Multiplying and Dividing Integers
Chapter 1: Variables, Expressions and Integers
Lesson 1.7: Multiplying and Dividing Integers
Using sign rules to find products and quotients
Learning Objectives
By the end of this lesson, you should be able to:
- Multiply and divide integers using the rules for positive and negative signs.
- Use integer multiplication and division to solve real-life problems.
God Gives Wisdom and Understanding
Multiplication and division with integers use clear patterns. When we look carefully for these patterns, mathematics becomes more understandable. God gives us the ability to learn, reason, and use knowledge wisely. We can develop these gifts through patience, practice, and careful work.
Memory verse: “For the Lord gives wisdom; from his mouth come knowledge and understanding.” — Proverbs 2:6
1. Multiplying Integers
When multiplying two integers, first find the product of their absolute values. Then determine the sign of the answer.
Rules for multiplying integers:
- The product of two integers with the same sign is positive.
- The product of two integers with different signs is negative.
- The product of any integer and 0 is 0.
Examples:
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Multiplying Integers
2. Products with More Than Two Factors
When multiplying more than two integers, count the negative factors:
- An even number of negative factors gives a positive product.
- An odd number of negative factors gives a negative product.
Examples:
![\[(-1)(-2)(-3)=-6\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-a8b037d15058149d7997c30d52bb41ce_l3.png?resize=175%2C20&ssl=1 "Rendered by QuickLaTeX.com")
There are three negative factors, so the product is negative.
![\[(-1)(-2)(-3)(-4)=24\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-d1c10e3279581a12919c494c0f58fea8_l3.png?resize=210%2C20&ssl=1 "Rendered by QuickLaTeX.com")
There are four negative factors, so the product is positive.
3. Dividing Integers
The sign rules for division are the same as the sign rules for multiplication.
- The quotient of two integers with the same sign is positive.
- The quotient of two integers with different signs is negative.
- The quotient of 0 and any nonzero integer is 0.
- Division by 0 is undefined.
Examples:
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Dividing Integers
Worked Example: Change in the Value of Shares
A person owns 200 computer-company shares. Each share decreases in value by €3. The person also owns 150 utility-company shares, and each of those shares increases in value by €2. Find the total change in value.
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The total change in value is -€300, so the value decreased by €300.
Common Mistakes to Avoid
- Do not add the signs as though they were ordinary numbers. Determine the sign of the answer from the sign rules.
- Do not forget that two negative signs in multiplication or division produce a positive answer.
- Never divide by zero.
Practice Questions
Show your working. First determine the sign, then calculate the absolute values.
1. Skill Practice
A. Find each product or quotient.
B. State whether each product or quotient is positive, negative, or zero.
C. Evaluate each expression when 
, 
, and 
.
D. Find the product and state its sign.
- Explain why the product of any integer and 0 is always 0.
2. Applications
- Shares: A person owns 200 computer-company shares. Each share loses €3 in value. The person also owns 150 utility-company shares, and each share gains €2.
- Write an expression for the total change.
- Find the total change in value.
- MIR submersible: A submersible changes its elevation by -100 feet each minute.
- What is its elevation after 200 minutes if it starts at sea level?
- How long will it take to reach 1,000 feet below sea level?
- Free diving: The positions of five free divers relative to sea level are -46 m, -40 m, -48 m, -51 m, and -60 m.
- Find the mean position.
- Find the median position.
- Which represents a lower position: the mean or the median?
- Home value: The value of a home changes by -2% each year. Using a simple repeated-rate model, what is the total percentage change after 10 years?
- Grocery store: A grocery store loses an average of €210 each day because of spoilage and other operating costs. How much does it lose in a 7-day week?
- Test scores: A student earns scores of 84, 91, 95, 78, and 92 on five mathematics tests. Find the mean score.
- Temperature: At noon, the temperature is 0°F. Six hours later, it is -18°F. What is the average change in temperature per hour?
- Business profits: A company has monthly profits of €1,200, €755, -€450, €210, and -€640. What is the average monthly profit?
- Challenge: A product has five negative integer factors and two positive integer factors. Is the product positive or negative? Explain.
Answers
Show answers: Skill Practice
- 60
- 2
- -80
- -13
- -132
- -350
- -2
- 2
- -360
- -7
- 17
- -120
- Positive
- Negative
- Zero
- Negative
- Positive
- Positive
- 11
- 24
- 240
- 50
- 90
- -60
- 30
- -30
- -42
- -60
- -6; negative
- 24; positive
- 0 multiplied by any integer is 0, so the product is 0.
Show answers: Applications
- a.
; b. 
. The total change is -€300. - a.
feet; b. 
minutes. - a.
m; b. Ordered positions: -60, -51, -48, -46, -40, so the median is -48 m; c. The mean represents a lower position because -49<-48. 
; the home value changes by -20%, a decrease of 20%.
; the store loses €1,470.
.
°F per hour.
.- Negative, because there are five negative factors, an odd number.
Show the key ideas to remember
- Same signs produce a positive product or quotient.
- Different signs produce a negative product or quotient.
- Any integer multiplied by zero equals zero.
- Zero divided by a nonzero integer equals zero.
- Division by zero is undefined.
- For several factors, an even number of negative factors gives a positive product; an odd number gives a negative product.
Remember Proverbs 2:6: “For the Lord gives wisdom.”