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Not all divisions end in zero. We will examine such divisions in a subsequent subsection.

Practice set a

Perform the following divisions.

126 ÷ 7 size 12{"126" div 7} {}

18

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324 ÷ 4 size 12{"324" div 4} {}

81

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2, 559 ÷ 3 size 12{2,"559" div 3} {}

853

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5, 645 ÷ 5 size 12{5,"645" div 5} {}

1,129

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757 , 125 ÷ 9 size 12{"757","125" div 9} {}

84,125

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Division with a multiple digit divisor

The process of division also works when the divisor consists of two or more digits. We now make educated guesses using the first digit of the divisor and one or two digits of the dividend.

Sample set b

Find 2, 232 ÷ 36 size 12{2,"232" div "36"} {} .

36 2232

Use the first digit of the divisor and the first two digits of the dividend to make the educated guess.

3 goes into 22 at most 7 times.

Try 7: 7 × 36 = 252 size 12{7 times "36"="252"} {} which is greater than 223. Reduce the estimate.

Try 6: 6 × 36 = 216 size 12{6 times "36"="216"} {} which is less than 223.

The first step of a long division problem. 2232 divided by 36. 36 goes into 223 approximately 6 times, with a remainder of 7. The ones digit of 2232 is then brought down to adjoin the 7. Multiply: 6 × 36 = 216. Write 216 below 223. Subtract: 223 - 216 = 7. Bring down the 2.

Divide 3 into 7 to estimate the number of times 36 goes into 72. The 3 goes into 7 at most 2 times.

Try 2: 2 × 36 = 72 size 12{2 times "36"="72"} {} .

Long division. 2232 divided by 36. 36 goes into 223 approximately 6 times, with a remainder of 7. The ones digit of 2232 is then brought down to adjoin the 7. 36 goes into 72 exactly twice, leaving a remainder of 0.

Check : Is 2232 equal to 36 time 62? Yes.

Thus, 2, 232 ÷ 36 = 62 size 12{2,"232" div "36"="62"} {} .

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Find 2, 417 , 228 ÷ 802 size 12{2,"417","228" div "802"} {} .

802 2417228

First, the educated guess: 24 ÷ 8 = 3 size 12{"24" div 8=3} {} . Then 3 × 802 = 2406 size 12{3 times "802"="2406"} {} , which is less than 2417. Use 3 as the guess. Since 3 × 802 = 2406 size 12{3 times "802"="2406"} {} , and 2406 has four digits, place the 3 above the fourth digit of the dividend.

The first step of a long division problem. 2417228 divided by 802. 802 goes into 2417 approximately 3 times, with a remainder of 11. The hundreds digit of 2417228 is then brought down to adjoin the 11.
Subtract: 2417 - 2406 = 11.
Bring down the 2.

The divisor 802 goes into 112 at most 0 times. Use 0.

The second step of a long division problem. 802 goes into 112 0 times, so a zero is placed above, and the next digit is brought down.
Multiply: 0 × 802 = 0. Subtract: 112 - 0 = 112. Bring down the 2.

The 8 goes into 11 at most 1 time, and 1 × 802 = 802 size 12{1 times "802"="802"} {} , which is less than 1122. Try 1.

The third step of a long division problem. 802 goes into 1122 once, so a 1 is placed above and the ones digit is brought down.
Subtract 1122 - 802 = 320
Bring down the 8.

8 goes into 32 at most 4 times.

4 × 802 = 3208 size 12{4 times "802"="3208"} {} .

Use 4.

The third step of a long division problem. 2417228 divided by 802. 802 goes into 2417 approximately 3 times, with a remainder of 11. The hundreds digit of 2417228 is then brought down to adjoin the 11. 802 goes into 112 0 times, so a zero is placed above, and the next digit is brought down. 802 goes into 1122 once, so a 1 is placed above and the ones digit is brought down. 802 goes into 3208 4 times, leaving a remainder of 0.

Check: Is 2417228 equal to 3014 times 802? Yes.

Thus, 2, 417 , 228 ÷ 802 = 3, 014 size 12{2,"417","228" div "802"=3,"014"} {} .

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Practice set b

Perform the following divisions.

1, 376 ÷ 32 size 12{1,"376" div "32"} {}

43

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6, 160 ÷ 55 size 12{6,"160" div "55"} {}

112

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18 , 605 ÷ 61 size 12{"18","605" div "61"} {}

305

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144 , 768 ÷ 48 size 12{"144","764" div "48"} {}

3,016

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Division with a remainder

We might wonder how many times 4 is contained in 10. Repeated subtraction yields

10 -   4 ̲ 6 - 4 ̲ 2

Since the remainder is less than 4, we stop the subtraction. Thus, 4 goes into 10 two times with 2 remaining. We can write this as a division as follows.

2 4 10 -   8 ̲ 2
Divide: 4 goes into 10 at most 2 times. Multiply: 2 × 4 = 8. Write 8 below 0. Subtract: 10 - 8 = 2.

Since 4 does not divide into 2 (the remainder is less than the divisor) and there are no digits to bring down to continue the process, we are done. We write

2 R 2 4    10 - 8 ̲ 2 or 10 ÷ 4 = 2 R 2 2 with remainder 2

Sample set c

Find 85 ÷ 3 size 12{"85" div 3} {} .

Long division. 85 divided by 3. 3 goes into 8 twice, with a remainder of 2. The ones digit is then brought down. 3 goes into 25 8 times, with a remainder of 1.
Divide: 3 goes into 8 at most 2 times. Multiply: 2 × 3 = 6. Write 6 below 8. Subtract: 8 - 6 = 2. Bring down the 5. Divide: 3 goes into 25 at most 8 times. Multiply: 3 × 8 = 24. Write 24 below 25. Subtract: 25 - 24 = 1.

There are no more digits to bring down to continue the process. We are done. One is the remainder.

Check: Multiply 28 and 3, then add 1.

28 ×   3 ̲ 84 +   1 ̲ 85

Thus, 85 ÷ 3 = 28 R1 size 12{"85" div 3="28"`R1} {} .

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Find 726 ÷ 23 size 12{"726" div "23"} {} .

Long division. 726 divided by 23. 23 goes into 72 three times, with a remainder of 3. The ones digit is then brought down. 23 goes into 36 once, with a remainder of 13.

Check: Multiply 31 by 23, then add 13.

31 times 23 equals 713. 713 plus 13 equals 726.

Thus, 726 ÷ 23 = 31 R 13 size 12{"726" div "23"="31"````R"13"} {} .

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Practice set c

Perform the following divisions.

75 ÷ 4 size 12{"75" div 4} {}

18 R3

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346 ÷ 8 size 12{"346" div 8} {}

43 R2

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489 ÷ 21 size 12{"489" div "21"} {}

23 R6

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5, 016 ÷ 82 size 12{5,"016" div "82"} {}

61 R14

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41 , 196 ÷ 67 size 12{"41","196" div "67"} {}

614 R58

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Calculators

The calculator can be useful for finding quotients with single and multiple digit divisors. If, however, the division should result in a remainder, the calculator is unable to provide us with the particular value of the remainder. Also, some calculators (most nonscientific) are unable to perform divisions in which one of the numbers has more than eight digits.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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Source:  OpenStax, Fundamentals of mathematics. OpenStax CNX. Aug 18, 2010 Download for free at http://cnx.org/content/col10615/1.4
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