1.1: Addition of Real Numbers
Addition of Integers
Objectives
The lesson objectives for The Addition of Real Numbers are:
- Addition of Integers Using Models
- Addition of Integers Using the Number Line
- Addition of Integers Using the Rules
Introduction
In this concept you will learn to add integers by using different representations. You will learn how to add integers by using appropriate models and by using the number line. These methods will lead to the formation of two rules for adding integers.
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Khan Academy Adding/Subtracting Negative Numbers
Guidance
On Monday, Marty borrows $50.00 from his father. On Thursday, he gives his father $28.00.
Write an addition statement to describe Marty’s financial transactions.
Marty borrowed $50.00 which he must repay to his father. Therefore Marty has \begin{align*}-\$50.00\end{align*}.
He returns $28.00 to his father. Now Marty has \begin{align*}-\$50.00+(\$28.00)=-\$22.00\end{align*}. He still owes his father $22.00.
Example A
\begin{align*}5+(-3)=?\end{align*}
The problem can be represented by using color counters. In this case, the red counters represent positive numbers and the yellow ones represent the negative numbers.
The above representation shows the addition of 5 positive counters and 3 negative counters.
One positive counter and one negative counter equals zero. \begin{align*}1+(-1)=0\end{align*}
Draw a line through the counters that equal zero.
The remaining counters represent the answer. Therefore, \begin{align*}5+(-3)=2\end{align*}. The answer is the difference between 5 and 3. The answer takes on the sign of the larger digit and in this case the five has a positive value and it is greater than 3.
Example B
\begin{align*}4+(-7)=?\end{align*}
The above representation shows the addition of 4 positive counters and 7 negative counters.
One positive counter and one negative counter equals zero. \begin{align*}1+(-1)=0\end{align*}
Draw a line through the counters that equal zero.
The remaining counters represent the answer. Therefore, \begin{align*}4+(-7)=-3\end{align*}. The answer is the difference between 7 and 4. The answer takes on the sign of the larger digit and in this case the seven has a negative value and it is greater than 4.
Example C
This same method can be extended to adding variables. Algebra tiles can be used to represent positive and negative values.
\begin{align*}6x+(-8x)=?\end{align*}
The green algebra tiles represent positive \begin{align*}x\end{align*} and the white tiles represent negative \begin{align*}x\end{align*}. There are 6 positive \begin{align*}x\end{align*} tiles and 8 negative \begin{align*}x\end{align*} tiles.
The remaining algebra tiles represent the answer. There are two negative \begin{align*}x\end{align*} tiles remaining. Therefore, \begin{align*}(6x)+(-8x)=-2x\end{align*}. The answer is the difference between \begin{align*}8x\end{align*} and \begin{align*}6x\end{align*}. The answer takes on the sign of the larger digit and in this case the eight has a negative value and it is greater than 6.
Example D
\begin{align*}(-3)+(-5)=?\end{align*}
The solution to this problem can be determined by using the number line.
Indicate the starting point of -3 by using a dot. From this point, add a -5 by moving five places to the left. You will stop at -8.
The point where you stopped is the answer to the problem. Therefore, \begin{align*}(-3)+(-5)=-8\end{align*}
The answer is the sum of 5 and 3. The answer takes on the sign of the digits being added. In this case the 3 and the 5 have negative signs. The answer will be a negative number.
From using models to add integers, there are two rules that become obvious. These rules are:
- When you add integers that have the same sign, you add the numbers and use the sign of those numbers.
- When you add integers that have different signs, you subtract the numbers and use the sign of the larger number.
Vocabulary
- Integer
- All natural numbers, their opposites, and zero are integers. A number in the list ..., -3, -2, -1, 0, 1, 2, 3...
- Irrational Numbers
- The irrational numbers are those that cannot be expressed as the ratio of two numbers. The irrational numbers include decimal numbers that are non-terminating decimals as well as those with digits that do not repeat with a pattern.
- Natural Numbers
- The natural numbers are the counting numbers and consist of all positive, whole numbers. The natural numbers are numbers in the list 1, 2, 3... and are often referred to as positive integers.
- Number Line
- A number line is a line that matches a set of points and a set of numbers one to one.
It is often used in mathematics to show mathematical computations.
- Rational Numbers
- The rational numbers are numbers that can be written as the ratio of two numbers \begin{align*}\frac{a}{b}\end{align*} and \begin{align*}b \neq 0\end{align*}. The rational numbers include all terminating decimals as well as those decimals that are non-terminating but have a repeating pattern of digits.
- Real Numbers
- The rational numbers and the irrational numbers make up the real numbers.
Guided Practice
- Use a model to answer the problem \begin{align*}(-7)+(+5)=?\end{align*}
- Use the number line to determine the answer to the problem \begin{align*}8+(-2)=?\end{align*}
- Determine the answer to \begin{align*}(-6)+(-3)=?\end{align*} and \begin{align*}(2)+(-5)=?\end{align*} by using the rules for adding integers.
Answers
1. \begin{align*}(-7)+(+5)=?\end{align*}
Cancel the counters that equal zero
There are 2 negative counters left. Therefore, \begin{align*}(-7)+(+5)=-2\end{align*}. The answer is the difference between 7 and 5. The answer takes on the sign of the larger digit and in this case the seven has a negative value and it is greater than 5.
2. \begin{align*}8+(-2)=?\end{align*}
You begin on 8 and move two places to the left. You stop at 6. Therefore \begin{align*}8+(-2)=6\end{align*}.
The answer is the difference between 8 and 2. The answer takes on the sign of the larger digit and in this case the eight has a positive value and it is greater than 2.
3. \begin{align*}(-6)+(-3)=?\end{align*}
Both numbers have negative signs. The numbers must be added and the sum will be a negative answer. Therefore \begin{align*}(-6)+(-3)=-9\end{align*}.
\begin{align*}(2)+(-5)=?\end{align*}
The numbers being added have different signs. The numbers must be subtracted and the answer will have the sign of the larger digit. Therefore \begin{align*}(2)+(-5)=-3\end{align*}.
Summary
The addition of integers can be represented with manipulative such as color counters and algebra tiles. A number line can also be used to show the addition of integers.
The addition of integers can be done by following two rules: Integers with unlike signs must be subtracted and the answer will have a sign the same as that of the higher digit. Integers with the same sign must be added and the answer will have a sign the same as that of the digits being added.
Problem Set
Use color counters to represent the following addition problems and use that model to determine the answer.
- \begin{align*}(-7)+(-2)\end{align*}
- \begin{align*}(6)+(-8)\end{align*}
- \begin{align*}(5)+(4)\end{align*}
- \begin{align*}(-7)+(9)\end{align*}
- \begin{align*}(-1)+(5)\end{align*}
Use a number line to represent the following addition problems and use the number line to determine the answer.
- \begin{align*}(8)+(-12)\end{align*}
- \begin{align*}(-2)+(-5)\end{align*}
- \begin{align*}(3)+(4)\end{align*}
- \begin{align*}(-6)+(10)\end{align*}
- \begin{align*}(-1)+(-7)\end{align*}
Use the rules that you have learned for adding integers to answer the following problems and state the rule that you used.
- \begin{align*}(-13)+(9)\end{align*}
- \begin{align*}(-3)+(-8)+(12)\end{align*}
- \begin{align*}(14)+(-6)+(5)\end{align*}
- \begin{align*}(15)+(-8)+(-9)\end{align*}
- \begin{align*}(7)+(6)+(-9)+(-8)\end{align*}
For each of the following models, write an addition problem and answer the problem.
Answers
Use color counters...
- \begin{align*}(-7)+(-2)\end{align*} There are no counters that cancel out. Add the counters. There are nine negative counters. \begin{align*}(-7)+(-2)=-9\end{align*}
- \begin{align*}(5)+(4)\end{align*} There are no counters that cancel out. Add the positive counters. There are nine positive counters. \begin{align*}(5)+(4)=9\end{align*}
- \begin{align*}(-1)+(5)\end{align*} One negative counter and one positive counter cancel out. There are four positive counters remaining. \begin{align*}(-1)+(5)=4\end{align*}
Use a number line...
- \begin{align*}(8)+(-12)\end{align*} \begin{align*}(8)+(-12)=-4\end{align*}
- \begin{align*}(3)+(4)\end{align*} \begin{align*}(3)+(4)=7\end{align*}
- \begin{align*}(-1)+(-7)\end{align*} \begin{align*}(-1)+(-7)=-8\end{align*}
Use the rules...
\begin{align*}&(-13)+(9)\\ & (-13)+(9)=-4\end{align*}
The numbers have unlike or opposite signs so they were subtracted and the answer has a negative sign which is the same sign as the larger digit 13.
\begin{align*}& (14)+(-6)+(5)\\ & (14)+(-6)+(5)=(19)+(-6)=13\end{align*}
The two positive numbers were added and the answer of 19 had a positive sign which was the sign of the numbers being added. The answer of 19 was then added to -6 and the answer was calculated by subtracting the two numbers. The answer of 13 has a positive sign which is the sign of the larger number 19.
\begin{align*}& (7)+(6)+(-9)+(-8)\\ & (7)+(6)=13\\ & (-9)+(-8)=-17\\ & (13)+(-17)=-4\end{align*}
The two positive numbers were added to give a positive answer of 13. The two negative numbers were added to give a negative answer of 17. The two answers were then added to give the final answer of -4. The answer was calculated by subtracting the two numbers and applying the negative sign of the larger number to the answer.
For each of the following models...
- \begin{align*}& (-6x)+(4x)\\ & (-6x)+(4x)=-2x\end{align*}
- \begin{align*}& (3x^2)+(-2x^2)+(3x)+(-5x)\\ & (3x^2)+(-2x^2)+(3x)+(-5x)=1x^2-2x\end{align*}
- \begin{align*}& (-12x)+(7x)\\ & (-12x)+(7x)=-5x\end{align*}
Addition of Fractions
Objectives
The lesson objectives for The Addition of Real Numbers are:
- Addition of Fractions Using Models
- Addition of Fractions Using the Number Line
- Addition of Fractions Using the Rules
Introduction
In this concept you will learn to add real numbers using different representations. You will learn to add fractions by using appropriate models and by using the number line. These methods will lead to the formation of rules for adding fractions.
Watch This
Khan Academy Adding and Subtracting Fractions
Guidance
\begin{align*}\frac{2}{5}+\frac{1}{5}=?\end{align*}
The problem can be represented by using fraction strips. You can create these fraction strips yourself or you can use commercial pieces called Fraction Factory pieces. Those being presented in the following examples are not the commercial type. Therefore, the colors used are simply a personal choice. This first example will explore adding positive fractions that have the same denominator.
\begin{align*}\boxed{\frac{2}{5} + \frac{1}{5} = \frac{2+1}{5} = \frac{3}{5}}\end{align*}
To add fractions, the fractions must have the same bottom numbers (denominators). Both fractions have a denominator of 5. The answer is the result of adding the top numbers (numerators). The numbers in the numerator are 1 and 2. The sum of 1 and 2 is 3. This sum is written in the numerator over the denominator of 5. Therefore \begin{align*}\frac{2}{5}+\frac{1}{5}=\frac{3}{5}\end{align*}.
Example A
\begin{align*}\frac{3}{7}+\frac{2}{7}=?\end{align*}
This first example will explore adding positive fractions that have the same denominator.
\begin{align*}\boxed{\frac{3}{7} + \frac{2}{7} = \frac{3+2}{7} = \frac{5}{7}}\end{align*}
To add fractions, the fractions must have the same bottom numbers (denominators). Both fractions have a denominator of 7. The answer is the result of adding the top numbers (numerators). The numbers in the numerator are 3 and 2. The sum of 3 and 2 is 5. This sum is written in the numerator over the denominator of 7. Therefore \begin{align*}\frac{3}{7}+\frac{2}{7}=\frac{5}{7}\end{align*}.
Example B
Louise is taking inventory of the amount of water in the water coolers located in the school. She estimates that one cooler is \begin{align*}\frac{2}{3}\end{align*} full and the other is \begin{align*}\frac{1}{4}\end{align*} full. What single fraction could Louise use to represent the amount of water of the two coolers together?
Use fraction strips to represent each fraction.
The strips have now been combined to represent \begin{align*}\frac{2}{3}+\frac{1}{4}\end{align*}.
\begin{align*}\frac{2}{3}\end{align*} and \begin{align*}\frac{8}{12}\end{align*} are equivalent fractions. \begin{align*}\frac{2}{3} \left(\frac{4}{4}\right)=\frac{8}{12}\end{align*}.
\begin{align*}\frac{1}{4}\end{align*} and \begin{align*}\frac{3}{12}\end{align*} are equivalent fractions. \begin{align*}\frac{1}{4} \left(\frac{3}{3}\right)=\frac{3}{12}\end{align*}.
The two green pieces will be replaced with eight purple pieces and the one blue piece will be replaced with three purple pieces.
The amount of water in the two coolers can be represented by the single fraction \begin{align*}\frac{11}{12}\end{align*}.
\begin{align*}& \frac{2}{3}+\frac{1}{4}\\ & \frac{8}{12}+\frac{3}{12}\\ & =\frac{11}{12}\end{align*}
The denominator of 12 is the LCD (least common denominator) of \begin{align*}\frac{2}{3}\end{align*} and \begin{align*}\frac{1}{4}\end{align*} because it is the LCM (least common multiple) of the denominators 3 and 4.
Example C
A number line can also be used to add fractions. In the following example, a mixed number which is a whole number and a fraction will be added to a fraction by using a \begin{align*}\frac{1}{4}\end{align*} number line.
\begin{align*}2\frac{3}{4}+\frac{1}{2}\end{align*}
The number line is labeled in intervals of 4 which indicates that each interval represents \begin{align*}\frac{1}{4}\end{align*}. From zero, move to the number 2 plus 3 more intervals to the right. Mark the location. This represents \begin{align*}2 \frac{3}{4}\end{align*}.
From here, move to the right \begin{align*}\frac{1}{2}\end{align*} or \begin{align*}\frac{1}{2}\end{align*} of 4, which is 2 intervals. An equivalent fraction for is \begin{align*}\frac{1}{2}\end{align*} is \begin{align*}\frac{2}{4}\end{align*}.
The sum of \begin{align*}2 \frac{3}{4}\end{align*} is \begin{align*}\frac{1}{2}\end{align*} is \begin{align*}3\frac{1}{4}\end{align*}.
From using models to add fractions, there are two rules that become obvious. These rules are:
- Fractions can be added if they have the same denominator. To add fractions that have the same denominator, add the numerators and write the sum over the common denominator.
- When you add fractions that have different denominators, you must express the fractions as equivalent fractions with a LCD. Now, add the numerators and write the sum over the common denominator.
Vocabulary
- Denominator
- The denominator of a fraction is the number on the bottom that indicates the total number of equal parts in the whole or the group. \begin{align*}\frac{5}{8}\end{align*} has denominator 8.
- Fraction
- A fraction is any rational number that is not an integer.
- Improper Fraction
- An improper fraction is a fraction in which the numerator is larger than the denominator.
- \begin{align*}\frac{8}{3}\end{align*} is an improper fraction.
- LCD
- The least common denominator is the lowest common multiple of the denominators of two or more fractions. The least common denominator of \begin{align*}\frac{3}{4}\end{align*} and \begin{align*}\frac{1}{5}\end{align*} is 20.
- LCM
- The least common multiple is the lowest common multiple that two or more numbers share. The least common multiple of 6 and 5 is 30.
- Mixed Number
- A mixed number is a number made up of a whole number and a fraction such as \begin{align*}4\frac{3}{5}\end{align*}.
- Numerator
- The numerator of a fraction is the number on top that is the number of equal parts being considered in the whole or the group. \begin{align*}\frac{5}{8}\end{align*} has numerator 5.
Guided Practice
- Use a model to answer the problem \begin{align*}\frac{1}{2}+\frac{1}{6}=?\end{align*}
- Use a number line to determine the answer to the problem \begin{align*}\frac{3}{4}+\frac{1}{2}\end{align*}.
- Determine the answer to \begin{align*}\frac{1}{6}+\frac{3}{4}=?\end{align*} and \begin{align*}\frac{2}{5}+\frac{2}{3}=?\end{align*} by using the rules for adding fractions.
Answers
1.
Use fraction strips to represent each fraction.
The strips have now been combined to represent \begin{align*}\frac{1}{2}+\frac{1}{6}\end{align*}.
\begin{align*}\frac{2}{3}\end{align*} and \begin{align*}\frac{3}{6}\end{align*} are equivalent fractions. \begin{align*}\frac{1}{2} \left(\frac{3}{3}\right)=\frac{3}{6}\end{align*}.
The two fractions now have the same denominator of 6.
The one yellow strip can be replaced with three green strips and the one orange strip can be replaced with one green strip.
\begin{align*}& \frac{1}{2}+\frac{1}{6}\\ & \frac{3}{6}+\frac{1}{6}\\ & = \frac{4}{6}\end{align*}
2. \begin{align*}\frac{3}{4}+\frac{1}{2}\end{align*}
Use a \begin{align*}\frac{1}{4}\end{align*} number line. The number line is labeled in intervals of 4. Place the starting point at \begin{align*}\frac{3}{4}\end{align*}.
From this point, move to the right a total of 2 intervals. \begin{align*}\frac{1}{2}\end{align*} of \begin{align*}4=2\end{align*}. An equivalent fraction for \begin{align*}\frac{1}{2}\end{align*} is \begin{align*}\frac{2}{4} \cdot \frac{1}{2} \left(\frac{2}{2}\right)=\frac{2}{4}\end{align*}. The point where you stop is the sum of \begin{align*}\frac{3}{4}+\frac{1}{2}\end{align*}.
\begin{align*}& \frac{3}{4}+\frac{1}{2}\\ & \frac{3}{4}+\frac{2}{4}\\ & =\frac{5}{4}\end{align*}
On the number line you stopped at the point \begin{align*}1 \frac{1}{4}\end{align*}. This is equal to \begin{align*}\frac{5}{4}\end{align*}.
3. \begin{align*}\frac{1}{6}+\frac{3}{4}=?\end{align*}
The least common multiple of 6 and 4 is 12. This means that both fractions must have a common denominator of 12 before they can be added.
\begin{align*}\frac{1}{6} \left(\frac{2}{2}\right)=\frac{2}{12}\end{align*} \begin{align*}\frac{1}{6}\end{align*} and \begin{align*}\frac{2}{12}\end{align*} are equivalent fractions.
\begin{align*}\frac{3}{4} \left(\frac{3}{3}\right)=\frac{9}{12}\end{align*} \begin{align*}\frac{3}{4}\end{align*} and \begin{align*}\frac{9}{12}\end{align*} are equivalent fractions.
\begin{align*}& \frac{1}{6}+\frac{3}{4}\\ & \frac{2}{12}+\frac{9}{12}\\ & =\frac{11}{12}\end{align*}
\begin{align*}\frac{2}{5}+\frac{2}{3}=?\end{align*}
The least common multiple of 5 and 3 is 15. This means that both fractions must have a common denominator of 15 before they can be added.
\begin{align*}\frac{2}{5} \left(\frac{3}{3}\right)=\frac{6}{15}\end{align*} \begin{align*}\frac{2}{5}\end{align*} and \begin{align*}\frac{6}{15}\end{align*} are equivalent fractions.
\begin{align*}\frac{2}{3} \left(\frac{5}{5}\right)=\frac{10}{15}\end{align*} \begin{align*}\frac{2}{3}\end{align*} and \begin{align*}\frac{10}{15}\end{align*} are equivalent fractions.
\begin{align*}& \frac{2}{5}+\frac{2}{3}\\ & \frac{6}{15}+\frac{10}{15}\\ & =\frac{16}{15}=1 \frac{1}{15}\end{align*}
\begin{align*}\frac{16}{15}\end{align*} is an improper fraction. An improper fraction is one which has a larger numerator than denominator. \begin{align*}\frac{15}{15}=1\end{align*} plus there is \begin{align*}\frac{1}{15}\end{align*} left over. This can be written as a whole number and a fraction \begin{align*}1 \frac{1}{15}\end{align*}. This representation is called a mixed number.
Summary
The addition of fractions can be represented with a manipulative such as a fraction strip. A number line can also be used to show the addition of fractions. The sum of two fractions will often result in an answer that is an improper fraction. An improper fraction is a fraction which has a larger numerator than denominator. This answer can be written as a mixed number. A mixed number is a number made up of a whole number and a fraction.
The addition of fractions can be done by following two rules:
Fractions can be added only if they have the same denominator. To add fractions that have the same denominator, add the numerators and write the sum over the common denominator.
In order to add fractions that have different denominators, the fractions must be expressed as equivalent fractions with a LCD. The sum of the numerators can be written over the common denominator.
Problem Set
Use fraction strips to represent the following addition problems and use that model to determine the answer.
- \begin{align*}\frac{1}{4}+\frac{5}{8}\end{align*}
- \begin{align*}\frac{2}{5}+\frac{1}{3}\end{align*}
- \begin{align*}\frac{2}{9}+\frac{2}{3}\end{align*}
- \begin{align*}\frac{3}{7}+\frac{2}{3}\end{align*}
- \begin{align*}\frac{7}{10}+\frac{1}{5}\end{align*}
Use a number line to represent the following addition problems and use the number line to determine the answer.
- \begin{align*}\frac{2}{3}+\frac{1}{2}\end{align*}
- \begin{align*}\frac{2}{5}+\frac{3}{10}\end{align*}
- \begin{align*}\frac{5}{9}+\frac{2}{3}\end{align*}
- \begin{align*}\frac{3}{8}+\frac{3}{4}\end{align*}
- \begin{align*}\frac{3}{5}+\frac{3}{10}\end{align*}
Use the rules that you have learned for adding integers to answer the following problems. Express all answers as proper fractions or as mixed numbers.
- \begin{align*}\frac{7}{11}+\frac{1}{2}\end{align*}
- \begin{align*}\frac{7}{8}+\frac{5}{12}\end{align*}
- \begin{align*}\frac{3}{4}+\frac{5}{6}\end{align*}
- \begin{align*}\frac{5}{6}+\frac{2}{5}\end{align*}
- \begin{align*}\frac{4}{5}+\frac{3}{4}\end{align*}
For each of the following questions, write an addition statement and find the result. Express all answers as either proper fraction or mixed numbers.
- Karen used \begin{align*}\frac{5}{8} \ cups\end{align*} of flour to make cookies. Jenny used \begin{align*}\frac{15}{16} \ cups\end{align*} of flour to make a cake. How much flour did they use altogether?
- Lauren used \begin{align*}\frac{3}{4} \ cup\end{align*} of milk, \begin{align*}1 \frac{1}{3} \ cups\end{align*} of flour and \begin{align*}\frac{3}{8} \ cup\end{align*} of oil to make pancakes. How many cups of ingredients did she use in total?
- Write two fractions with different denominators whose sum is \begin{align*}\frac{5}{6}\end{align*}. Use fraction strips to model your answer.
- Allan’s cat ate \begin{align*}2 \frac{2}{3} \ cans\end{align*} of food in one week and \begin{align*}3 \frac{1}{4} \ cans\end{align*} the next week. How many cans of food did the cat eat in two weeks?
- Amanda and Justin each solved the same problem. Amanda’s Solution \begin{align*}& \frac{1}{6}+\frac{3}{4}\\ & \frac{2}{12}+\frac{9}{12}\\ & =\frac{11}{24}\end{align*} Justin’s Solution \begin{align*}& \frac{1}{6}+\frac{3}{4}\\ & \frac{2}{12}+\frac{9}{12}\\ & =\frac{11}{12}\end{align*} Who is correct? What would you tell the person who has the wrong answer?
Answers
Use fraction strips...
- \begin{align*}\frac{1}{4} + \frac{5}{8}\end{align*} The strips have been combined to represent \begin{align*}\frac{1}{4}+\frac{5}{8}\end{align*}. The least common multiple of 4 and 8 is 8. Therefore both fractions must have a common denominator of 8. \begin{align*}\frac{1}{4} \left(\frac{2}{2}\right)=\frac{2}{8}\end{align*}. The blue strip can be replaced with 2 orange strips and the 5 pink strips will be replaced with 5 orange strips. \begin{align*}& \frac{1}{4}+\frac{5}{8}\\ & \frac{2}{8}+\frac{5}{8}\\ & =\frac{7}{8}\end{align*}
- \begin{align*}\frac{2}{9}+\frac{2}{3}\end{align*} The strips have been combined to represent \begin{align*}\frac{2}{9}+\frac{2}{3}\end{align*}. The least common multiple of 9 and 3 is 9. Therefore both fractions must have a common denominator of 9. \begin{align*}\frac{2}{3} \left(\frac{3}{3}\right)=\frac{6}{9}\end{align*}. The gold strips can be replaced with 2 red strips and the 2 green strips will be replaced with 6 red strips. \begin{align*}& \frac{2}{9}+\frac{2}{3}\\ & \frac{2}{9}+\frac{6}{9}\\ & =\frac{8}{9}\end{align*}
- The strips have been combined to represent \begin{align*}\frac{7}{10}+\frac{1}{5}\end{align*}. The least common multiple of 10 and 5 is 10. Therefore both fractions must have a common denominator of 10. \begin{align*}\frac{1}{5} \left(\frac{2}{2}\right)=\frac{2}{10}\end{align*}. The blue strips can be replaced with 7 green strips and the 1 yellow strip will be replaced with 2 green strips. \begin{align*}& \frac{7}{10}+\frac{1}{5}\\ & \frac{7}{10}+\frac{2}{10}\\ & =\frac{9}{10}\end{align*}
Use a number line...
- \begin{align*}\frac{2}{3}+\frac{1}{2}\end{align*} The least common multiple of 3 and 2 is 6. Use a number line that is marked in intervals of 6. \begin{align*}\frac{2}{3} & \rightarrow \frac{2}{3} \left(\frac{2}{2}\right)=\frac{4}{6}\\ \frac{1}{2} & \rightarrow \frac{1}{2} \left(\frac{3}{3}\right)=\frac{3}{6}\end{align*} \begin{align*}& \frac{2}{3}+\frac{1}{2}\\ & \frac{4}{6}+\frac{3}{6}\\ & =\frac{7}{6}=1 \frac{1}{6}\end{align*}
- \begin{align*}\frac{5}{9}+\frac{2}{3}\end{align*} Use a \begin{align*}\frac{1}{9}\end{align*} number line since the least common multiple of 9 and 3 is 9. \begin{align*}\frac{2}{3} \rightarrow \frac{2}{3} \left(\frac{3}{3}\right)=\frac{6}{9}\end{align*} \begin{align*}& \frac{5}{9}+\frac{2}{3}\\ & \frac{5}{9}+\frac{6}{9}\\ & =\frac{11}{9}=1 \frac{2}{9}\end{align*}
- \begin{align*}\frac{3}{5}+\frac{3}{10}\end{align*} The least common multiple of 5 and 10 is 10. Use a number line that is labeled in intervals of 10. \begin{align*}\frac{3}{5}\rightarrow \frac{3}{5} \left(\frac{2}{2}\right)=\frac{6}{10}\end{align*} \begin{align*}& \frac{3}{5}+\frac{3}{10}\\ & \frac{6}{10}+\frac{3}{10}\\ & =\frac{9}{10}\end{align*}
Use the rules...
- \begin{align*}\frac{7}{11}+\frac{1}{2}\end{align*} The least common multiple of 11 and 2 is 22. \begin{align*}& \frac{7}{11} \left(\frac{2}{2}\right)+\frac{1}{2}\left(\frac{11}{11}\right)\\ & \frac{14}{22}+\frac{11}{22}\\ & =\frac{25}{22}=1 \frac{3}{22}\end{align*}
- \begin{align*}\frac{3}{4}+\frac{5}{6}\end{align*} The least common multiple of 4 and 6 is 12. \begin{align*}& \frac{3}{4} \left(\frac{3}{3}\right)+\frac{5}{6} \left(\frac{2}{2}\right)\\ & \frac{9}{12}+\frac{10}{12}\\ & =\frac{19}{12}=1 \frac{7}{12}\end{align*}
- \begin{align*}\frac{4}{5}+\frac{3}{4}\end{align*} The least common multiple of 5 and 4 is 20. \begin{align*}& \frac{4}{5} \left(\frac{4}{4}\right)+\frac{3}{4}\left(\frac{5}{5}\right)\\ & \frac{16}{20}+\frac{15}{20}\\ & =\frac{31}{20}=1 \frac{11}{20}\end{align*}
For each of the following questions...
- \begin{align*}\frac{5}{8}+\frac{15}{16}\end{align*} The least common multiple of 8 and 16 is 16. \begin{align*}& \frac{5}{8} \left(\frac{2}{2}\right)+\frac{15}{16}\\ & \frac{10}{16}+\frac{15}{16}\\ & =\frac{25}{16}=1 \frac{9}{16}\end{align*} They used \begin{align*}1 \frac{9}{16} \ cups\end{align*} of flour altogether.
- To obtain an answer of \begin{align*}\frac{5}{6}\end{align*} by adding two fractions, the fractions with the denominator of 6 must be \begin{align*}\frac{3}{6}\end{align*} and \begin{align*}\frac{2}{6}\end{align*}. Two fractions with different denominators whose sum is \begin{align*}\frac{5}{6}\end{align*} are \begin{align*}\frac{1}{2}\end{align*} and \begin{align*}\frac{1}{3}\end{align*}.
- Amanda’s Solution \begin{align*}& \frac{1}{6}+\frac{3}{4}\\ & \frac{2}{12}+\frac{9}{12}\\ & =\frac{11}{24}\end{align*} Justin’s Solution \begin{align*}& \frac{1}{6}+\frac{3}{4}\\ & \frac{2}{12}+\frac{9}{12}\\ & =\frac{11}{12}\end{align*} Justin’s solution is correct. I would tell Amanda that when fractions are added, the fractions must have a common denominator. The sum of the numerators is placed over the common denominator. The denominators are NOT added.
Addition of Decimals
Objectives
The lesson objectives for The Addition of Real Numbers are:
- Addition of Positive Decimal Numbers.
- Addition of Positive and Negative Decimal Numbers
- Addition of Decimals Using the Rules
Introduction
In this concept you will learn to add decimal numbers. You will learn first to add decimal numbers that are positive values. Then, you will add decimal numbers that are both negative and positive values. Mastering these concepts will lead to the formation of rules for adding decimal numbers.
Watch This
Guidance
Stephen went shopping to buy some new school supplies. He bought a backpack that cost $28.67 and a scientific calculator for $34.88. How much money did Stephen spend altogether?
Stephen bought two items. To determine the total amount of money he spent, add the prices of the items.
\begin{align*}\$ 28.67+\$34.88\end{align*}
Adding numbers that are written horizontally is often difficult. To add the given decimal numbers, the problem should be written using the vertical alignment method. The decimal points must be kept directly under each other as well as the digits must be kept in the same place value in line with each other. This means that digits in the ones place must be directly below digits in the ones place, digits in the tenths place must be in the tenths column, digits in the hundredths place must be in the hundredths column and so on. Once the numbers have been correctly aligned, the addition process is the same as adding whole numbers.
The numbers and the decimal points have been correctly aligned. Now add the numbers.
Stephen spent $63.55 altogether.
Example A
\begin{align*}2.23+5.34\end{align*}
Because decimal numbers represent fractions with denominators equal to multiples of ten, addition is very easy.
Begin by expressing the each decimal number as the sum of the whole number and its fraction parts.
\begin{align*}2.23 &= 2+\frac{2}{10}+\frac{3}{100}\\ 5.34 &= 5+\frac{3}{10}+\frac{4}{100}\end{align*}
Add the whole numbers and the like fractions.
\begin{align*}2.23 &= 2+\frac{2}{10}+\frac{3}{100}\\ 5.34 &= \frac{5+\frac{3}{10}+\frac{4}{100}}{7+\frac{5}{10}+\frac{7}{10}=7.57}\end{align*}
Instead of using this method to add decimals, simply write the decimals using the vertical alignment method and add the digits in each column.
Example B
\begin{align*}87.296+48.6\end{align*}
Begin by writing the question using the vertical alignment method.
\begin{align*}& \quad 87.296\\ & \underline{+48.6\;\;\;\;}\end{align*}
The decimal points must be kept directly under each other as well as the digits must be kept in the same place value in line with each other. This means that digits in the ones place must be directly below digits in the ones place, digits in the tenths place must be in the tenths column, digits in the hundredths place must be in the hundredths column and so on. To ensure that the digits are aligned correctly, add zeros to 48.6.
\begin{align*}& \quad 87.296\\ & \underline{+48.6{\color{blue}00}}\end{align*}
Add the numbers.
\begin{align*}& \quad 87.296\\ & \underline{+ 48.6{\color{blue}00}}\\ & 135.896\end{align*}
Example C
\begin{align*}(97.38)+(-45.17)\end{align*}
The first step is to write the problem using the vertical alignment method. The two decimal numbers that are being added have opposite signs. Apply the same rule that you used when adding integers that had opposite signs – subtract the numbers and use the sign of the larger number in the answer.
\begin{align*}& \quad 97.38\\ & \underline{- 45.17}\\ & \quad 52.21\end{align*}
The larger decimal number is 97.38 and it has a positive sign. This means that the sign of the answer will also be a positive value.
Example D
\begin{align*}(-168.8)+(-217.4536)\end{align*}
The first step is to write the problem using the vertical alignment method. The two decimal numbers that are being added have the same signs. Apply the same rule that you used when adding integers that had same signs – add the numbers and use the sign of the numbers in the answer.
\begin{align*}& \quad -168.8\\ & \underline{+-217.4536}\end{align*}
To ensure that the digits are aligned correctly, add zeros to 168.8. Add the numbers.
\begin{align*}& \quad -168.8{\color{blue}000}\\ & \underline{+-217.4536}\end{align*}
Add the numbers.
\begin{align*}& \quad -168.8{\color{blue}000}\\ & \underline{+-217.4536}\\ & \ \ -386.2536\end{align*}
The decimal numbers being added both had negative signs. This means that the sign of the answer is also a negative value.
Vocabulary
- Decimal Number
- A decimal number is a fraction whose denominator is 10 or some multiple of 10.
- Decimal Point
- A decimal point is the place marker in a decimal number that separates the whole number and the fraction part. The decimal number 326.45 has the decimal point between the six and the four.
Guided Practice
- Add these decimal numbers by using the expanded fraction form: \begin{align*}14.68+39.217\end{align*}
- \begin{align*}45.36+15+137.692+32.8\end{align*}
- \begin{align*}(53.69)+(-33.7)+(6.298)\end{align*}
Answers
1. \begin{align*}14.68+39.21\end{align*}
\begin{align*}14.68 &= 14+\frac{6}{10}+\frac{8}{100}\\ 39.217 &= 39+\frac{2}{10}+\frac{1}{100}+\frac{7}{1000}\end{align*}
As in adding decimals using the vertical alignment method, add a zero to 14.68 so that both addends have the same number of fraction parts.
\begin{align*}14.68{\color{blue}0} &= 14+\frac{6}{10}+\frac{8}{100}+\frac{{\color{blue}0}}{1000}\\ 39.217 &= 39+\frac{2}{10}+\frac{1}{100}+\frac{7}{1000}\end{align*}
Add the whole numbers and the like fractions.
\begin{align*}14.68{\color{blue}0} &= 14+\frac{6}{10}+\frac{8}{100}+\frac{{\color{blue}0}}{1000}\\ 39.217 &= \frac{39+\frac{2}{10}+\frac{1}{100}+\frac{7}{1000}}{53+\frac{8}{10}+\frac{9}{100}+\frac{7}{1000}=53.897}\end{align*}
2. \begin{align*}45.36+15+137.692+32.8\end{align*}
Write the decimal numbers using the vertical alignment method.
\begin{align*}& \quad 45.36\\ & \quad 15\\ & \ 137.692\\ & \underline{+32.8}\end{align*}
Attach zeros to provide the same number of decimal digits in all of the addends. In a whole number, the decimal point is not written but it is understood as being at the end of the number. \begin{align*}15=15\end{align*}.
\begin{align*}& \quad 45.36{\color{blue}0}\\ & \quad 15.{\color{blue}000}\\ & \ 137.692\\ & \underline{+32.8{\color{blue}00}}\end{align*}
Add the numbers in each vertical column.
\begin{align*}& \overset{2 1 \ \ 1}{\quad 45.36{\color{blue}0}}\\ & \quad 15.{\color{blue}000}\\ & \ 137.692\\ & \underline{+ 32.8{\color{blue}00}}\\ & \ 230.852\end{align*}
3. \begin{align*}(53.69)+(-33.7)+(6.298)\end{align*}
Add the two positive decimal numbers. The answer will have a positive value – add the numbers with the same sign and the answer will have the same sign as the number being added.
\begin{align*}53.69+6.298\end{align*}
Write the numbers using the vertical alignment method and add zeros so that all addends will have the same number of decimal digits. Add the numbers in each vertical column.
\begin{align*}& \quad 53.69{\color{blue}0}\\ & \underline{+ \ \ 6.298}\\ & \quad 59.988\end{align*}
Add the negative decimal number to this answer. When adding numbers with opposite signs, subtract the numbers and the answer will have the sign of the larger number. In this case, the larger number is 59.988, so the answer will have a positive value. Don’t forget the zeros.
\begin{align*}& \quad 59.988\\ & \underline{-33.7{\color{blue}00}}\\ & \ \ 26.288\end{align*}
Summary
The addition of decimal numbers is simply the addition of whole numbers and like fraction parts. To make this process simpler, the decimal numbers are written using the vertical alignment method. The decimal points are aligned as well as the numbers are aligned according to their place value. The numbers in each vertical column are then added. If the decimal numbers are signed numbers, the rules for adding integers are applied to the problem.
Problem Set
Add the following decimal numbers by using the expanded fraction form:
- \begin{align*}14.36+9.42\end{align*}
- \begin{align*}52.72+27.163\end{align*}
- \begin{align*}0.26+4.5+1.137\end{align*}
- \begin{align*}37.231+14.567\end{align*}
- \begin{align*}78.32+6.2+19.46\end{align*}
Add the following decimal numbers:
- \begin{align*}65.23+12.75\end{align*}
- \begin{align*}148.067+53.78+6.9\end{align*}
- \begin{align*}56.75+14.9294+17.854\end{align*}
- \begin{align*}18+26.87+65.358\end{align*}
- \begin{align*}23.067+268.93+9.4\end{align*}
Add the following signed decimal numbers:
- \begin{align*}(-24.69)+(-39.87)\end{align*}
- \begin{align*}(76.35)+(-36.68)\end{align*}
- \begin{align*}(-12.5)+(47.97)+(-21.653)\end{align*}
- \begin{align*}(62.462)+(254.69)+(-427.9)\end{align*}
- \begin{align*}(-37.76)+(-45.8)+(53.92)\end{align*}
Determine the answer to the following problems.
- When the owners of the Finest Fixer Co. completed a small construction job, they found that the following expenses had been incurred: labour, $975.75; gravel, $88.79; sand, $43.51; cement, $284.96; and bricks $2214.85. What bill should they give the customer if they want to make a profit of $225 for the job?
- A tile setter purchases the following supplies for the day:
- One bag of thin-set mortar - @$5.67 per bag
- 44 sq ft of tile - @$107.80 for 44 sq ft of tile
- One gallon of grout - @$17.97 per gallon
- One container of grout sealer - @$32.77 per container
- 3 containers of grout and tile cleaner - @$5.99 per container
- 4 scrub pads - @$2.78 each
- One trowel - @ 3.95 each
- 2 packages of tile spacers - @2.27 each
- One grout bag - @2.79 each
- One grout float - @10.45 each
What is the cost of these items before tax is added?
- The four employees of the Broken Body Shop earned the following amounts last week: $815.86, $789.21, $804.18 and $888.35. What is the average weekly pay for the employees?
- Jennifer bought the following school supplies:
- 1000 sheets of paper - @$14.67
- 36 pencils - @ $6.55
- 1 binder - @$18.48
- 1 backpack - @ $22.74
- 1 lunch bag - @ 4.64
How much did Jennifer spend on these supplies before taxes?
- A local seamstress needs to purchase fabric to sew curtains for the local theatre. She needs 123.75 yd. of black cotton for a backdrop, 217.4 yd. of white linen for stage curtains, 75 yd. for accessory curtains and 98.5 yd. for costumes. How many yards of fabric must be purchased to fill this order?
Answers
Add using the expanded fraction form:
- \begin{align*}14.36+9.42\end{align*} \begin{align*}& 14.36=14+\frac{3}{10}+\frac{6}{100}\\ & \ 9.42=9+\frac{4}{10}+\frac{2}{100}\\ \\ & 14.36=14+\frac{3}{10}+\frac{6}{100}\\ & \ \underline{9.42=9+\frac{4}{10}+\frac{2}{100}\;\;\;}\\ & \qquad =23+\frac{7}{10}+\frac{8}{100}=23.78\end{align*}
- \begin{align*}0.26+4.5+1.137\end{align*} \begin{align*}& \ \ 0.26=0+\frac{2}{10}+\frac{6}{100}\\ & \quad 4.5 = 4+\frac{5}{10}\\ & \underline{1.137=1+\frac{1}{10}+\frac{3}{100}+\frac{7}{1000}}\\ \\ & 0.26{\color{blue}0}=0+\frac{2}{10}+\frac{6}{100}+\frac{{\color{blue}0}}{100}\\ & 4.5{\color{blue}00}=4+\frac{5}{10}+\frac{{\color{blue}0}}{100}+\frac{{\color{blue}0}}{1000}\\ & \underline{1.137=1+\frac{1}{10}+\frac{3}{100}+\frac{7}{1000}}\\ & \qquad \ =5+\frac{8}{10}+\frac{9}{100}+\frac{7}{1000}=5.897\end{align*}
- \begin{align*}78.32+6.2+19.46\end{align*} \begin{align*}& 78.32=78+\frac{3}{10}+\frac{2}{100}\\ & \quad 6.2=6+\frac{2}{10}\\ & \underline{19.46=19+\frac{4}{10}+\frac{6}{100}}\\ \\ & 78.32 = 78+\frac{3}{10}+\frac{2}{100}\\ & \ 6.2{\color{blue}0}=6+\frac{2}{10}+\frac{{\color{blue}0}}{100}\\ & \underline{19.46=19+\frac{4}{10}+\frac{6}{100}}\\ & \qquad \ =103+\frac{9}{10}+\frac{8}{100}=103.98\end{align*}
Add the following decimal numbers:
- \begin{align*}65.23+12.75\end{align*} \begin{align*}& \quad 65.23\\ &\underline{+12.75}\end{align*} \begin{align*}& \quad 65.23\\ & \underline{+12.75}\\ & \ \ 77.98\end{align*}
- \begin{align*}56.75+14.9294+17.854\end{align*} \begin{align*}& \quad 56.75\\ & \ \ 14.9294\\ & \underline{+17.854\;}\end{align*} \begin{align*}& \quad 56.75{\color{blue}00}\\ & \quad 14.9294\\ & \underline{+17.854{\color{blue}0}\;}\end{align*} \begin{align*}& \quad 56.75{\color{blue}00}\\ & \quad 14.9294\\ & \underline{+17.854{\color{blue}0}\;}\\ & \quad 89.5334\end{align*}
- \begin{align*}23.067+268.93+9.4\end{align*} \begin{align*}& \quad 23.067\\ & \ 268.93\\ & \underline{+ \ 9.4 \;\;\;\;\;}\end{align*} \begin{align*}& \quad \ 23.067\\ & \quad 268.93{\color{blue}0}\\ & \underline{+ \ \ \ 9.4{\color{blue}00}\;}\end{align*} \begin{align*}& \ 23.067\\ & 268.93{\color{blue}0}\\ & \underline{+ 9.4{\color{blue}00}\;\;}\\ & 301.397\end{align*}
Add the following signed decimal numbers:
- \begin{align*}(-24.69)+(-39.87)\end{align*} \begin{align*}& \ \ -24.69\\ & \underline{+-39.87}\end{align*} \begin{align*}& \ \ -24.69\\ & \underline{+-39.87}\\ & \ \ -64.56\end{align*}
- \begin{align*}(-12.5)+(47.97)+(-21.653)\end{align*} \begin{align*}& \ \ -12.5\\ & \underline{+-21.653}\end{align*} \begin{align*}& \ \ -12.5{\color{blue}00}\\ & \underline{+-21.653}\\ & \quad -34.153\end{align*} \begin{align*}(47.97)+(-34.153)\end{align*} \begin{align*}& \quad \quad 47.97\\ & \underline{+-34.153}\end{align*} \begin{align*}& \quad \quad 47.97{\color{blue}0}\\ & \underline{+-34.153}\\ & \quad \quad 13.817\end{align*}
- \begin{align*}(-37.76)+(-45.8)+(53.92)\end{align*} \begin{align*}& \quad \quad 37.76\\ & \underline{+-45.8 \;\;}\end{align*} \begin{align*}& \ \ -37.76\\ & \underline{+-45.8{\color{blue}0}\;}\end{align*} \begin{align*}& \ \ -37.76\\ & \underline{+-45.8{\color{blue}0}\;}\\ & \quad -83.56\end{align*} \begin{align*}(-83.56)+(53.92)\end{align*} \begin{align*}& -83.56\\ & \underline{+ \ 53.92}\end{align*} \begin{align*}& -83.56\\ & \underline{+ \ 53.92}\\ & -29.64\end{align*}
Determine the answer to the following problems.
- When the owners of the Finest Fixer Co. completed a small construction job, they found that the following expenses had been incurred: labour, $975.75; gravel, $88.79; sand, $43.51; cement, $284.96; and bricks $2214.85. What bill should they give the customer if they want to make a profit of $225 for the job? \begin{align*}& \$ \ \ 975.75\\ & \$ \ \ \ 88.79\\ & \$ \ \ \ 43.51\\ & \$ \ 284.96\\ & \underline{\$ 2214.85}\\ & \$3604.86\end{align*} The total expenses incurred by the company was $3607.86 \begin{align*}& \ \$ 3607.86\\ & \underline{+ \$ 225.00}\\ & \ \$ 3832.86\end{align*} The total bill that must be given to the customer is $3832.86
- The four employees of the Broken Body Shop earned the following amounts last week: $815.86, $789.21, $804.18 and $888.35. What is the average weekly pay for the employees? \begin{align*}& \ \ \$ 815.86\\ & \ \ \$ 789.21\\ & \ \ \$ 804.18\\ & \underline{+ \$ 888.35}\\ & \ \$ 3297.60\end{align*} \begin{align*}\frac{\$3297.60}{4}=\$824.40\end{align*} The average weekly pay is $824.40.
- A local seamstress needs to purchase fabric to sew curtains for the local theatre. She needs 123.75 yd. of black cotton for a backdrop, 217.4 yd. of white linen for stage curtains, 75 yd. for accessory curtains and 98.5 yd. for costumes. How many yards of fabric must be purchased to fill this order? \begin{align*}123.75 +217.4+75+98.5\end{align*} \begin{align*}& \ \ 123.75\\ & \ \ 217.4\\ & \ \ 75\\ & \ \underline{+98.5\;\;}\end{align*} \begin{align*}& \ \ 123.75\\ & \ \ 217.4{\color{blue}0}\\ & \quad 75.{\color{blue}00}\\ & \underline{+98.5{\color{blue}0}\;}\end{align*} \begin{align*}& \ \ 123.75\\ & \ \ 217.4{\color{blue}0}\\ & \quad 75.{\color{blue}00}\\ & \underline{+98.5{\color{blue}0}\;}\\ & \ 514.65\end{align*} The amount of fabric that must be purchased to fill this order is 514.65 yards.
Summary
In this lesson you have learned how to add real numbers by using a variety of models. The real numbers that you added were integers, fractions and decimal numbers. The models that were used to add integers were color counters, algebra tiles and a number line. By using these models, you learned the two rules for adding integers. These rules were: Integers with unlike signs must be subtracted and the answer will have a sign the same as that of the higher number. Integers with the same sign must be added and the answer will have a sign the same as that of the numbers being added. After adding integers, you then learned how to add fractions by using fraction strips and a number line. You learned that fractions can only be added if they have a common denominator. The sum of the numerators of the fractions being added is placed over the common denominator. If the numerator of the fraction is larger than the denominator, the answer could be expressed as a mixed number. The last real numbers that were added were decimal numbers. When adding decimal numbers that were signed numbers, the rules for the addition of integers were applied to the problems.
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