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4.3: Add, Subtract, Multiply and Divide Fractions - Mathematics


Partner Activity 1

Which expression would you rather add?

(dfrac{51}{684}+dfrac{43}{684}+dfrac{738}{684}) OR (dfrac{1}{8}+dfrac{4}{5}+dfrac{1}{9})

Explain to a 3rd grader why:

____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Now, we will explore why fractions behave the way they do for adding, subtracting, multiplying and dividing:

Example (PageIndex{1}): Add Fractions with a Drawing, Number Line, then with Common Denominators

Add (dfrac{1}{2}+dfrac{1}{3}). Why 6 boxes?

Example (PageIndex{2}): Subtract Fractions with a Drawing, Number Line, then with Common Denominators

Subtract (dfrac{1}{2} - dfrac{1}{3}). Why 6 boxes?

Example (PageIndex{3}): Multiply Fractions with a Drawing, Number Line, then with Common Denominators

Add (dfrac{1}{2} imes dfrac{1}{3}). Why 6 boxes?

Example (PageIndex{4}): Divide Fractions with a Drawing, Number Line, then with Common Denominators

Divide (dfrac{1}{2} div dfrac{1}{3}). “Think portions, when it comes to division! Why 6 boxes?

Example (PageIndex{5})

Why does “multiply and flip the second fraction” work when dividing fractions?

Solution

We know that:

[dfrac{a}{b} imes dfrac{b}{a}=dfrac{a b}{a b}=1 onumber ]

[egin{aligned}
dfrac{3}{4} div dfrac{2}{7}&= left(dfrac{3}{4} imes dfrac{7}{2} ight) divleft(dfrac{2}{7} imes dfrac{7}{2} ight)
&=left(dfrac{3}{4} imes dfrac{7}{2} ight) div 1
&=dfrac{3}{4} imes dfrac{7}{2}
&=dfrac{21}{8}
end{aligned} onumber ]

Partner Activity 2

Which Operation is Correct?

A stretch of highway is (3 dfrac{1}{2}) miles long. Each day, (dfrac{2}{3}) of a mile is repaved. How many days are needed to repave the entire section? How would you explain to a 5th grader which operation is correct?

Do we add?

(3 dfrac{1}{2}+dfrac{2}{3}=dfrac{7}{2}+dfrac{2}{3}=dfrac{21}{6}+dfrac{4}{6}=dfrac{25}{6}=4 dfrac{1}{6}) days

Do we subtract?

(3 dfrac{1}{2}-dfrac{2}{3}=dfrac{7}{2}-dfrac{2}{3}=dfrac{21}{6}-dfrac{4}{6}=dfrac{17}{6}=2 dfrac{5}{6}) days

Do we multiply?

(3 dfrac{1}{2} imes dfrac{2}{3}=dfrac{7}{2} imes dfrac{2}{3}=dfrac{14}{6}=2 dfrac{2}{6}=2 dfrac{1}{3}) days

Do we divide?

(3 dfrac{1}{2} div dfrac{2}{3}=dfrac{7}{2} imes dfrac{3}{2}=dfrac{21}{4}=5 dfrac{1}{4}) days

Practice Problems

Add, subtract, multiply or divide the expressions. Use any method.

  1. (dfrac{3}{4}+dfrac{8}{7})
  2. (dfrac{8}{7}-dfrac{3}{4})
  3. (dfrac{3}{4} imes dfrac{8}{7})
  4. (dfrac{3}{4} div dfrac{8}{7})
  5. (5 dfrac{2}{5} div 3 dfrac{1}{6})
  6. (5 dfrac{2}{5}-3 dfrac{1}{6})
  7. (5 dfrac{2}{5} imes 3 dfrac{1}{6})
  8. (5 dfrac{2}{5} div 3 dfrac{1}{6})

Students sometimes find it hard to know how to do fractions.

So, first of all, we will talk about the basics.

Fractions consist of two numbers.

The number on the top is called the numerator.

The number on the bottom is called the denominator.

Rules for Multiplying Fractions

The size of a fraction does not change when you multiply the numerator and denominator by the same number.

Look at the example below.

When we take 1 from the top of the original fraction and multiply it by 2, we get 2 in the top of the new fraction.

When we take 4 from the bottom of the original fraction and multiply it by 2, we get 8 in the bottom of the new fraction.

As a result, we know that, one-fourth is equal to two-eighths.

Multiplying Fractions − Exercises

Find the missing numbers in the fractions below. Then check your answers.

Multiplying Fractions − Answers and Explanations:

The denominator in the new fraction is 9. We know that 3 from the bottom of the first fraction has been multiplied by 3. So, we multiply the numerator of the first fraction by 3 in order to get the numerator of the new fraction.

The denominator in the new fraction is 24, so 6 from the denominator of the first fraction has been multiplied by 4. We take the numerator of the first fraction times 4 in order to get the numerator of the new fraction.

The numerator in the new fraction is 3, so 1 from the top of the first fraction has been multiplied by 3. We multiply the denominator of the first fraction by 3 in order to get the denominator of the new fraction.

The denominator in the new fraction is 20, so 5 from the denominator of the first fraction has been multiplied by 4. We take the numerator of the first fraction times 4 in order to get the numerator of the new fraction.

The numerator in the new fraction is 12, so 4 from the top of the first fraction has been multiplied by 3. We multiply the denominator of the first fraction by 3 in order to get the denominator of the new fraction.

Rules for Dividing Fractions

When you see the division symbol in a fraction problem, you need to invert the second fraction.

"Invert" means to interchange the positions of the numerator and the denominator.

For example, if we invert 2 /3, we get 3 /2.

Once you have inverted the second fraction, need to multiply the two fractions.

Remember to multiply the numerator of the first fraction by the numerator of the second fraction.

Then multiply the denominator of the first fraction by the denominator of the second fraction.


4.3 Multiply and Divide Mixed Numbers and Complex Fractions

Multiply and write the answer in simplified form: 1 8 · 2 3 1 8 · 2 3 .
If you missed this problem, review Example 4.25.

Multiply and Divide Mixed Numbers

In the previous section, you learned how to multiply and divide fractions. All of the examples there used either proper or improper fractions. What happens when you are asked to multiply or divide mixed numbers? Remember that we can convert a mixed number to an improper fraction . And you learned how to do that in Visualize Fractions.

Example 4.37

Solution

Multiply, and write your answer in simplified form: 5 2 3 · 6 17 . 5 2 3 · 6 17 .

Multiply, and write your answer in simplified form: 3 7 · 5 1 4 . 3 7 · 5 1 4 .

How To

Multiply or divide mixed numbers.

  1. Step 1. Convert the mixed numbers to improper fractions.
  2. Step 2. Follow the rules for fraction multiplication or division.
  3. Step 3. Simplify if possible.

Example 4.38

Multiply, and write your answer in simplified form: 2 4 5 ( − 1 7 8 ) . 2 4 5 ( − 1 7 8 ) .

Solution

Multiply, and write your answer in simplified form. 5 5 7 ( − 2 5 8 ) . 5 5 7 ( − 2 5 8 ) .

Multiply, and write your answer in simplified form. −3 2 5 · 4 1 6 . −3 2 5 · 4 1 6 .

Example 4.39

Divide, and write your answer in simplified form: 3 4 7 ÷ 5 . 3 4 7 ÷ 5 .

Solution

Divide, and write your answer in simplified form: 4 3 8 ÷ 7 . 4 3 8 ÷ 7 .

Divide, and write your answer in simplified form: 2 5 8 ÷ 3 . 2 5 8 ÷ 3 .

Example 4.40

Solution

Divide, and write your answer in simplified form: 2 2 3 ÷ 1 1 3 . 2 2 3 ÷ 1 1 3 .

Divide, and write your answer in simplified form: 3 3 4 ÷ 1 1 2 . 3 3 4 ÷ 1 1 2 .

Translate Phrases to Expressions with Fractions

Example 4.41

Translate the phrase into an algebraic expression: “the quotient of 3 x 3 x and 8 .” 8 .”

Solution

The keyword is quotient it tells us that the operation is division. Look for the words of and and to find the numbers to divide.

This tells us that we need to divide 3 x 3 x by 8 . 8 . 3 x 8 3 x 8

Translate the phrase into an algebraic expression: the quotient of 9 s 9 s and 14 . 14 .

Translate the phrase into an algebraic expression: the quotient of 5 y 5 y and 6 . 6 .

Example 4.42

Translate the phrase into an algebraic expression: the quotient of the difference of m m and n , n , and p . p .

Solution

Translate the phrase into an algebraic expression: the quotient of the difference of a a and b , b , and c d . c d .

Translate the phrase into an algebraic expression: the quotient of the sum of p p and q , q , and r . r .

Simplify Complex Fractions

Our work with fractions so far has included proper fractions, improper fractions, and mixed numbers. Another kind of fraction is called complex fraction , which is a fraction in which the numerator or the denominator contains a fraction.

Some examples of complex fractions are:

To simplify a complex fraction, remember that the fraction bar means division. So the complex fraction 3 4 5 8 3 4 5 8 can be written as 3 4 ÷ 5 8 . 3 4 ÷ 5 8 .

Example 4.43

Solution

How To

Simplify a complex fraction.

  1. Step 1. Rewrite the complex fraction as a division problem.
  2. Step 2. Follow the rules for dividing fractions.
  3. Step 3. Simplify if possible.

Example 4.44

Solution

Example 4.45

Solution

Example 4.46

Solution

Simplify Expressions with a Fraction Bar

Where does the negative sign go in a fraction? Usually, the negative sign is placed in front of the fraction, but you will sometimes see a fraction with a negative numerator or denominator. Remember that fractions represent division. The fraction − 1 3 − 1 3 could be the result of dividing −1 3 , −1 3 , a negative by a positive, or of dividing 1 −3 , 1 −3 , a positive by a negative. When the numerator and denominator have different signs, the quotient is negative.

If both the numerator and denominator are negative, then the fraction itself is positive because we are dividing a negative by a negative.

Placement of Negative Sign in a Fraction

For any positive numbers a and b , a and b ,

Example 4.47

Which of the following fractions are equivalent to 7 −8 ? 7 −8 ?

Solution

Which of the following fractions are equivalent to −3 5 ? −3 5 ?

Which of the following fractions are equivalent to − 2 7 ? − 2 7 ?

Fraction bars act as grouping symbols. The expressions above and below the fraction bar should be treated as if they were in parentheses. For example, 4 + 8 5 − 3 4 + 8 5 − 3 means ( 4 + 8 ) ÷ ( 5 − 3 ) . ( 4 + 8 ) ÷ ( 5 − 3 ) . The order of operations tells us to simplify the numerator and the denominator first—as if there were parentheses—before we divide.

We’ll add fraction bars to our set of grouping symbols from Use the Language of Algebra to have a more complete set here.

Grouping Symbols

How To

Simplify an expression with a fraction bar.

  1. Step 1. Simplify the numerator.
  2. Step 2. Simplify the denominator.
  3. Step 3. Simplify the fraction.

Example 4.48

Solution

Example 4.49

Solution

Example 4.50

Simplify: ( 8 − 4 ) 2 8 2 − 4 2 . ( 8 − 4 ) 2 8 2 − 4 2 .

Solution

Simplify: ( 11 − 7 ) 2 11 2 − 7 2 . ( 11 − 7 ) 2 11 2 − 7 2 .

Simplify: ( 6 + 2 ) 2 6 2 + 2 2 . ( 6 + 2 ) 2 6 2 + 2 2 .

Example 4.51

Simplify: 4 ( −3 ) + 6 ( −2 ) −3 ( 2 ) −2 . 4 ( −3 ) + 6 ( −2 ) −3 ( 2 ) −2 .

Solution

Simplify: 8 ( −2 ) + 4 ( −3 ) −5 ( 2 ) + 3 . 8 ( −2 ) + 4 ( −3 ) −5 ( 2 ) + 3 .

Simplify: 7 ( −1 ) + 9 ( −3 ) −5 ( 3 ) −2 . 7 ( −1 ) + 9 ( −3 ) −5 ( 3 ) −2 .

Media

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Section 4.3 Exercises

Practice Makes Perfect

Multiply and Divide Mixed Numbers

In the following exercises, multiply and write the answer in simplified form.

In the following exercises, divide, and write your answer in simplified form.

Translate Phrases to Expressions with Fractions

In the following exercises, translate each English phrase into an algebraic expression.

Simplify Complex Fractions

In the following exercises, simplify the complex fraction.

Simplify Expressions with a Fraction Bar

In the following exercises, identify the equivalent fractions.

Which of the following fractions are equivalent to 5 −11 ? 5 −11 ?
−5 −11 , −5 11 , 5 11 , − 5 11 −5 −11 , −5 11 , 5 11 , − 5 11

Which of the following fractions are equivalent to −4 9 ? −4 9 ?
−4 −9 , −4 9 , 4 9 , − 4 9 −4 −9 , −4 9 , 4 9 , − 4 9

Which of the following fractions are equivalent to − 11 3 ? − 11 3 ?
−11 3 , 11 3 , −11 −3 , 11 −3 −11 3 , 11 3 , −11 −3 , 11 −3

Which of the following fractions are equivalent to − 13 6 ? − 13 6 ?
13 6 , 13 −6 , −13 −6 , −13 6 13 6 , 13 −6 , −13 −6 , −13 6

In the following exercises, simplify.

7 ⋅ 4 − 2 ( 8 − 5 ) 9 ⋅ 3 − 3 ⋅ 5 7 ⋅ 4 − 2 ( 8 − 5 ) 9 ⋅ 3 − 3 ⋅ 5

9 ⋅ 7 − 3 ( 12 − 8 ) 8 ⋅ 7 − 6 ⋅ 6 9 ⋅ 7 − 3 ( 12 − 8 ) 8 ⋅ 7 − 6 ⋅ 6

9 ( 8 − 2 ) −3 ( 15 − 7 ) 6 ( 7 − 1 ) −3 ( 17 − 9 ) 9 ( 8 − 2 ) −3 ( 15 − 7 ) 6 ( 7 − 1 ) −3 ( 17 − 9 )

8 ( 9 − 2 ) −4 ( 14 − 9 ) 7 ( 8 − 3 ) −3 ( 16 − 9 ) 8 ( 9 − 2 ) −4 ( 14 − 9 ) 7 ( 8 − 3 ) −3 ( 16 − 9 )

Everyday Math

Baking A recipe for chocolate chip cookies calls for 2 1 4 2 1 4 cups of flour. Graciela wants to double the recipe.

  1. ⓐ How much flour will Graciela need? Show your calculation. Write your result as an improper fraction and as a mixed number.
  2. ⓑ Measuring cups usually come in sets with cups for 1 8 , 1 4 , 1 3 , 1 2 , and 1 1 8 , 1 4 , 1 3 , 1 2 , and 1 cup. Draw a diagram to show two different ways that Graciela could measure out the flour needed to double the recipe.

Baking A booth at the county fair sells fudge by the pound. Their award winning “Chocolate Overdose” fudge contains 2 2 3 2 2 3 cups of chocolate chips per pound.

  1. ⓐ How many cups of chocolate chips are in a half-pound of the fudge?
  2. ⓑ The owners of the booth make the fudge in 10 10 -pound batches. How many chocolate chips do they need to make a 10 10 -pound batch? Write your results as improper fractions and as a mixed numbers.

Writing Exercises

Explain how to find the reciprocal of a mixed number.

Explain how to multiply mixed numbers.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

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    Math Fraction Games

    Do you want to play fun and interactive math fraction games? The list of games on this webpage is designed to reinforce basic skills, while making learning a breeze.

    The math games on this page are suitable for elementary and middle school students.

    Before adding and subtracting fractions, students must understand basic facts about factors and divisibility. This fun game can be used to reinforce these important math concepts.

    In this interactive jeopardy game, students will practice finding the greatest common factor (GCF) and the least common multiple (LCM) of two or more numbers.

    In this fun soccer game, students will practice reducing fractions to the simplest form.

    This is a fun baseball math game about game about simplifying fractions. Students will have to first hit a homerun to be able to see and answer a math problem.

    Have fun adding and subtracting fractions by playing this interactive board game against another player, or in two teams. This game is an excellent review activity that promotes student engagement.

    In this concentration game, students click on two cards to match equivalent fractions. If there is a match, the problems remain on the page if not, the cards are turned over.

    In this game, students will practice adding and subtracting fractions with common and different denominators.

    Young students can use this fun millionaire game to practice changing decimals and fractions to percents.

    In this jeopardy game, students will convert fractions to decimals and percents and vice-versa.

    In this fun and interactive soccer game, students will test their ability to multiply different fractions.

    How many points can you earn? Practice multiplying fractions by playing this fun game alone, with a partner, or in two teams.

    Do you know how to divide fractions? Show your friends how many points you can score when playing this fun game.

    Dividing fractions does not have to be boring anymore. Play this fun and interactive game and score as many points as possible.

    This online jeopardy game is is a fun way to review the four operations with fractions: addition, subtraction, multiplication, and division.

    In this concentration game, students will match different fractions with the equivalent percents.

    This Fraction Jeopardy Game has two rounds and three categories: fractions, decimals, and percents. The points are doubled in round two. Have fun earning points!


    4.3: Add, Subtract, Multiply and Divide Fractions - Mathematics

    Beginning Algebra
    Tutorial 3: Fractions

    1. Know what the numerator and denominator of a fraction are.
    2. Find the prime factorization of a number.
    3. Simplify a fraction.
    4. Find the least common denominator of given fractions.
    5. Multiply, divide, add and subtract fractions.

    Do you ever feel like running and hiding when you see a fraction? If so, you are not alone. But don't fear help is here. Hey that rhymes. Anyway, in this tutorial we will be going over how to simplify, multiply, divide, add, and subtract fractions. Sounds like we have our work cut out for us. I think you are ready to tackle these fractions.

    b = denominator

    Examples of prime numbers are 2, 3, 5, 7, 11, and 13. The list can go on and on.

    Be careful, 1 is not a prime number because it only has one distinct factor which is 1.

    When you rewrite a number using prime factorization, you write that number as a product of prime numbers.

    For example, the prime factorization of 12 would be

    That last product is 12 and is made up of all prime numbers.

    When is a Fraction Simplified?

    Writing the Fraction in Lowest Terms
    (or Simplifying the Fraction)

    Make sure that you do reduce your answers, as shown above. You may do this before you multiply or after.

    *Div. the common factor of 5 out of both num. and den.

    In other words, you flip the number upside down. The numerator becomes the denominator and vice versa.

    For example, 5 (which can be written as 5/1) and 1/5 are reciprocals. 3/4 and 4/3 are also reciprocals of each other.

    *Write as prod. of num. over prod. of den.

    *Div. the common factor of 2 out of both num. and den.

    Step 2: Put the sum or difference found in step 1 over the common denominator.

    Why do we have to have a common denominator when we add or subtract fractions.
    Another good question. The denominator indicates what type of fraction that you have and the numerator is counting up how many of that type you have. You can only directly combine fractions that are of the same type (have the same denominator). For example if 2 was my denominator, I would be counting up how many halves I had, if 3 was my denominator, I would be counting up how many thirds I had. But, I would not be able to add a fraction with a denominator of 2 directly with a fraction that had a denominator of 3 because they are not the same type of fraction. I would have to find a common denominator first before I could combine, which we will cover after this example.

    Least Common Denominator (LCD)

    You can achieve this by multiplying the top and bottom by the same number. This is like taking it times 1. You can write 1 as any non zero number over itself. For example 5/5 or 7/7. 1 is the identity number for multiplication. In other words, when you multiply a number by 1, it keeps its identity or stays the same value.

    *Multiply num. and den. by 4.

    An improper fraction is a fraction in which the numerator is larger than the denominator.

    *Mult. den. 4 times whole number 7 and add it to num. 3.


    Adding or Subtracting Fractions
    Without Common Denominators

    Therefore, the LCD is 4.

    *Multiply num. and den. by 4.

    Therefore, the LCD is 15.

    *Multiply num. and den. by 5.

    *Multiply num. and den. by 3.

    *Write the sum and difference over the common den.

    *Div. the common factor of 3 out of both num. and den.

    To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that problem. At the link you will find the answer as well as any steps that went into finding that answer.

    Practice Problem 1a: Write the number as a product of primes.

    Practice Problems 2a - 2b: Write the fraction in lowest terms.

    Practice Problems 3a - 3e: Perform the following operations. Write answers in the lowest terms.


    Add Subtract Multiply Divide Fractions Worksheet Generator

    Here is our random worksheet generator for fraction addition, subtraction, multiplication and division worksheets.

    Using this generator will let you create your own worksheets for:

    • Adding, Subtracting, Multiplying and dividing proper fractions
    • Adding, Subtracting, Multiplying and dividing improper fractions
    • Choose exactly which fractions you want to use
    • Choose which part of the equation you want to be missing

    To start creating your sheet, choose an option from the Number values box below.

    Add Subtract Multiply Divide Fractions Worksheet Generator

    4 Steps to Your Worksheets.

    1. Choose your fractions values
    2. Choose your answer values
    3. Choose the number of questions
    4. View your sheet
    5. Print your sheet

    Fraction Values

    You can choose fraction values up to a given denominator value, so that if you choose fifths, then you can get any denominator up to fifths, i.e. halves, thirds, quarters, fifths.

    Fraction Types

    For the generator you can either choose proper fractions, which is the default, or improper fractions. Proper fractions have a numerator less than the denominator.

    You can also select Unit Fractions which have a numerator of 1.

    Fraction Types

    You can select either Unlike Denominators, which is the default, Like Denominators or Denominator Multiples.

    If you select Like Denominators, then the denominators will be the same.

    If you select Denominator Multiples, then one of the denominators will be a multiple of the other denominator.

    If you select 'Improper Fractions' the generator will create both proper and improper fractions.

    Missing Factors

    Sometimes, you may want to add an extra layer of difficulty by providing a fraction equation where one of the other fraction values is missing (instead of the fraction after the = sign).

    Having this feature turned on will create a missing fraction somewhere in the equation - it may be the first number, second number, or third number.


    Applying formulas to multiple cells quickly

    You will probably want to be able to apply calculations in your spreadsheet to multiple cells easily and quickly, and so I want to show you two ways of doing this. (The final examples in this lesson will demonstrate both of these methods with an image included).

    Option 1: After entering a math formula in a cell, copy and paste the cell/formula into the cell(s) below, and the formula will be copied into each individual cell. If there are cell references in your formula, they will adjust automatically as they are copied into the rows below.

    Option 2: You can also use the ARRAYFORMULA function to make your mathematical formulas apply to multiple cells. With ARRAYFORMULA, you can apply a single formula to multiple cells or an entire column.


    4.3: Add, Subtract, Multiply and Divide Fractions - Mathematics

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    Multiplying and Dividing Fractions

    When multiplying fractions, simply multiply across the numerators and across the denominators. Then simplify. You can also simplify first before multiplying.

    Multiply 2 and 4 to get 8. Then multiply 9 and 7 to get 63. The result is (frac<8><63>). There is no simplification needed since the greatest common factor is 1.

    Now suppose we wish to divide (frac<2> <9>div frac<4><7>).

    When dividing fractions, take the first fraction and multiply by the reciprocal of the second. The reciprocal is just interchanging the numerator and denominator. The division problem turns into a multiplication problem.

    2 × 7 = 14 and 9 × 4 = 36. So the answer is (frac<14><36>). But notice this isn’t in simplest form. The greatest common factor is 2, so dividing both by 2 gives the simplified answer of (frac<7><18>).
    Multiplying and Dividing Fractions Video


    4.3: Add, Subtract, Multiply and Divide Fractions - Mathematics

    For the number line above, "1" corresponds or is related to the red point, "2" is related to the green point, "3" is related to the blue point, and so forth. When we move to the right on the number line, we increase in numbers. We defined this as addition. When we move to the left, we decrease. And we defined this as subtraction. This is generally the way a number line works.

    When we add two positive numbers or multiply two positive numbers, we get a positive number. However, we can subtract a positive number from a positive number and suddenly, we don't get a positive number!

    For example, if we subtract 7 from 4, we would start at 4 on the number line and move left 7 places. That brings us to -3. Because -3 is to the left of 0, it is less than zero.

    Looking at the inverse operation, we can say that if 4 – 7 = -3, then -3 + 7 = 4. And that is correct. If we start at -3 and move 7 spaces right, we will be at 4.

    Positive numbers are not just the integers to the right of the zero, but all types of numbers like fractions, decimals, and radicals. Negative numbers also include various forms and various types of numbers that appear left of the zero.

    We don't always have a number line with which to work, so we need to learn a few rules about working with negative numbers. First, we need to define absolute value. The absolute value of a number is the number of units the number is from zero. It is always expressed as positive but without a ‘plus' sign.

    The absolute value of 3 is 3. The absolute value of -3 is also 3. Both 3 and -3 are three units from zero. Absolute value is denoted by writing the number between two vertical bars.

    When adding numbers of the same sign (either positive or negative), add their absolute values and give the result the same sign.

    6 + 5 = 11 (6 and 5 are both positive 6 + 5 is 11 which is positive)

    (-7 and -8 are both negative add |7| + |8| which is 7 + 8 to get 15 the answer is -15)

    If all the numbers in a group to be added are negative: -2 + -3 + -4 = -9, again add the absolute values of 2 + 3 + 4 to get 9 and give it a negative sign.

    Adding Positive and Negative Numbers

    When adding numbers of the opposite sign, take their absolute values, subtract the smaller from the larger, and give the result the sign of the number with the larger absolute value.

    -8 + 6 = |8| which is 8 and |6| which is 6. Subtract the smaller from the larger:

    8 – 6 which gives a result of 2 and give it the sign of the larger number which is 8.

    Subtracting Positive Numbers and Negative Numbers

    When subtracting a positive number from a negative number, use the same rule as for adding two negative numbers: add the absolute values and give the difference a negative sign.

    -5 - 4 = |5| + |4| = |9| = -9 (this is like -5 + -4 = -9)

    When subtracting a negative number from a positive number, the double negative of subtracting a negative becomes a positive, so use the same rule as for adding two positive numbers: add the absolute values and give the difference a positive sign.

    If you were to use a number line, you would go left for subtract and then reverse (to the right) for the negative number, so the final answer is to the right of the original number.

    16 – -10 = |16| + |10| = 16 + 10 = 26

    Additive Inverse

    A number's additive inverse is the number of the opposite sign such that when the two are added, the result is zero.

    As you can see this is the positive and negative numbers of the same absolute value.

    Multiplying Positive and Negative Numbers

    When multiplying a positive number and a negative number (or a negative number and a positive number), multiply the absolute values, and give the answer a negative sign.

    8 x -5 = |8| x |5| = 8 x 5 = 40, but give it the negative sign, making it -40

    To multiply multiple numbers, count the number of negative signs on the numbers to be multiplied. If it is an even number, the product will be positive, and if it is an odd number, the product will be negative.

    6 x -2 x -3 x 5 = |6| x |2| x |3| x |5|

    6 x 2 = 12, 12 x 3 = 36, and 36 x 5 = 180

    There are two negative signs (an even number), so the answer is positive.

    If it had been -6 x -2 x -3 x 5, the answer would have been -180

    Multiplying Two Negative Numbers

    When multiplying two negative numbers, the two negatives cancel each other, so multiply the absolute values, and give the answer a positive sign.

    -21 x -3 = |21| x |3| = 63 (which remains positive)

    Dividing a Negative Number by a Negative Number

    To divide two numbers that have the same sign (either two positives or two negatives), use the absolute values and the result will be positive.

    Dividing a Positive Number by a Negative Number or a Negative Number by a Positive Number

    To divide a pair of numbers that have different signs (negative by positive or positive by negative), use the absolute values of the two numbers, and give the result the negative sign.

    -12 ¸ 3 = |12| ¸ |3| = 4, but it is -4

    18 ¸ -3 = |18| ¸ |3| = 6, but it is -6

    Uses of Negative Numbers

    Negative numbers are used to show cold temperatures. Numbers below 0°C are negative and are below freezing. (Remember that numbers below 32°F are below freezing, but temperatures often go below 0°F.)

    Negative numbers are used to show measurements below sea level. Sea level is given as 0.

    Negative numbers are used with money to show indebtedness, or money owed. If a person or household spends more money than it makes, we say that they are "negative by a certain amount" or we call it "in the red" because accountants use red ink to show negative numbers.

    Greater Than and Less Than and Sets of Numbers

    A set of numbers is a group of numbers that answers a given description. For example, the set of integers less than 0 would be expressed as n < 0. In this sentence, the set of numbers that satisfies the conditions would be all the negative integers.

    All of the integers greater than 0 would be expressed as n > 0. The set of numbers that satisfies these conditions would be the set of all the positive integers. Each of these integers would be called a member or element of that set.

    What are the integers between 3 and 8? That would be 4, 5, 6, and 7. Another way to express this would be the set of numbers that is greater than 3 but less than 8, which can be put into a mathematical sentence that looks like:

    Read this: n such that n is greater than 3 and less than 8

    Because 3 < n or 3 is less than n or n is greater than 3

    And n < 8 or n is less than 8 or 8 is greater than n

    We could have said 3 n < 8 in which case, 3 would have been included in the answer, so n = 3, 4, 5, 6, 7. The sign means ‘less than or equal to' and the sign means ‘greater than or equal to.'

    We make an open circle on the number if it is a ‘greater than' (>) or ‘less than' (<) and we make a filled circle on the number if it is a ‘greater than or equal to' () or a ‘less than or equal to' (). We darken the line from the beginning circle to either an ending circle or the point on the line.

    Using two or more operators in a formula

    When a formula contains two or more mathematical operators, InfoPath performs the calculation according to the precedence of the operator. The following list shows the order in which operations are performed:

    Calculations within parentheses

    Multiplication and division calculations

    Addition and subtraction calculations

    If the formula contains two operators that have the same precedence, the calculations are performed from left to right.

    For example, imagine that you are creating a permit application form template that has a text box that displays the total cost of all of the permits submitted in an application. The values required by this formula come from other text boxes in the form. The text box that shows the total cost contains the following formula:

    txtPermit1Qty * txtPermitCost1 + txtPermit2Qty * txtPermitCost2 /txtNumberOfPermits

    This formula includes addition (+), multiplication (*), and division (/) operators. According to the precedence of operators, the multiplication and division calculations are performed before the addition calculation. Since the multiplication and division operators have the same precedence, the multiplication calculation is done before the division operator because the multiplication operator is located to the left of the division operator. The formula is calculated in this way:

    The value in txtPermit1Qty is multiplied with the value in txtPermitCost1.

    The value in txtPermit2Qty is multiplied with the value in txtPermitCost2, and that result is then divided by the value in txtNumberOfPermits.

    The result of the calculation in step 1 is added to the results of the calculation in step 2.

    To control calculation order, enclose the calculation that you want done first in parentheses. The calculations in the parentheses are performed before calculations outside the parentheses. Put calculations with parentheses to the left of calculations without parentheses. Calculations in nested parentheses are performed working from the inner to the outer parentheses.

    For example, consider the following formula:

    ((txtPermit1Qty * txtPermitCost1) + (txtPermit2Qty * txtPermitCost2))/txtNumberOfPermits

    In this calculation, the value that results from multiplying the values in txtPermit1Qty and txtPermitCost1 is added to the value that results from multiplying the values in txtPermit2Qty and txtPermitCost2. The sum of that calculation is then divided by the value in txtNumberOfPermits.


    Watch the video: Alg II - Adding, Subtracting, Multiplying Polynomials (December 2021).