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Division of Rational Numbers

Multiply the first fraction by the reciprocal of the second fraction

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Division of Real Numbers

The meteorologist on the local radio station just announced that a cold front caused the temperature to drop 12C\begin{align*}12^{\circ}C\end{align*} in just four hours. What was the mean temperature change per hour over these four hours?

Guidance

There are two rules for dividing real numbers:

1. When you divide two integers that have the same signs, the answer is always positive.
2. When you divide two integers that have opposite signs, the answer is always negative.

Dividing fractions is like multiplying fractions with one additional step. To divide fractions, multiply the first fraction by the reciprocal of the second fraction. For example, 35÷29=35×92\begin{align*}\frac{3}{5}\div\frac{2}{9}=\frac{3}{5}\times\frac{9}{2}\end{align*}.

To divide decimals, use the following steps:

1. Write the divisor and the dividend in standard long-division form.
2. Move the decimal point of the divisor to the right so that the divisor is a whole number.
3. 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.
4. Place the decimal point in the quotient directly above the new decimal point in the dividend.
5. 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 15F\begin{align*}-15^{\circ}F\end{align*} and a mean hourly change of 3C\begin{align*}-3^{\circ}C\end{align*}. How many hourly temperature changes did Miguel record?

Solution: The result of (15)÷(3)\begin{align*}(-15)\div(-3)\end{align*} is 5.

Example B

i) (611)÷(57)\begin{align*}\left(\frac{6}{11}\right) \div \left(\frac{5}{7}\right)\end{align*}

ii) (413)÷(257)\begin{align*}\left(4 \frac{1}{3}\right) \div \left(2 \frac{5}{7}\right)\end{align*}

Solution:

i)

(611)÷(57)611×756×711×5=4255\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)

(413)÷(257) Write the mixed numbers as improper fractions.(133)÷(197) Multiply by the reciprocal of 197.133×719=9157=13457 Simplify the fraction.\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) (0.68)÷(1.7)\begin{align*}(0.68)\div(1.7)\end{align*}

ii) 0.365÷18.25\begin{align*}0.365 \div -18.25\end{align*}

Solution:

i) (0.68)÷(1.7)\begin{align*}(0.68) \div (1.7)\end{align*}

1.7  )0.6   8¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ 0.4  68 0\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) 0.365÷18.25\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.

18.25  )0.36   50¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯.02 36500\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 12C\begin{align*}12^{\circ}C\end{align*} in just four hours.

The mean temperature change per hour is the result of (12)÷(+4)\begin{align*}(-12)\div(+4)\end{align*}, which is –3.

Vocabulary

Dividend
In a division problem, the dividend is the number that is being divided. The dividend is written under the division sign. In 4)38¯¯¯¯¯¯¯\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. In 4)38¯¯¯¯¯¯¯\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 435\begin{align*}4 \frac{3}{5}\end{align*}.
Reciprocal

The reciprocal of a number is the inverse of that number. If ab\begin{align*}\frac{a}{b}\end{align*} is a nonzero number, then ba\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. (710)÷(56)=?\begin{align*}\left(\frac{7}{10}\right) \div \left ( \frac{5}{6} \right )=?\end{align*}

2. (625)÷(123)=?\begin{align*}\left(6 \frac{2}{5}\right) \div \left(1 \frac{2}{3}\right)=?\end{align*}

3. How many pieces of plywood 0.375 in. thick are in a stack of 30 in. high?

1. (710)÷(56)\begin{align*}\left(\frac{7}{10}\right) \div \left(\frac{5}{6}\right)\end{align*}

=710×65\begin{align*}=\frac{7}{10} \times \frac{6}{5}\end{align*}

=7×610×5\begin{align*}=\frac{7 \times 6}{10 \times 5}\end{align*}

=4250=2125\begin{align*}=\frac{42}{50}=\frac{21}{25}\end{align*}

2. (625)÷(123)\begin{align*}\left(6\frac{2}{5}\right) \div \left(1\frac{2}{3}\right)\end{align*}

=(325)÷(53)\begin{align*}=\left(\frac{32}{5}\right) \div \left(\frac{5}{3}\right)\end{align*}

=(325)×(35)\begin{align*}=\left(\frac{32}{5}\right) \times \left(\frac{3}{5}\right)\end{align*}

=32×35×5\begin{align*}=\frac{32 \times 3}{5 \times 5}\end{align*}

=9625=32125\begin{align*}& =\frac{96}{25}\\ & =3\frac{21}{25}\end{align*}

3. To determine the number of pieces of plywood in the stack, divide the thickness of one piece into the height of the pile.

0.375  )30.000  ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ 8030000 00\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

Find each quotient or product.

1. (+14)÷(+2)\begin{align*}(+14)\div (+2)\end{align*}
2. (14)÷(+2)\begin{align*}(-14) \div (+2)\end{align*}
3. (9)÷(3)\begin{align*}(-9)\div (-3)\end{align*}
4. (+16)÷(+4)\begin{align*}(+16) \div (+4)\end{align*}
5. (+25)÷(5)\begin{align*}(+25) \div (-5)\end{align*}
6. (9)×(7)\begin{align*}(-9)\times (7)\end{align*}
7. (8)×(8)\begin{align*}(-8)\times (-8)\end{align*}
8. (+4)×(7)\begin{align*}(+4)\times (-7)\end{align*}
9. (10)×(3)\begin{align*}(-10) \times (-3)\end{align*}
10. (+5)×(+2)\begin{align*}(+5) \times (+2)\end{align*}
11. (516)÷(37)\begin{align*}\left(\frac{5}{16}\right) \div \left(\frac{3}{7}\right)\end{align*}
12. (8.8)÷(3.2)\begin{align*}(-8.8)\div (-3.2)\end{align*}
13. (7.23)÷(0.6)\begin{align*}(7.23)\div (0.6)\end{align*}
14. (234)÷(118)\begin{align*}\left(2\frac{3}{4}\right) \div \left(1\frac{1}{8}\right)\end{align*}
15. (30.24)÷(0.42)\begin{align*}(-30.24) \div (-0.42)\end{align*}

For each of the following questions, write a division statement and find the result.

1. 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?
2. A wedding invitation must be printed on card stock measuring 414 in\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.)
3. 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.
4. 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?
5. 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?

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Color Highlighted Text Notes

Vocabulary Language: English

Dividend

In a division problem, the dividend is the number or expression that is being divided.

divisor

In a division problem, the divisor is the number or expression that is being divided into the dividend. For example: In the expression $152 \div 6$, 6 is the divisor and 152 is the dividend.

identity element

An identity element is a value which, when combined with an operation on another number, leaves that other number unchanged. The identity element for addition is zero, the identity element for multiplication is one.

Mixed Number

A mixed number is a number made up of a whole number and a fraction, such as $4\frac{3}{5}$.

Multiplicative Inverse

The multiplicative inverse of a number is the reciprocal of the number. The product of a real number and its multiplicative inverse will always be equal to 1 (which is the multiplicative identity for real numbers).

Quotient

The quotient is the result after two amounts have been divided.

Real Number

A real number is a number that can be plotted on a number line. Real numbers include all rational and irrational numbers.

reciprocal

The reciprocal of a number is the number you can multiply it by to get one. The reciprocal of 2 is 1/2. It is also called the multiplicative inverse, or just inverse.