1.4: Division of Real Numbers
Division of Real Numbers
Objectives
The lesson objectives for The Division of Real Numbers are:
- Division of integers Using Rules
- Division of Fractions Using Rules
- Division of Decimals Using Rules
Introduction
In this concept you will learn to divide real numbers using the rules. You will learn to divide integers, fractions and decimal numbers. Since all of these topics will be presented in one lesson, you will begin by watching the videos that are listed below.
Watch This
Khan Academy Multiplying and Dividing Negative Numbers
Khan Academy Dividing Fractions
Khan Academy Dividing Decimals
Guidance
The meteorologist on the local radio station just announced that a cold front caused the temperature to drop \begin{align*}12^{\circ}C\end{align*} in just four hours. Use counters to show the mean temperature change per hour over these four hours.
Since it took four hours for the temperature to drop, share the counters into four equal groups.
There are three negative counters in each group. The mean temperature change per hour was \begin{align*}-3^{\circ} C\end{align*}. The result of \begin{align*}(-12)\div(+4)\end{align*} is -3. When you divide by a positive integer, you can think of sharing equally by the magnitude of the positive integer.
Example A
Miguel was doing a science project on weather and he reported a total temperature change of \begin{align*}-15^{\circ}F\end{align*} and a mean hourly change of \begin{align*}-3^{\circ}C\end{align*}. How many hourly temperature changes did Miguel record?
You can use color counters to determine the number of changes that Miguel recorded. Start with zero and add a group of 3 yellow counters. Continue adding groups of three counters until you have 15 yellow counters.
How many times did you need to add -3 to get -15?
You added 5 groups of three yellow counters. When you divide integers with the same sign, you can think of repeated addition. The result of \begin{align*}(-15)\div(-3)\end{align*} is 5.
Why can you NOT model \begin{align*}(+18)\div(-3)\end{align*} with color counters?
Example B
Multiplication and division are related. The relationship can be shown by using a triangle.
\begin{align*}(+6) \times (+3)=+18\end{align*}
The arrows and the operation signs can be changed to produce the related division statements.
Dividing a positive integer by a positive integer results in a positive answer. The answer to a division question is called the quotient.
Copy the triangle and illustrate the division statements for the multiplication statement
\begin{align*}(-6)\times(+8)=-48\end{align*}
The rules for dividing integers are:
- The quotient of integers with same or like signs is always positive.
- The quotient of integers with opposite or unlike signs is always negative.
Example C
i) \begin{align*}\left(\frac{6}{11}\right) \div \left(\frac{5}{7}\right)\end{align*}
i) \begin{align*}\left(4 \frac{1}{3}\right) \div \left(2 \frac{5}{7}\right)\end{align*}
The division of fractions involves one more step than the multiplication of fractions. This step is to begin the solution by inverting the fraction after the operation sign. If \begin{align*}\frac{5}{7}\end{align*} is inverted, the new fraction is \begin{align*}\frac{7}{5}\end{align*}. This new fraction \begin{align*}\frac{7}{5}\end{align*} is called the reciprocal of \begin{align*}\frac{5}{7}\end{align*}. The product of a number and its reciprocal is one. When you have completed this step, change the division sign to a multiplication sign and follow the rules for multiplying fractions.
\begin{align*}& \left(\frac{6}{11}\right) \div \left(\frac{5}{7}\right)\\ & \frac{6}{11} \times \frac{7}{5}\\ & \frac{6 \times 7}{11 \times 5}\\ & =\frac{42}{55}\end{align*}
ii) \begin{align*}& \left(4\frac{1}{3}\right) \div \left(2\frac{5}{7}\right) \ \text{Write the mixed numbers as improper fractions.}\\ & \left(\frac{13}{3}\right) \div \left(\frac{19}{7}\right)\ \text{Multiply by the reciprocal of} \ \frac{19}{7}.\\ & \frac{13}{3} \times \frac{7}{19}\\ & = \frac{91}{57}= 1 \frac{34}{57} \ \text{Simplify the fraction.}\end{align*}
Example D
i) \begin{align*}(0.68)\div(1.7)\end{align*}
ii) \begin{align*}0.365 \div -18.25\end{align*}
To divide decimal numbers, use the following steps:
- \begin{align*} \overset{ \qquad \qquad \quad \#({\color{blue}\text{quotient}})}{\#({\color{blue}\text{divisor}}) \overline{ ) {\#({\color{blue}\text{dividend}}) }}}\end{align*} Write the divisor and the dividend in standard long-division form.
- Move the decimal point of the divisor to the right so that the divisor is a whole number.
- Move the decimal point of the dividend to the right the same number of places that you moved the decimal point of the divisor. If necessary, add zeros in the dividend.
- Place the decimal point in the quotient directly above the new decimal point in the dividend.
- The decimal points can now be ignored. Divide the numbers the same as you would divide whole numbers.
\begin{align*}(0.68) \div (1.7)\end{align*}
\begin{align*}& \overset{ \qquad \ 0.4}{\underset{\ \ \rightarrow}{1.7} \overline{ ) {\underset{\ \ \rightarrow}{0.6} \ 8 \;}}}\\ & \quad \ \ \underline{- 6 \; 8}\\ & \qquad \quad \ 0\end{align*}
The decimal point of the divisor was moved one place to the right. The decimal point of the dividend was moved one place to the right. The decimal point was placed in the quotient directly above the new decimal point of the dividend.
ii) \begin{align*}0.365 \div -18.25\end{align*}
You have learned that when you divide a positive number by a negative number, the answer will always be negative.
\begin{align*}& \overset{ \qquad \qquad -.02}{\underset{\quad \ \ \longrightarrow}{-18.25} \overline{ ) {\underset{\ \ \longrightarrow}{0.36} \ 5{\color{blue}0}}}}\\ & \qquad \quad \ \underline{- 36 \; \; 50}\\ & \qquad \qquad \qquad 0\end{align*}
The decimal point of the divisor was moved two places to the right. The decimal point of the dividend was moved two place to the right. The decimal point was placed in the quotient directly above the new decimal point of the dividend.
Vocabulary
- Dividend
- In a division problem, the dividend is the number that is being divided. The dividend is written under the division sign. \begin{align*} \overset{}{4 \overline{ ) {38}}}\end{align*}, 38 is the dividend.
- Divisor
- In a division problem, the divisor is the number that is being divided into the dividend. The divisor is written in front of the division sign. \begin{align*} \overset{}{4 \overline{ ) {38}}}\end{align*} 4 is the divisor.
- 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*}.
- Reciprocal
The reciprocal of a number is the inverse of that number. If \begin{align*}\frac{a}{b}\end{align*} is a nonzero number, then \begin{align*}\frac{b}{a}\end{align*} is its reciprocal. The product of a number and its reciprocal is one.
- Quotient
The quotient is the answer of a division problem.
Guided Practice
1. Use color counters to represent the division problem \begin{align*}(-24) \div(+6)\end{align*}.
2. Use a triangle to represent the division statements related to \begin{align*}(-6) \times (+7)=-42\end{align*}.
3. Determine the answer to \begin{align*}\left(\frac{7}{10}\right) \div \left ( \frac{5}{6} \right )=?\end{align*} and \begin{align*}\left(6 \frac{2}{5}\right) \div \left(1 \frac{2}{3}\right)=?\end{align*} by using the rules for dividing fractions.
4. How many pieces of plywood 0.375 in. thick are in a stack of 30 in. high?
Answers
1. \begin{align*}(-24) \div (+6)\end{align*}
Begin with 24 yellow counters. Create 6 groups of yellow counters until you use all of the counters.
There are 6 groups of yellow counters. Each group contains four yellow counters.
\begin{align*}(-24) \div (+6)=-4\end{align*}
2. \begin{align*}(-6) \times (+7) =-42\end{align*}
3. i) \begin{align*}\left(\frac{7}{10}\right) \div \left(\frac{5}{6}\right)\end{align*}
\begin{align*}\frac{7}{10} \times \frac{6}{5}\end{align*}
Multiply by the reciprocal.
\begin{align*}=\frac{7 \times 6}{10 \times 5}\end{align*}
Multiply numerators. Multiply denominators.
\begin{align*}=\frac{42}{50}=\frac{21}{25}\end{align*}
Simplify the fraction.
ii) \begin{align*}\left(6\frac{2}{5}\right) \div \left(1\frac{2}{3}\right)\end{align*}
\begin{align*}\left(\frac{32}{5}\right) \div \left(\frac{5}{3}\right)\end{align*}
Change the mixed numbers to improper fractions.
\begin{align*}\left(\frac{32}{5}\right) \times \left(\frac{3}{5}\right)\end{align*}
Multiply by the reciprocal.
\begin{align*}\frac{32 \times 3}{5 \times 5}\end{align*}
Multiply numerators. Multiply denominators.
\begin{align*}& =\frac{96}{25}\\ & =3\frac{21}{25}\end{align*}
Simplify the fraction.
4. To determine the number of pieces of plywood in the stack, divide the thickness of one piece into the height of the pile.
\begin{align*}& \overset{ \qquad \qquad \ 80}{\underset{\ \ \longrightarrow}{0.375} \overline{ ) {\underset{\ \ \longrightarrow}{30.000}}}}\\ & \qquad \underline{- 3000\;}\\ & \qquad \qquad \quad 0\\ & \qquad \ \underline{- \;\;\;\;\;\;\;0}\\ & \qquad \qquad \quad 0\end{align*}
There are 80 pieces of plywood in the pile.
Summary
The division of integers can be represented using color counters. By using these counters, the rules for dividing integers were obvious.
- When you divided two integers that had the same signs, the answer was always positive.
- When you divided two integers that had opposite signs, the answer was always negative.
Another way to demonstrate these rules was to use a triangle. The triangle showed the relationship between multiplication statements and division statements. The rules for dividing integers and for multiplying integers are the same.
You also learned that the division of fractions is simply the multiplication of one fraction by the reciprocal of another. You also learned that the reciprocal of a fraction is the fraction written upside down. The numerator was written as the denominator and the denominator was written as the numerator.
The final topic that you learned was the division of decimal numbers. The decimal point of the divisor was placed to the right of the divisor so that it became a whole number. The number of places that the decimal point was moved in the divisor determined the number of places that the decimal point had to be moved in the dividend. When the decimal point was placed in the quotient above the new decimal point in the dividend, the process of dividing was the same as long-division of whole numbers.
The rules for dividing integers apply to the division of all real numbers.
Problem Set
Use color counters to find each quotient.
- \begin{align*}(+14)\div (+2)\end{align*}
- \begin{align*}(-14) \div (+2)\end{align*}
- \begin{align*}(-9)\div (-3)\end{align*}
- \begin{align*}(+16) \div (+4)\end{align*}
- \begin{align*}(+25) \div (-5)\end{align*}
Use a triangle to represent the division statements for each of the multiplication statements.
- \begin{align*}(-9)\times (7)=-63\end{align*}
- \begin{align*}(-8)\times (-8)=64\end{align*}
- \begin{align*}(+4)\times (-7)=-28\end{align*}
- \begin{align*}(-10) \times (-3)=30\end{align*}
- \begin{align*}(+5) \times (+2)=+10\end{align*}
Calculate the quotient for each of the following:
- \begin{align*}\left(\frac{5}{16}\right) \div \left(\frac{3}{7}\right)\end{align*}
- \begin{align*}(-8.8)\div (-3.2)\end{align*}
- \begin{align*}(7.23)\div (0.6)\end{align*}
- \begin{align*}\left(2\frac{3}{4}\right) \div \left(1\frac{1}{8}\right)\end{align*}
- \begin{align*}(-30.24) \div (-0.42)\end{align*}
For each of the following questions, write a division statement and find the result.
- A truck is delivering fruit baskets to the local food banks for the patrons. Each fruit basket weighs 3.68 lb. How many baskets are in a load weighing 5888 lb?
- A wedding invitation must be printed on card stock measuring \begin{align*}4 \frac{1}{4} \ in\end{align*}. wide. If the area of the invitation is \begin{align*}23 \frac{3}{8} \ in^2\end{align*}, what is its length? (Hint: The area of a rectangle is found by multiplying the length times the width.)
- A seamstress needs to divide \begin{align*}32 \frac{5}{8} \ ft\end{align*}. of piping into 3 equal pieces. Calculate the length of each piece.
- The floor area of a room on a house plan measures 3.5 in. by 4.625 in. If the house plan is drawn to the scale 0.25 in. represents 1 ft, what is the actual size of the room?
- How many hair bows of \begin{align*}3 \frac{1}{2} \ in\end{align*}. can be cut from \begin{align*}24 \frac{3}{4} \ in\end{align*}. of ribbon?
Sample Answers
Use color counters...
- \begin{align*}(+14) \div (+2)\end{align*} Begin with 14 red counters. Create 2 groups of red counters until you have used all of the counters. There are 7 red counters in each group when the 14 counters were equally shared. The result of \begin{align*}(+14) \div (+2)\end{align*} is +7.
- \begin{align*}(-9) \div (-3)\end{align*} Begin with 9 yellow counters. How many times do you have to add (-3) to get (-9)? You have to add (-3) three times to get (-9). The result of \begin{align*}(-9)\div (-3)\end{align*} is +3.
- \begin{align*}(+25) \div (-5)\end{align*} This problem cannot be done using color counters. You cannot share counters a negative number of times. You cannot add (-5) to give a result of (+25). When you add numbers with the same signs, the answer will have the sign of the numbers being added.
Use a triangle...
- \begin{align*}(-9) \times (7) = -63\end{align*}
- \begin{align*}(+4) \times (-7)=-28\end{align*}
- \begin{align*}(+5) \times (+2)=+10\end{align*}
Calculate the quotient...
- \begin{align*}\left(\frac{5}{16}\right) \div \left(\frac{3}{7}\right)\end{align*} \begin{align*}\frac{5}{16} \times \frac{7}{3}\end{align*} Multiply by the reciprocal of \begin{align*}\left(\frac{3}{7}\right)\end{align*}. \begin{align*}\frac{5 \times 7}{16 \times 3}\end{align*} Multiply numerators. Multiply denominators. \begin{align*}=\frac{35}{48}\end{align*}
- \begin{align*}(7.23) \div (0.6)\end{align*} \begin{align*}& \overset{\quad 12.05}{\underset{\ \ \rightarrow}{0.6} \overline{ ) {\underset{\ \ \rightarrow}{7.2} \; 3{\color{blue}0} \;}}}\\ & \ \ \underline{- \ 6}\\ & \quad \ \ 12\\ & \underline{\;\;\;-12}\\ & \ \qquad 03\\ & \underline{\;\;\;\;\;\;\; -0}\\ & \ \qquad \ 3{\color{blue}0}\\ & \underline{\;\;\;\;\;\;\; -30}\\ & \ \qquad \quad 0\\\end{align*} Move the decimal point to the right of the divisor. Move the decimal point of the dividend one place to the right. Divide as you would whole numbers.
- \begin{align*}(-30.24) \div (-0.42)\end{align*} When you divide numbers that have the same signs, the result is always positive. \begin{align*}& \overset{ \qquad \qquad \ 72}{\underset{\quad \ \ \longrightarrow}{-0.42} \overline{ ) {\underset{\qquad \longrightarrow}{-30.24} }}}\\ & \qquad \quad \underline{+294 }\\ & \qquad \quad \ \ -84 \quad \boxed{-42 \times 7 =-294. \ \text{When this value is subtracted it becomes} \ -(-294)=+(294)}\\ & \qquad \quad \ \ \underline{\;+84}\\ & \qquad \qquad \quad \ 0\end{align*} Move the decimal point to the right of the divisor. Move the decimal point of the dividend two places to the right. Divide as you would whole numbers. The decimal point of all whole numbers is understood at the end of the number. The quotient of 72 does not have to be written with the decimal point at the end.
For each of the following questions...
- \begin{align*}5888 \div 3.68\end{align*} The total weight must be divided by the weight of each fruit basket. \begin{align*}& \overset{\qquad 1600}{\underset{\quad \longrightarrow}{3.68} \overline{ ) {5888\underset{\longrightarrow}{{\color{blue}00}}\;}}}\\ & \quad \ \underline{- 368\;}\\ & \qquad 2208\\ & \underline{\;\;\;\;-2208}\\ & \ \qquad \quad 00\\ & \underline{\;\;\;\;\;\;\;\;\;-00}\\ & \ \qquad \quad 00\\ & \underline{\;\;\;\;\;\;\;\;\; -00}\\ & \ \qquad \quad \ 0\\\end{align*} To move the decimal point of the dividend two places to the right, two zeros were added. There are 1600 fruit baskets in the truck.
\begin{align*}& \left(32 \frac{5}{8}\right) \div 3\\ & \left(\frac{261}{8}\right) \times \left(\frac{1}{3}\right)\end{align*}
Change the mixed number to an improper fraction and multiply by the reciprocal of 3.
\begin{align*}\left(\frac{261 \times 1}{8 \times 3}\right)\end{align*}
Multiply numerators. Multiply denominators.
\begin{align*}\left(\frac{261}{24}\right)=10 \frac{7}{8}\end{align*}
Simplify the fraction.
The length of each piece of piping is \begin{align*}10 \frac{7}{8} \ ft\end{align*}.
\begin{align*}& 24 \frac{3}{4} \div 3 \frac{1}{2}\\ & \left(\frac{99}{4}\right) \div \left(\frac{7}{2}\right)\end{align*}
Change the mixed numbers to improper fractions.
\begin{align*}\frac{99}{4} \times \frac{2}{7}\end{align*}
Multiply by the reciprocal of \begin{align*}\left(\frac{7}{2}\right)\end{align*}.
Multiply numerators. Multiply denominators. Simplify the fraction.
Seven hair bows can be made and there will be \begin{align*}\frac{1}{14} \ in\end{align*}. of ribbon left over.
Summary
The division of integers can be represented with manipulative such as color counters. By using the color counters, you learned that division by a positive integer can be thought of as sharing equally and division of a negative integer by a negative integer can be thought of as repeated addition. You also learned that a positive integer divided by a negative integer could not be modeled by using color counters. A positive integer cannot be shared in a negative number of ways and the repeated addition of negative integers would result in a negative sum. The rules for dividing integers were obvious from the color counter models:
- The quotient of two positive integers is always positive.
- The quotient of two negative integers is always positive.
- The quotient of a positive integer and a negative integer is always negative.
These rules can be applied to the division of all real numbers. In this lesson you also learned to demonstrate the relationship between multiplication statements and division statements by using a triangle. The factors that were multiplied to get the product were the same factors that resulted when the factors were divided into the product.
After you learned to divide integers, you learned to divide fractions. The rules used to multiply fractions were applied to dividing fractions after you changed the problem to a multiplying question and the divisor to its reciprocal. To divide fractions you simply multiplied by the reciprocal of the divisor (the fraction following the division sign).
The final topic that was presented in this lesson was the division of decimals. You learned that dividing decimal numbers was the same as dividing whole numbers after the decimal point of the divisor was moved to the right and as well moved the same number places in the dividend.
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