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Techniques of Estimation: Estimation by Clustering

Module by: Wade Ellis, Denny Burzynski. E-mail the authorsEdited By: Math Editors

Summary: 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 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. (a) Both 68 and 73 cluster around 70, so 68 + 7368 + 73 size 12{"68 "+" 73"} {} is close to 80+70=2(70)=14080+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. (b) Both 32 and 29 cluster around 30, so 32 + 2932 + 29 size 12{"32 "+" 29"} {} is close to 30 + 30=2(30)=6030 + 30=2(30)=60 size 12{"30 "+" 30"=2 \( "30" \) ="60"} {}.

The sum may be estimated by

2 30 + 2 70 = 6 + 140 = 200 2 30 + 2 70 = 6 + 140 = 200

In fact, 32 + 68 + 29 + 73 = 20232 + 68 + 29 + 73 = 202 size 12{"32 "+" 68 "+" 29 "+" 73 "=" 202"} {}.

Sample Set A

Estimate each sum. Results may vary.

Example 1

27 + 48 + 31 + 5227 + 48 + 31 + 52 size 12{"27 "+" 48 "+" 31 "+" 52"} {}.

27 and 31 cluster near 30. Their sum is about 2 30 = 602 30 = 60 size 12{"2 " cdot " 30 "=" 60"} {}.

48 and 52 cluster near 50. Their sum is about 2 50 =1002 50 =100 size 12{"2 " cdot " 50 "="100"} {}.

Thus, 27+48+31+52 27+48+31+52 size 12{"27"+"48"+"31"+5"2 "} {} is about (230)+(250) = 60+100 = 160 (230)+(250) = 60+100 = 160

In fact, 27 + 48 + 31 + 52 = 15827 + 48 + 31 + 52 = 158 size 12{"27 "+" 48 "+" 31 "+" 52 "=" 158"} {}.

Example 2

88 + 21 + 19 + 9188 + 21 + 19 + 91 size 12{"88 "+" 21 "+" 19 "+" 91"} {}.

88 and 91 cluster near 90. Their sum is about 2 90 = 1802 90 = 180 size 12{"2 " cdot " 90 "=" 180"} {}.

21 and 19 cluster near 20. Their sum is about 2 20 = 402 20 = 40 size 12{"2 " cdot " 20 "=" 40"} {}.

Thus, 88 + 21 + 19 + 9188 + 21 + 19 + 91 size 12{"88 "+" 21 "+" 19 "+" 91"} {} is about (290)+(220) = 180+40 = 220 (290)+(220) = 180+40 = 220

In fact, 88 + 21 + 19 + 91 = 21988 + 21 + 19 + 91 = 219 size 12{"88 "+" 21 "+" 19 "+" 91 "=" 219"} {}.

Example 3

17 + 21 + 48 + 1817 + 21 + 48 + 18 size 12{"17 "+" 21 "+" 48 "+" 18"} {}.

17, 21, and 18 cluster near 20. Their sum is about 3 20 = 603 20 = 60 size 12{"3 " cdot " 20 "=" 60"} {}.

48 is about 50.

Thus, 17 + 21 + 48 + 1817 + 21 + 48 + 18 size 12{"17 "+" 21 "+" 48 "+" 18"} {} is about (320)+50 = 60+50 = 110 (320)+50 = 60+50 = 110

In fact, 17 + 21 + 48 + 18 = 10417 + 21 + 48 + 18 = 104 size 12{"17 "+" 21 "+" 48 "+" 18 "=" 104"} {}.

Example 4

61 + 48 + 49 + 57 + 5261 + 48 + 49 + 57 + 52 size 12{"61 "+" 48 "+" 49 "+" 57 "+" 52"} {}.

61 and 57 cluster near 60. Their sum is about 2 60 = 1202 60 = 120 size 12{"2 " cdot " 60 "=" 120"} {}.

48, 49, and 52 cluster near 50. Their sum is about 3 50 = 1503 50 = 150 size 12{3 cdot " 50 "=" 150"} {}.

Thus, 61 + 48 + 49 + 57 + 5261 + 48 + 49 + 57 + 52 size 12{"61 "+" 48 "+" 49 "+" 57 "+" 52"} {} is about (260)+(350) = 120+150 = 270 (260)+(350) = 120+150 = 270

In fact, 61 + 48 + 49 + 57 + 52 = 26761 + 48 + 49 + 57 + 52 = 267 size 12{"61 "+" 48 "+" 49 "+" 57 "+" 52 "=" 267"} {}.

Example 5

706 + 321 + 293 + 684706 + 321 + 293 + 684 size 12{"706 "+" 321 "+" 293 "+" 684"} {}.

706 and 684 cluster near 700. Their sum is about 2 700 = 1,4002 700 = 1,400 size 12{"2 " cdot " 700 "=" 1,400"} {}.

321 and 293 cluster near 300. Their sum is about 2 300 = 6002 300 = 600 size 12{"2 " cdot " 300 "=" 600"} {}.

Thus, 706 + 321 + 293 + 684706 + 321 + 293 + 684 size 12{"706 "+" 321 "+" 293 "+" 684"} {} is about (2700)+(2300) = 1,400+600 = 2,000 (2700)+(2300) = 1,400+600 = 2,000

In fact, 706 + 321 + 293 + 684 = 2,004706 + 321 + 293 + 684 = 2,004 size 12{"706 "+" 321 "+" 293 "+" 684 "=" 2,004"} {}.

Practice Set A

Use the clustering method to estimate each sum.

Exercise 1

28 + 51 + 31 + 4728 + 51 + 31 + 47 size 12{"28 "+" 51 "+" 31 "+" 47"} {}

Solution

( 2 30 ) + ( 2 50 ) = 60 + 100 = 160 (230)+(250)=60+100=160

Exercise 2

42 + 39 + 68 + 4142 + 39 + 68 + 41 size 12{"42 "+" 39 "+" 68 "+" 41"} {}

Solution

( 3 40 ) + 70 = 120 + 70 = 190 (340)+70=120+70=190

Exercise 3

37 + 39 + 83 + 42 + 7937 + 39 + 83 + 42 + 79 size 12{"37 "+" 39 "+" 83 "+" 42 "+" 79"} {}

Solution

( 3 40 ) + ( 2 80 ) = 120 + 160 = 280 (340)+(280)=120+160=280

Exercise 4

612 + 585 + 830 + 794612 + 585 + 830 + 794 size 12{"612 "+" 585 "+" 830 "+" 794"} {}

Solution

( 2 600 ) + ( 2 800 ) = 1,200 + 1,600 = 2,800 (2600)+(2800)=1,200+1,600=2,800

Exercises

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

Exercise 5

28+51+31+4728+51+31+47 size 12{"28"+"51"+"31"+"47"} {}

Solution

230+250=160  157230+250=160  157 size 12{2 left ("30" right )+2 left ("50" right )="160" left ("157" right )} {}

Exercise 6

42+19+39+2342+19+39+23 size 12{"42"+"19"+"39"+"23"} {}

Exercise 7

88+62+59+9088+62+59+90 size 12{"88"+"62"+"59"+"90"} {}

Solution

290+260=300  299290+260=300  299 size 12{2 left ("90" right )+2 left ("60" right )="300" left ("299" right )} {}

Exercise 8

76+29+33+8276+29+33+82 size 12{"76"+"29"+"33"+"82"} {}

Exercise 9

19+23+87+2119+23+87+21 size 12{"19"+"23"+"87"+"21"} {}

Solution

320+90=150  150320+90=150  150 size 12{3 left ("20" right )+"90"="150" left ("150" right )} {}

Exercise 10

41+28+42+3741+28+42+37 size 12{"41"+"28"+"42"+"37"} {}

Exercise 11

89+32+89+9389+32+89+93 size 12{"89"+"32"+"89"+"93"} {}

Solution

390+30=300  303390+30=300  303 size 12{3 left ("90" right )+"30"="300" left ("303" right )} {}

Exercise 12

73+72+27+7173+72+27+71 size 12{"73"+"72"+"27"+"71"} {}

Exercise 13

43+62+61+5543+62+61+55 size 12{"43"+"62"+"61"+"55"} {}

Solution

40+360=220  22140+360=220  221 size 12{"40"+3 left ("60" right )="220" left ("221" right )} {}

Exercise 14

31+77+31+2731+77+31+27 size 12{"31"+"77"+"31"+"27"} {}

Exercise 15

57+34+28+61+6257+34+28+61+62 size 12{"57"+"34"+"28"+"61"+"62"} {}

Solution

360+230=240  242360+230=240  242 size 12{3 left ("60" right )+2 left ("30" right )="240" left ("242" right )} {}

Exercise 16

94+18+23+91+1994+18+23+91+19 size 12{"94"+"18"+"23"+"91"+"19"} {}

Exercise 17

103+72+66+97+99103+72+66+97+99 size 12{"103"+"72"+"66"+"97"+"99"} {}

Solution

3100+270=440  4373100+270=440  437 size 12{3 left ("100" right )+2 left ("70" right )="440" left ("437" right )} {}

Exercise 18

42+121+119+124+4142+121+119+124+41 size 12{"42"+"121"+"119"+"124"+"41"} {}

Exercise 19

19+24+87+23+91+9319+24+87+23+91+93 size 12{"19"+"24"+"87"+"23"+"91"+"93"} {}

Solution

320+390=330  337320+390=330  337 size 12{3 left ("20" right )+3 left ("90" right )="330" left ("337" right )} {}

Exercise 20

108+61+63+96+57+99108+61+63+96+57+99 size 12{"108"+"61"+"63"+"96"+"57"+"99"} {}

Exercise 21

518+721+493+689518+721+493+689 size 12{"518"+"721"+"493"+"689"} {}

Solution

2500+2700=2,400  2,4212500+2700=2,400  2,421 size 12{2 left ("500" right )+2 left ("700" right )=2,"400" left (2,"421" right )} {}

Exercise 22

981+1208+1214+1006981+1208+1214+1006 size 12{"981"+"1208"+"1214"+"1006"} {}

Exercise 23

23+81+77+79+19+8123+81+77+79+19+81 size 12{"23"+"81"+"77"+"79"+"19"+"81"} {}

Solution

220+480=360  360220+480=360  360 size 12{2 left ("20" right )+4 left ("80" right )="360" left ("360" right )} {}

Exercise 24

94+68+66+101+106+71+11094+68+66+101+106+71+110 size 12{"94"+"68"+"66"+"101"+"106"+"71"+"110"} {}

Exercises for Review

Exercise 25

((Reference)) Specify all the digits greater than 6.

Solution

7, 8, 9

Exercise 26

((Reference)) Find the product: 2391471223914712 size 12{ { {2} over {3} } cdot { {9} over {"14"} } cdot { {7} over {"12"} } } {}.

Exercise 27

((Reference)) Convert 0.06 to a fraction.

Solution

350350 size 12{ { {3} over {"50"} } } {}

Exercise 28

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

Exercise 29

((Reference)) Estimate the sum using the method of rounding: 4,882 + 2,7044,882 + 2,704 size 12{"4,882 "+" 2,704"} {}.

Solution

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

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