<< Chapter < Page Chapter >> Page >
This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses division of fractions. By the end of the module students should be able to determine the reciprocal of a number and divide one fraction by another.

Section overview

  • Reciprocals
  • Dividing Fractions

Reciprocals

Reciprocals

Two numbers whose product is 1 are called reciprocals of each other.

Sample set a

The following pairs of numbers are reciprocals.

3 4 and 4 3 3 4 4 3 = 1

Got questions? Get instant answers now!

7 16 and 16 7 7 16 16 7 = 1

Got questions? Get instant answers now!

1 6 and 6 1 1 6 6 1 = 1

Got questions? Get instant answers now!

Notice that we can find the reciprocal of a nonzero number in fractional form by inverting it (exchanging positions of the numerator and denominator).

Practice set a

Find the reciprocal of each number.

3 10 size 12{ { {3} over {"10"} } } {}

10 3 size 12{ { {"10"} over {3} } } {}

Got questions? Get instant answers now!

2 3 size 12{ { {2} over {3} } } {}

3 2 size 12{ { {3} over {2} } } {}

Got questions? Get instant answers now!

7 8 size 12{ { {7} over {8} } } {}

8 7 size 12{ { {8} over {7} } } {}

Got questions? Get instant answers now!

1 5 size 12{ { {1} over {5} } } {}

5

Got questions? Get instant answers now!

2 2 7 size 12{2 { {2} over {7} } } {}

Write this number as an improper fraction first.

7 16 size 12{ { {7} over {"16"} } } {}

Got questions? Get instant answers now!

5 1 4 size 12{5 { {1} over {4} } } {}

4 21 size 12{ { {4} over {"21"} } } {}

Got questions? Get instant answers now!

10 3 16 size 12{"10" { {3} over {"16"} } } {}

16 163 size 12{ { {"16"} over {"163"} } } {}

Got questions? Get instant answers now!

Dividing fractions

Our concept of division is that it indicates how many times one quantity is con­tained in another quantity. For example, using the diagram we can see that there are 6 one-thirds in 2.

Two rectangles, each divided into three parts. The rectangles are connected to each other. There are 6 one-thirds in 2.

Since 2 contains six 1 3 size 12{ { {1} over {3} } } {} 's we express this as

Two divided by one-third is equal to six. Note also that two times three is equal to six, because one-third and three are reciprocals.

Using these observations, we can suggest the following method for dividing a number by a fraction.

Dividing one fraction by another fraction

To divide a first fraction by a second, nonzero fraction, multiply the first traction by the reciprocal of the second fraction.

Invert and multiply

This method is commonly referred to as "invert the divisor and multiply."

Sample set b

Perform the following divisions.

1 3 ÷ 3 4 size 12{ { {1} over {3} } div { {3} over {4} } } {} . The divisor is 3 4 size 12{ { {3} over {4} } } {} . Its reciprocal is 4 3 size 12{ { {4} over {3} } } {} . Multiply 1 3 size 12{ { {1} over {3} } } {} by 4 3 size 12{ { {4} over {3} } } {} .

1 3 4 3 = 1 4 3 3 = 4 9 size 12{ { {1} over {3} } cdot { {4} over {3} } = { {1 cdot 4} over {3 cdot 3} } = { {4} over {9} } } {}

1 3 ÷ 3 4 = 4 9 size 12{ { {1} over {3} } div { {3} over {4} } = { {4} over {9} } } {}

Got questions? Get instant answers now!

3 8 ÷ 5 4 size 12{ { {3} over {8} } div { {5} over {4} } } {} The divisor is 5 4 size 12{ { {5} over {4} } } {} . Its reciprocal is 4 5 size 12{ { {4} over {5} } } {} . Multiply 3 8 size 12{ { {3} over {8} } } {} by 4 5 size 12{ { {4} over {5} } } {} .

3 3 2 4 1 5 = 3 1 2 5 = 3 10 size 12{ { {3} over { { { {3}}} cSub { size 8{2} } } } cdot { { { { {4}}} cSup { size 8{1} } } over {5} } = { {3 cdot 1} over {2 cdot 5} } = { {3} over {"10"} } } {}

3 8 ÷ 5 4 = 3 10 size 12{ { {3} over {8} } div { {5} over {4} } = { {3} over {"10"} } } {}

Got questions? Get instant answers now!

5 6 ÷ 5 12 size 12{ { {5} over {6} } div { {5} over {"12"} } } {} . The divisor is 5 12 size 12{ { {5} over {"12"} } } {} . Its reciprocal is 12 5 size 12{ { {"12"} over {5} } } {} . Multiply 5 6 size 12{ { {5} over {6} } } {} by 12 5 size 12{ { {"12"} over {5} } } {} .

5 1 6 1 12 2 5 1 = 1 2 1 1 = 2 1 = 2 size 12{ { { { { {5}}} cSup { size 8{1} } } over { { { {6}}} cSub { size 8{1} } } } cdot { { {"12"} cSup { size 8{2} } } over { {5} cSub { size 8{1} } } } = { {1 cdot 2} over {1 cdot 1} } = { {2} over {1} } =2} {}

5 6 ÷ 5 12 = 2

Got questions? Get instant answers now!

2 2 9 ÷ 3 1 3 size 12{2 { {2} over {9} } div 3 { {1} over {3} } } {} . Convert each mixed number to an improper fraction.

2 2 9 = 9 2 + 2 9 = 20 9 size 12{2 { {2} over {9} } = { {9 cdot 2+2} over {9} } = { {"20"} over {9} } } {} .

3 1 3 = 3 3 + 1 3 = 10 3 size 12{3 { {1} over {3} } = { {3 cdot 3+1} over {3} } = { {10} over {3} } } {} .

20 9 ÷ 10 3 size 12{ { {"20"} over {9} } div { {"10"} over {3} } } {} The divisor is 10 3 size 12{ { {"10"} over {3} } } {} . Its reciprocal is 3 10 size 12{ { {3} over {"10"} } } {} . Multiply 20 9 size 12{ { {"20"} over {9} } } {} by 3 10 size 12{ { {3} over {"10"} } } {} .

20 2 9 3 3 1 10 1 = 2 1 3 1 = 2 3 size 12{ { { { { {2}} { {0}}} cSup { size 8{2} } } over { { { {9}}} cSub { size 8{3} } } } cdot { { { { {3}}} cSup { size 8{1} } } over { { { {1}} { {0}}} cSub { size 8{1} } } } = { {2 cdot 1} over {3 cdot 1} } = { {2} over {3} } } {}

2 2 9 ÷ 3 1 3 = 2 3 size 12{2 { {2} over {9} } div 3 { {1} over {3} } = { {2} over {3} } } {}

Got questions? Get instant answers now!

12 11 ÷ 8 size 12{ { {"12"} over {"11"} } div 8} {} . First conveniently write 8 as 8 1 size 12{ { {8} over {1} } } {} .

12 11 ÷ 8 1 size 12{ { {"12"} over {"11"} } div { {8} over {1} } } {} The divisor is 8 1 size 12{ { {8} over {1} } } {} . Its reciprocal is 1 8 size 12{ { {1} over {8} } } {} . Multiply 12 11 size 12{ { {"12"} over {"11"} } } {} by 1 8 size 12{ { {1} over {8} } } {} .

12 3 11 1 8 2 = 3 1 11 2 = 3 22 size 12{ { { { { {1}} { {2}}} cSup { size 8{3} } } over {"11"} } cdot { {1} over { { { {8}}} cSub { size 8{2} } } } = { {3 cdot 1} over {"11" cdot 2} } = { {3} over {"22"} } } {}

12 11 ÷ 8 = 3 22 size 12{ { {"12"} over {"11"} } div 8= { {3} over {"22"} } } {}

Got questions? Get instant answers now!

7 8 ÷ 21 20 3 35 size 12{ { {7} over {8} } div { {"21"} over {"20"} } cdot { {3} over {"35"} } } {} . The divisor is 21 20 size 12{ { {"21"} over {"20"} } } {} . Its reciprocal is 20 21 size 12{ { {"20"} over {"21"} } } {} .

7 1 8 2 20 5 1 21 3 1 3 1 35 7 = 1 1 1 2 1 7 = 1 14 size 12{ { { {7} cSup { size 8{1} } } over { {8} cSub { size 8{2} } } } cdot { { {"20"} cSup { size 8{ {5} cSup { size 6{1} } } } } over { {"21"} cSub { {3} cSub { size 6{1} } } } } size 12{ cdot { { {3} cSup {1} } over { size 12{ {"35"} cSub {7} } } } } size 12{ {}= { {1 cdot 1 cdot 1} over {2 cdot 1 cdot 7} } = { {1} over {"14"} } }} {}

7 8 ÷ 21 20 3 25 = 1 14 size 12{ { {7} over {8} } div { {"21"} over {"20"} } cdot { {3} over {"25"} } = { {1} over {"14"} } } {}

Got questions? Get instant answers now!

How many 2 3 8 size 12{2 { {3} over {8} } } {} -inch-wide packages can be placed in a box 19 inches wide?

The problem is to determine how many two and three eighths are contained in 19, that is, what is 19 ÷ 2 3 8 size 12{"19" div 2 { {3} over {8} } } {} ?

2 3 8 = 19 8 size 12{2 { {3} over {8} } = { {"19"} over {8} } } {} Convert the divisor 2 3 8 size 12{2 { {3} over {8} } } {} to an improper fraction.

19 = 19 1 size 12{"19"= { {"19"} over {1} } } {} Write the dividend 19 as 19 1 size 12{ { {"19"} over {1} } } {} .

19 1 ÷ 19 8 size 12{ { {"19"} over {1} } div { {"19"} over {8} } } {} The divisor is 19 8 size 12{ { {"19"} over {8} } } {} . Its reciprocal is 8 19 size 12{ { {8} over {"19"} } } {} .

19 1 1 8 19 1 = 1 8 1 1 = 8 1 = 8 size 12{ { { {"19"} cSup { size 8{1} } } over {1} } cdot { {8} over { {"19"} cSub { size 8{1} } } } = { {1 cdot 8} over {1 cdot 1} } = { {8} over {1} } =8} {}

Thus, 8 packages will fit into the box.

Got questions? Get instant answers now!

Practice set b

Perform the following divisions.

1 2 ÷ 9 8 size 12{ { {1} over {2} } div { {9} over {8} } } {}

4 9 size 12{ { {4} over {9} } } {}

Got questions? Get instant answers now!

3 8 ÷ 9 24 size 12{ { {3} over {8} } div { {9} over {"24"} } } {}

1

Got questions? Get instant answers now!

7 15 ÷ 14 15 size 12{ { {7} over {"15"} } div { {"14"} over {"15"} } } {}

1 2 size 12{ { {1} over {2} } } {}

Got questions? Get instant answers now!

8 ÷ 8 15 size 12{8 div { {8} over {"15"} } } {}

15

Got questions? Get instant answers now!

6 1 4 ÷ 5 12 size 12{6 { {1} over {4} } div { {5} over {"12"} } } {}

15

Got questions? Get instant answers now!

3 1 3 ÷ 1 2 3 size 12{3 { {1} over {3} } div 1 { {2} over {3} } } {}

2

Got questions? Get instant answers now!

5 6 ÷ 2 3 8 25 size 12{ { {5} over {6} } div { {2} over {3} } cdot { {8} over {"25"} } } {}

2 5 size 12{ { {2} over {5} } } {}

Got questions? Get instant answers now!

A container will hold 106 ounces of grape juice. How many 6 5 8 size 12{6 { {5} over {8} } } {} -ounce glasses of grape juice can be served from this container?

16 glasses

Got questions? Get instant answers now!

Determine each of the following quotients and then write a rule for this type of division.

1 ÷ 2 3 size 12{1 div { {2} over {3} } } {}

3 2 size 12{ { {3} over {2} } } {}

Got questions? Get instant answers now!

1 ÷ 3 8 size 12{1 div { {3} over {8} } } {}

8 3 size 12{ { {8} over {3} } } {}

Got questions? Get instant answers now!

1 ÷ 3 4 size 12{1 div { {3} over {4} } } {}

4 3 size 12{ { {4} over {3} } } {}

Got questions? Get instant answers now!

1 ÷ 5 2 size 12{1 div { {5} over {2} } } {}

2 5 size 12{ { {2} over {5} } } {}

Got questions? Get instant answers now!

When dividing 1 by a fraction, the quotient is the .

is the reciprocal of the fraction.

Got questions? Get instant answers now!

Exercises

For the following problems, find the reciprocal of each number.

4 5 size 12{ { {4} over {5} } } {}

5 4 size 12{ { {5} over {4} } } {} or 1 1 4 size 12{1 { {1} over {4} } } {}

Got questions? Get instant answers now!

8 11 size 12{ { {8} over {"11"} } } {}

Got questions? Get instant answers now!

2 9 size 12{ { {2} over {9} } } {}

9 2 size 12{ { {9} over {2} } } {} or 4 1 2 size 12{4 { {1} over {2} } } {}

Got questions? Get instant answers now!

1 5 size 12{ { {1} over {5} } } {}

Got questions? Get instant answers now!

3 1 4 size 12{3 { {1} over {4} } } {}

4 13 size 12{ { {4} over {"13"} } } {}

Got questions? Get instant answers now!

8 1 4 size 12{8 { {1} over {4} } } {}

Got questions? Get instant answers now!

3 2 7 size 12{3 { {2} over {7} } } {}

7 23 size 12{ { {7} over {"23"} } } {}

Got questions? Get instant answers now!

5 3 4 size 12{5 { {3} over {4} } } {}

Got questions? Get instant answers now!

For the following problems, find each value.

3 8 ÷ 3 5 size 12{ { {3} over {8} } div { {3} over {5} } } {}

5 8 size 12{ { {5} over {8} } } {}

Got questions? Get instant answers now!

5 9 ÷ 5 6 size 12{ { {5} over {9} } div { {5} over {6} } } {}

Got questions? Get instant answers now!

9 16 ÷ 15 8 size 12{ { {9} over {"16"} } div { {"15"} over {8} } } {}

3 10 size 12{ { {3} over {"10"} } } {}

Got questions? Get instant answers now!

4 9 ÷ 6 15 size 12{ { {4} over {9} } div { {6} over {"15"} } } {}

Got questions? Get instant answers now!

25 49 ÷ 4 9 size 12{ { {"25"} over {"49"} } div { {4} over {9} } } {}

225 196 size 12{ { {"225"} over {"196"} } } {} or 1 29 196 size 12{1 { {"29"} over {"196"} } } {}

Got questions? Get instant answers now!

15 4 ÷ 27 8 size 12{ { {"15"} over {4} } div { {"27"} over {8} } } {}

Got questions? Get instant answers now!

24 75 ÷ 8 15 size 12{ { {"24"} over {"75"} } div { {8} over {"15"} } } {}

3 5 size 12{ { {3} over {5} } } {}

Got questions? Get instant answers now!

5 7 ÷ 0 size 12{ { {5} over {7} } div 0} {}

Got questions? Get instant answers now!

7 8 ÷ 7 8 size 12{ { {7} over {8} } div { {7} over {8} } } {}

1

Got questions? Get instant answers now!

0 ÷ 3 5 size 12{0 div { {3} over {5} } } {}

Got questions? Get instant answers now!

4 11 ÷ 4 11 size 12{ { {4} over {"11"} } div { {4} over {"11"} } } {}

1

Got questions? Get instant answers now!

2 3 ÷ 2 3 size 12{ { {2} over {3} } div { {2} over {3} } } {}

Got questions? Get instant answers now!

7 10 ÷ 10 7 size 12{ { {7} over {"10"} } div { {"10"} over {7} } } {}

49 100 size 12{ { {"49"} over {"100"} } } {}

Got questions? Get instant answers now!

3 4 ÷ 6 size 12{ { {3} over {4} } div 6} {}

Got questions? Get instant answers now!

9 5 ÷ 3 size 12{ { {9} over {5} } div 3} {}

3 5 size 12{ { {3} over {5} } } {}

Got questions? Get instant answers now!

4 1 6 ÷ 3 1 3 size 12{4 { {1} over {6} } div 3 { {1} over {3} } } {}

Got questions? Get instant answers now!

7 1 7 ÷ 8 1 3 size 12{7 { {1} over {7} } div 8 { {1} over {3} } } {}

6 7 size 12{ { {6} over {7} } } {}

Got questions? Get instant answers now!

1 1 2 ÷ 1 1 5 size 12{1 { {1} over {2} } div 1 { {1} over {5} } } {}

Got questions? Get instant answers now!

3 2 5 ÷ 6 25 size 12{3 { {2} over {5} } div { {6} over {"25"} } } {}

85 6 size 12{ { {"85"} over {6} } } {} or 14 1 6 size 12{"14" { {1} over {6} } } {}

Got questions? Get instant answers now!

5 1 6 ÷ 31 6 size 12{5 { {1} over {6} } div { {"31"} over {6} } } {}

Got questions? Get instant answers now!

35 6 ÷ 3 3 4 size 12{ { {"35"} over {6} } div 3 { {3} over {4} } } {}

28 18 = 14 9 size 12{ { {"28"} over {"18"} } = { {"14"} over {9} } } {} or 1 5 9 size 12{1 { {5} over {9} } } {}

Got questions? Get instant answers now!

5 1 9 ÷ 1 18 size 12{5 { {1} over {9} } div { {1} over {"18"} } } {}

Got questions? Get instant answers now!

8 3 4 ÷ 7 8 size 12{8 { {3} over {4} } div { {7} over {8} } } {}

10

Got questions? Get instant answers now!

12 8 ÷ 1 1 2 size 12{ { {"12"} over {8} } div 1 { {1} over {2} } } {}

Got questions? Get instant answers now!

3 1 8 ÷ 15 16 size 12{3 { {1} over {8} } div { {"15"} over {"16"} } } {}

10 3 size 12{ { {"10"} over {3} } } {} or 3 1 3 size 12{3 { {1} over {3} } } {}

Got questions? Get instant answers now!

11 11 12 ÷ 9 5 8 size 12{"11" { {"11"} over {"12"} } div 9 { {5} over {8} } } {}

Got questions? Get instant answers now!

2 2 9 ÷ 11 2 3 size 12{2 { {2} over {9} } div "11" { {2} over {3} } } {}

4 21 size 12{ { {4} over {"21"} } } {}

Got questions? Get instant answers now!

16 3 ÷ 6 2 5 size 12{ { {"16"} over {3} } div 6 { {2} over {5} } } {}

Got questions? Get instant answers now!

4 3 25 ÷ 2 56 75 size 12{4 { {3} over {"25"} } div 2 { {"56"} over {"75"} } } {}

3 2 size 12{ { {3} over {2} } } {} or 1 1 2 size 12{1 { {1} over {2} } } {}

Got questions? Get instant answers now!

1 1000 ÷ 1 100 size 12{ { {1} over {"1000"} } div { {1} over {"100"} } } {}

Got questions? Get instant answers now!

3 8 ÷ 9 16 6 5 size 12{ { {3} over {8} } div { {9} over {"16"} } cdot { {6} over {5} } } {}

4 5 size 12{ { {4} over {5} } } {}

Got questions? Get instant answers now!

3 16 9 8 6 5 size 12{ { {3} over {"16"} } cdot { {9} over {8} } cdot { {6} over {5} } } {}

Got questions? Get instant answers now!

4 15 ÷ 2 25 9 10 size 12{ { {4} over {"15"} } div { {2} over {"25"} } cdot { {9} over {"10"} } } {}

3

Got questions? Get instant answers now!

21 30 1 1 4 ÷ 9 10 size 12{ { {"21"} over {"30"} } cdot 1 { {1} over {4} } div { {9} over {"10"} } } {}

Got questions? Get instant answers now!

8 1 3 36 75 ÷ 4 size 12{8 { {1} over {3} } cdot { {"36"} over {"75"} } div 4} {}

1

Got questions? Get instant answers now!

Exercises for review

( [link] ) What is the value of 5 in the number 504,216?

Got questions? Get instant answers now!

( [link] ) Find the product of 2,010 and 160.

321,600

Got questions? Get instant answers now!

( [link] ) Use the numbers 8 and 5 to illustrate the commutative property of multiplication.

Got questions? Get instant answers now!

( [link] ) Find the least common multiple of 6, 16, and 72.

144

Got questions? Get instant answers now!

( [link] ) Find 8 9 size 12{ { {8} over {9} } } {} of 6 3 4 size 12{6 { {3} over {4} } } {} .

Got questions? Get instant answers now!

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
7hours 36 min - 4hours 50 min
Tanis Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Fundamentals of mathematics. OpenStax CNX. Aug 18, 2010 Download for free at http://cnx.org/content/col10615/1.4
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Fundamentals of mathematics' conversation and receive update notifications?

Ask