1.8: Division of Real Numbers
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. What was the mean temperature change per hour over these four hours?
Watch This
Khan Academy Dividing Positive and Negative Numbers
Khan Academy Dividing Fractions
Khan Academy Dividing Decimals
Guidance
There are two rules for dividing integers:
- When you divide two integers that have the same signs, the answer is always positive.
- When you divide two integers that have opposite signs, the answer iss always negative.
We can demonstrate these rules using counters. The rules for dividing integers apply to the division of all real numbers.
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.
To divide decimal numbers, use the following steps:
- 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.
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.
Example B
i) \begin{align*}\left(\frac{6}{11}\right) \div \left(\frac{5}{7}\right)\end{align*}
ii) \begin{align*}\left(4 \frac{1}{3}\right) \div \left(2 \frac{5}{7}\right)\end{align*}
Solution:
i) \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 C
i) \begin{align*}(0.68)\div(1.7)\end{align*}
ii) \begin{align*}0.365 \div -18.25\end{align*}
Solution:
i) \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.
Concept Problem Revisited
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. You can use counters to show the mean temperature change per hour over these four hours.
Since it took four hours for the temperature to drop, put 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.
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. 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.
3. 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. 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.
3. 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.
Practice
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*}
Calculate the quotient for each of the following:
- \begin{align*}(-9)\times (7)\end{align*}
- \begin{align*}(-8)\times (-8)\end{align*}
- \begin{align*}(+4)\times (-7)\end{align*}
- \begin{align*}(-10) \times (-3)\end{align*}
- \begin{align*}(+5) \times (+2)\end{align*}
- \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?
Image Attributions
Here you will learn to divide real numbers using rules. You will learn to divide integers, fractions and decimal numbers.