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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

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
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
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Cied
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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