    \[\begin{array} {ll} {} &{\text{the product of }-2 \text{ and } 14} \\ {\text{Translate.}} &{(-2)(14)} \\{\text{Simplify.}} &{-28} \end{array}\]

    Exercise \(\PageIndex{44}\)

    Translate to an algebraic expression and simplify if possible: the product of \(−5\) and \(12\).

    Answer

    \(-5(12); -60\)

    Exercise \(\PageIndex{45}\)

    Translate to an algebraic expression and simplify if possible: the product of \(8\) and \(-13\).

    Answer

    \(-8(13); -104\)

    Exercise \(\PageIndex{46}\)

    Translate to an algebraic expression and simplify if possible: the quotient of \(−56\) and \(−7\).

    Answer

    \[\begin{array} {ll} {} &{\text{the quotient of }-56 \text{ and } -7} \\ {\text{Translate.}} &{-56\div(-7)} \\{\text{Simplify.}} &{8} \end{array}\]

    Exercise \(\PageIndex{47}\)

    Translate to an algebraic expression and simplify if possible: the quotient of \(−63\) and \(−9\).

    Answer

    \(-63\div (-9); 7\)

    Exercise \(\PageIndex{48}\)

    Translate to an algebraic expression and simplify if possible: the quotient of \(−72\) and \(−9\).

    Answer

    \(-72\div (-9); 8\)

    Use Integers in Applications

    We’ll outline a plan to solve applications. It’s hard to find something if we don’t know what we’re looking for or what to call it! So when we solve an application, we first need to determine what the problem is asking us to find. Then we’ll write a phrase that gives the information to find it. We’ll translate the phrase into an expression and then simplify the expression to get the answer. Finally, we summarize the answer in a sentence to make sure it makes sense.

    How to Apply a Strategy to Solve Applications with Integers

    Exercise \(\PageIndex{49}\)

    The temperature in Urbana, Illinois one morning was \(11\) degrees. By mid-afternoon, the temperature had dropped to \(−9\) degrees. What was the difference of the morning and afternoon temperatures?

    Answer
    Step 1. Read the problem. Make sure all the words and ideas are understood.  
    Step 2. Identify what we are asked to find. the difference of the morning and afternoon temperatures
    Step 3. Write a phrase that gives the information to find it. the difference of \(11\) and \(-9\)
    Step 4. Translate the phrase to an expression. \(11 - (-9)\)
    Step 5. Simplify the expression. \(20\)
    Step 6. Write a complete sentence that answers the question. The difference in temperatures was 20 degrees.

    Exercise \(\PageIndex{50}\)

    The temperature in Anchorage, Alaska one morning was \(15\) degrees. By mid-afternoon the temperature had dropped to \(30\) degrees below zero. What was the difference in the morning and afternoon temperatures?

    Answer

    The difference in temperatures was \(45\) degrees.

    Exercise \(\PageIndex{51}\)

    The temperature in Denver was \(−6\) degrees at lunchtime. By sunset the temperature had dropped to \(−15\) degrees. What was the difference in the lunchtime and sunset temperatures?

    Answer

    The difference in temperatures was \(9\) degrees.

    APPLY A STRATEGY TO SOLVE APPLICATIONS WITH INTEGERS.

    1. Read the problem. Make sure all the words and ideas are understood
    2. Identify what we are asked to find.
    3. Write a phrase that gives the information to find it.
    4. Translate the phrase to an expression.
    5. Simplify the expression.
    6. Answer the question with a complete sentence.

    Exercise \(\PageIndex{52}\)

    The Mustangs football team received three penalties in the third quarter. Each penalty gave them a loss of fifteen yards. What is the number of yards lost?

    Answer
    Step 1. Read the problem. Make sure all the words and ideas are understood.  
    Step 2. Identify what we are asked to find. the number of yards lost
    Step 3. Write a phrase that gives the information to find it. three times a \(15\)-yard penalty
    Step 4. Translate the phrase to an expression. \(3(-15)\)
    Step 5. Simplify the expression. \(-45\)
    Step 6. Write a complete sentence that answers the question. The team lost \(45\) yards.

    Exercise \(\PageIndex{53}\)

    The Bears played poorly and had seven penalties in the game. Each penalty resulted in a loss of \(15\) yards. What is the number of yards lost due to penalties?

    Answer

    The Bears lost \(105\) yards.

    Exercise \(\PageIndex{54}\)

    Bill uses the ATM on campus because it is convenient. However, each time he uses it he is charged a $2 fee. Last month he used the ATM eight times. How much was his total fee for using the ATM?

    Answer

    A $16 fee was deducted from his checking account.

    Key Concepts

    • Multiplication and Division of Two Signed Numbers
      • Same signs—Product is positive
      • Different signs—Product is negative
    • Strategy for Applications
      1. Identify what you are asked to find.
      2. Write a phrase that gives the information to find it.
      3. Translate the phrase to an expression.
      4. Simplify the expression.
      5. Answer the question with a complete sentence.