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This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses how to estimate by clustering. By the end of the module students should understand the concept of clustering and be able to estimate the result of adding more than two numbers when clustering occurs using the clustering technique.

Section overview

  • Estimation by Clustering

Cluster

When more than two numbers are to be added, the sum may be estimated using the clustering technique. The rounding technique could also be used, but if several of the numbers are seen to cluster (are seen to be close to) one particular number, the clustering technique provides a quicker estimate. Consider a sum such as

32 + 68 + 29 + 73 size 12{"32 "+" 68 "+" 29 "+" 73"} {}

Notice two things:

  1. There are more than two numbers to be added.
  2. Clustering occurs.
    1. Both 68 and 73 cluster around 70, so 68 + 73 size 12{"68 "+" 73"} {} is close to 80 + 70 = 2 ( 70 ) = 140 size 12{"80"+"70"=2 \( "70" \) ="140"} {} .
      The expression 32 + 68 + 29 + 71. 32 and 29 are grouped together with a line, and 68 and 71 are grouped in the same way.
    2. Both 32 and 29 cluster around 30, so 32 + 29 size 12{"32 "+" 29"} {} is close to 30 + 30 = 2 ( 30 ) = 60 size 12{"30 "+" 30"=2 \( "30" \) ="60"} {} .

The sum may be estimated by

2 30 + 2 70 = 6 + 140 = 200

In fact, 32 + 68 + 29 + 73 = 202 size 12{"32 "+" 68 "+" 29 "+" 73 "=" 202"} {} .

Sample set a

Estimate each sum. Results may vary.

27 + 48 + 31 + 52 size 12{"27 "+" 48 "+" 31 "+" 52"} {} .

27 and 31 cluster near 30. Their sum is about 2 30 = 60 size 12{"2 " cdot " 30 "=" 60"} {} .

48 and 52 cluster near 50. Their sum is about 2 50 = 100 size 12{"2 " cdot " 50 "="100"} {} .

Thus, 27 + 48 + 31 + 5 2 size 12{"27"+"48"+"31"+5"2 "} {} is about ( 2 30 ) + ( 2 50 ) = 60 + 100 = 160

In fact, 27 + 48 + 31 + 52 = 158 size 12{"27 "+" 48 "+" 31 "+" 52 "=" 158"} {} .

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88 + 21 + 19 + 91 size 12{"88 "+" 21 "+" 19 "+" 91"} {} .

88 and 91 cluster near 90. Their sum is about 2 90 = 180 size 12{"2 " cdot " 90 "=" 180"} {} .

21 and 19 cluster near 20. Their sum is about 2 20 = 40 size 12{"2 " cdot " 20 "=" 40"} {} .

Thus, 88 + 21 + 19 + 91 size 12{"88 "+" 21 "+" 19 "+" 91"} {} is about ( 2 90 ) + ( 2 20 ) = 180 + 40 = 220

In fact, 88 + 21 + 19 + 91 = 219 size 12{"88 "+" 21 "+" 19 "+" 91 "=" 219"} {} .

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17 + 21 + 48 + 18 size 12{"17 "+" 21 "+" 48 "+" 18"} {} .

17, 21, and 18 cluster near 20. Their sum is about 3 20 = 60 size 12{"3 " cdot " 20 "=" 60"} {} .

48 is about 50.

Thus, 17 + 21 + 48 + 18 size 12{"17 "+" 21 "+" 48 "+" 18"} {} is about ( 3 20 ) + 50 = 60 + 50 = 110

In fact, 17 + 21 + 48 + 18 = 104 size 12{"17 "+" 21 "+" 48 "+" 18 "=" 104"} {} .

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61 + 48 + 49 + 57 + 52 size 12{"61 "+" 48 "+" 49 "+" 57 "+" 52"} {} .

61 and 57 cluster near 60. Their sum is about 2 60 = 120 size 12{"2 " cdot " 60 "=" 120"} {} .

48, 49, and 52 cluster near 50. Their sum is about 3 50 = 150 size 12{3 cdot " 50 "=" 150"} {} .

Thus, 61 + 48 + 49 + 57 + 52 size 12{"61 "+" 48 "+" 49 "+" 57 "+" 52"} {} is about ( 2 60 ) + ( 3 50 ) = 120 + 150 = 270

In fact, 61 + 48 + 49 + 57 + 52 = 267 size 12{"61 "+" 48 "+" 49 "+" 57 "+" 52 "=" 267"} {} .

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706 + 321 + 293 + 684 size 12{"706 "+" 321 "+" 293 "+" 684"} {} .

706 and 684 cluster near 700. Their sum is about 2 700 = 1,400 size 12{"2 " cdot " 700 "=" 1,400"} {} .

321 and 293 cluster near 300. Their sum is about 2 300 = 600 size 12{"2 " cdot " 300 "=" 600"} {} .

Thus, 706 + 321 + 293 + 684 size 12{"706 "+" 321 "+" 293 "+" 684"} {} is about ( 2 700 ) + ( 2 300 ) = 1,400 + 600 = 2,000

In fact, 706 + 321 + 293 + 684 = 2,004 size 12{"706 "+" 321 "+" 293 "+" 684 "=" 2,004"} {} .

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Practice set a

Use the clustering method to estimate each sum.

28 + 51 + 31 + 47 size 12{"28 "+" 51 "+" 31 "+" 47"} {}

( 2 30 ) + ( 2 50 ) = 60 + 100 = 160

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42 + 39 + 68 + 41 size 12{"42 "+" 39 "+" 68 "+" 41"} {}

( 3 40 ) + 70 = 120 + 70 = 190

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37 + 39 + 83 + 42 + 79 size 12{"37 "+" 39 "+" 83 "+" 42 "+" 79"} {}

( 3 40 ) + ( 2 80 ) = 120 + 160 = 280

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612 + 585 + 830 + 794 size 12{"612 "+" 585 "+" 830 "+" 794"} {}

( 2 600 ) + ( 2 800 ) = 1,200 + 1,600 = 2,800

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Exercises

Use the clustering method to estimate each sum. Results may vary.

28 + 51 + 31 + 47 size 12{"28"+"51"+"31"+"47"} {}

2 30 + 2 50 = 160   157 size 12{2 left ("30" right )+2 left ("50" right )="160" left ("157" right )} {}

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42 + 19 + 39 + 23 size 12{"42"+"19"+"39"+"23"} {}

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88 + 62 + 59 + 90 size 12{"88"+"62"+"59"+"90"} {}

2 90 + 2 60 = 300   299 size 12{2 left ("90" right )+2 left ("60" right )="300" left ("299" right )} {}

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76 + 29 + 33 + 82 size 12{"76"+"29"+"33"+"82"} {}

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19 + 23 + 87 + 21 size 12{"19"+"23"+"87"+"21"} {}

3 20 + 90 = 150   150 size 12{3 left ("20" right )+"90"="150" left ("150" right )} {}

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41 + 28 + 42 + 37 size 12{"41"+"28"+"42"+"37"} {}

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89 + 32 + 89 + 93 size 12{"89"+"32"+"89"+"93"} {}

3 90 + 30 = 300   303 size 12{3 left ("90" right )+"30"="300" left ("303" right )} {}

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73 + 72 + 27 + 71 size 12{"73"+"72"+"27"+"71"} {}

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43 + 62 + 61 + 55 size 12{"43"+"62"+"61"+"55"} {}

40 + 3 60 = 220   221 size 12{"40"+3 left ("60" right )="220" left ("221" right )} {}

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31 + 77 + 31 + 27 size 12{"31"+"77"+"31"+"27"} {}

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57 + 34 + 28 + 61 + 62 size 12{"57"+"34"+"28"+"61"+"62"} {}

3 60 + 2 30 = 240   242 size 12{3 left ("60" right )+2 left ("30" right )="240" left ("242" right )} {}

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94 + 18 + 23 + 91 + 19 size 12{"94"+"18"+"23"+"91"+"19"} {}

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103 + 72 + 66 + 97 + 99 size 12{"103"+"72"+"66"+"97"+"99"} {}

3 100 + 2 70 = 440   437 size 12{3 left ("100" right )+2 left ("70" right )="440" left ("437" right )} {}

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42 + 121 + 119 + 124 + 41 size 12{"42"+"121"+"119"+"124"+"41"} {}

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19 + 24 + 87 + 23 + 91 + 93 size 12{"19"+"24"+"87"+"23"+"91"+"93"} {}

3 20 + 3 90 = 330   337 size 12{3 left ("20" right )+3 left ("90" right )="330" left ("337" right )} {}

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108 + 61 + 63 + 96 + 57 + 99 size 12{"108"+"61"+"63"+"96"+"57"+"99"} {}

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518 + 721 + 493 + 689 size 12{"518"+"721"+"493"+"689"} {}

2 500 + 2 700 = 2, 400   2, 421 size 12{2 left ("500" right )+2 left ("700" right )=2,"400" left (2,"421" right )} {}

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981 + 1208 + 1214 + 1006 size 12{"981"+"1208"+"1214"+"1006"} {}

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23 + 81 + 77 + 79 + 19 + 81 size 12{"23"+"81"+"77"+"79"+"19"+"81"} {}

2 20 + 4 80 = 360   360 size 12{2 left ("20" right )+4 left ("80" right )="360" left ("360" right )} {}

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94 + 68 + 66 + 101 + 106 + 71 + 110 size 12{"94"+"68"+"66"+"101"+"106"+"71"+"110"} {}

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Exercises for review

( [link] ) Specify all the digits greater than 6.

7, 8, 9

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( [link] ) Find the product: 2 3 9 14 7 12 size 12{ { {2} over {3} } cdot { {9} over {"14"} } cdot { {7} over {"12"} } } {} .

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( [link] ) Convert 0.06 to a fraction.

3 50 size 12{ { {3} over {"50"} } } {}

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( [link] ) Write the proportion in fractional form: "5 is to 8 as 25 is to 40."

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( [link] ) Estimate the sum using the method of rounding: 4,882 + 2,704 size 12{"4,882 "+" 2,704"} {} .

4, 900 + 2, 700 = 7, 600   ( 7, 586 ) size 12{4,"900"+2,"700"=7,"600" \( 7,"586" \) } {}

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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
learn
Google
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
Smarajit Reply
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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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