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Estimation by Rounding

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 rounding. By the end of the module students should understand the reason for estimation and be able to estimate the result of an addition, multiplication, subtraction, or division using the rounding technique.

Section Overview

  • Estimation By Rounding

When beginning a computation, it is valuable to have an idea of what value to expect for the result. When a computation is completed, it is valuable to know if the result is reasonable.

In the rounding process, it is important to note two facts:

  1. The rounding that is done in estimation does not always follow the rules of rounding discussed in (Reference) (Rounding Whole Numbers). Since estima­tion is concerned with the expected value of a computation, rounding is done using convenience as the guide rather than using hard-and-fast rounding rules. For example, if we wish to estimate the result of the division 80 ÷ 2680 ÷ 26 size 12{"80 " div " 26"} {}, we might round 26 to 20 rather than to 30 since 80 is more conveniently divided by 20 than by 30.
  2. Since rounding may occur out of convenience, and different people have differ­ent ideas of what may be convenient, results of an estimation done by rounding may vary. For a particular computation, different people may get different estimated results. Results may vary.

Estimation

Estimation is the process of determining an expected value of a computation.

Common words used in estimation are about, near, and between.

Estimation by Rounding

The rounding technique estimates the result of a computation by rounding the numbers involved in the computation to one or two nonzero digits.

Sample Set A

Example 1

Estimate the sum: 2,357 + 6,1062,357 + 6,106 size 12{"2,357 "+" 6,106"} {}.

Notice that 2,357 is near 2,400,︸two nonzerodigits2,400,︸two nonzerodigits and that 6,106 is near 6,100.︸two nonzerodigits6,100.︸two nonzerodigits

The sum can be estimated by 2,400 + 6,100 = 8,5002,400 + 6,100 = 8,500 size 12{"2,400 "+" 6,100 "=" 8,500"} {}. (It is quick and easy to add 24 and 61.)

Thus, 2,357 + 6,1062,357 + 6,106 size 12{"2,357 "+" 6,106"} {} is about 8,400. In fact, 2,357 + 6,106=8,4632,357 + 6,106=8,463 size 12{"2,357 "+" 6,106"="8,463"} {}.

Practice Set A

Exercise 1

Estimate the sum: 4,216 + 3,9424,216 + 3,942 size 12{"4,216 "+" 3,942"} {}.

Solution

4,216 + 3,942 : 4,200 + 3,900 4,216+3,942:4,200+3,900. About 8,100. In fact, 8,158.

Exercise 2

Estimate the sum: 812 + 514812 + 514 size 12{"812 "+" 514"} {}.

Solution

812 + 514 : 800 + 500 812+514:800+500. About 1,300. In fact, 1,326.

Exercise 3

Estimate the sum: 43,892 + 92,10643,892 + 92,106 size 12{"43,892 "+" 92,106"} {}.

Solution

43,892 + 92,106 : 44,000 + 92,000 43,892+92,106:44,000+92,000. About 136,000. In fact, 135,998.

Sample Set B

Example 2

Estimate the difference: 5,203-3,0155,203-3,015 size 12{"5,203 - 3,015"} {}.

Notice that 5,203 is near 5,200,︸two nonzerodigits5,200,︸two nonzerodigits and that 3,015 is near 3,000.︸one nonzerodigit3,000.︸one nonzerodigit

The difference can be estimated by 5,200 - 3,000= 2,2005,200 - 3,000= 2,200 size 12{"5,200 - 3,000 "=" 2,200"} {}.

Thus, 5,203 - 3,0155,203 - 3,015 size 12{"5,203 - 3,015"} {} is about 2,200. In fact, 5,203 - 3,015 = 2,1885,203 - 3,015 = 2,188 size 12{"5,203 - 3,015 "=" 2,188"} {}.

We could make a less accurate estimation by observing that 5,203 is near 5,000. The number 5,000 has only one nonzero digit rather than two (as does 5,200). This fact makes the estimation quicker (but a little less accurate). We then estimate the difference by 5,000 - 3,000= 2,0005,000 - 3,000= 2,000 size 12{"5,000 - 3,000 "=" 2,000"} {}, and conclude that 5,203 - 3,0155,203 - 3,015 size 12{"5,203 - 3,015"} {} is about 2,000. This is why we say "answers may vary."

Practice Set B

Exercise 4

Estimate the difference: 628 - 413628 - 413 size 12{"628 - 413"} {}.

Solution

628 - 413 : 600 - 400 628-413:600-400. About 200. In fact, 215.

Exercise 5

Estimate the difference: 7,842 - 5,2097,842 - 5,209 size 12{"7,842 - 5,209"} {}.

Solution

7,842 - 5,209 : 7,800 - 5,200 7,842-5,209:7,800-5,200. About 2,600. In fact, 2,633.

Exercise 6

Estimate the difference: 73,812 - 28,49273,812 - 28,492 size 12{"73,812 - 28,492"} {}.

Solution

73,812 - 28,492 : 74,000 - 28,000 73,812-28,492:74,000-28,000. About 46,000. In fact, 45,320.

Sample Set C

Example 3

Estimate the product: 73 ⋅ 4673 ⋅ 46 size 12{"73 " cdot " 46"} {}.

Notice that 73 is near 70,︸one nonzerodigit70,︸one nonzerodigit and that 46 is near 50.︸one nonzerodigit50.︸one nonzerodigit

The product can be estimated by 70 ⋅ 50 = 3,50070 ⋅ 50 = 3,500 size 12{"70 " cdot " 50 "=" 3,500"} {}. (Recall that to multiply numbers ending in zeros, we multiply the nonzero digits and affix to this product the total number of ending zeros in the factors. See (Reference) for a review of this technique.)

Thus, 73 ⋅ 4673 ⋅ 46 size 12{"73 " cdot " 46"} {} is about 3,500. In fact, 73 ⋅ 46 = 3,35873 ⋅ 46 = 3,358 size 12{"73 " cdot " 46 "=" 3,358"} {}.

Example 4

Estimate the product: 87 ⋅ 4,31687 ⋅ 4,316 size 12{"87 " cdot " 4,316"} {}.

Notice that 87 is close to 90,︸one nonzerodigit90,︸one nonzerodigit and that 4,316 is close to 4,000.︸one nonzerodigit4,000.︸one nonzerodigit

The product can be estimated by 90 ⋅ 4,000 = 360,00090 ⋅ 4,000 = 360,000 size 12{"90 " cdot " 4,000 "=" 360,000"} {}.

Thus, 87 ⋅ 4,31687 ⋅ 4,316 size 12{"87 " cdot " 4,316"} {} is about 360,000. In fact, 87 ⋅ 4,316 = 375,49287 ⋅ 4,316 = 375,492 size 12{"87 " cdot " 4,316 "=" 375,492"} {}.

Practice Set C

Exercise 7

Estimate the product: 31 ⋅ 8731 ⋅ 87 size 12{"31 " cdot " 87"} {}.

Solution

31 ⋅ 87 : 30 ⋅ 90 31⋅87:30⋅90. About 2,700. In fact, 2,697.

Exercise 8

Estimate the product: 18 ⋅ 4218 ⋅ 42 size 12{"18 " cdot " 42"} {}.

Solution

18 ⋅ 42 : 20 ⋅ 40 18⋅42:20⋅40. About 800. In fact, 756.

Exercise 9

Estimate the product: 16 ⋅ 9416 ⋅ 94 size 12{"16 " cdot " 94"} {}.

Solution

16 ⋅ 94 : 15 ⋅ 100 16⋅94:15⋅100. About 1,500. In fact, 1,504.

Sample Set D

Example 5

Estimate the quotient: 153 ÷ 17153 ÷ 17 size 12{"153 " div " 17"} {}.

Notice that 153 is close to 150,︸two nonzerodigits150,︸two nonzerodigits and that 17 is close to 15.︸two nonzerodigits15.︸two nonzerodigits

The quotient can be estimated by 150 ÷ 15 = 10150 ÷ 15 = 10 size 12{"150 " div " 15 "=" 10"} {}.

Thus, 153 ÷ 17153 ÷ 17 size 12{"153 " div " 17"} {} is about 10. In fact, 153 ÷ 17 = 9153 ÷ 17 = 9 size 12{"153 " div " 17 "=" 9"} {}.

Example 6

Estimate the quotient: 742,000 ÷ 2,400742,000 ÷ 2,400 size 12{"742,000 " div " 2,400"} {}.

Notice that 742,000 is close to 700,000︸one nonzerodigit700,000︸one nonzerodigit, and that 2,400 is close to 2,000.︸one nonzerodigit2,000.︸one nonzerodigit

The quotient can be estimated by 700,000 ÷ 2,000 = 350700,000 ÷ 2,000 = 350 size 12{"700,000 " div " 2,000 "=" 350"} {}.

Thus, 742,000 ÷ 2,400742,000 ÷ 2,400 size 12{"742,000 " div " 2,400"} {} is about 350. In fact, 742,000 ÷ 2,400 = 309.16¯742,000 ÷ 2,400 = 309.16¯ size 12{"742,000 " div " 2,400 "=" 309" "." 1 {overline {6}} } {}.

Practice Set D

Exercise 10

Estimate the quotient: 221 ÷ 18221 ÷ 18 size 12{"221 " div " 18"} {}.

Solution

221 ÷ 18 : 200 ÷ 20 221÷18:200÷20. About 10. In fact, 12.27.

Exercise 11

Estimate the quotient: 4,079 ÷ 3814,079 ÷ 381 size 12{"4,079 " div " 381"} {}.

Solution

4,079 ÷ 381 : 4,000 ÷ 400 4,079÷381:4,000÷400. About 10. In fact, 10.70603675...

Exercise 12

Estimate the quotient: 609,000 ÷ 16,000609,000 ÷ 16,000 size 12{"609,000 " div " 16,000"} {}.

Solution

609,000 ÷ 16,000 : 600,000 ÷ 15,000 609,000÷16,000:600,000÷15,000. About 40. In fact, 38.0625.

Sample Set E

Example 7

Estimate the sum: 53.82 + 41.653.82 + 41.6 size 12{"53" "." "82 "+" 41" "." 6} {}.

Notice that 53.82 is close to 54,︸two nonzerodigits54,︸two nonzerodigits and that 41.6 is close to 42.︸two nonzerodigits42.︸two nonzerodigits

The sum can be estimated by 54 + 42 = 9654 + 42 = 96 size 12{"54 "+" 42 "=" 96"} {}.

Thus, 53.82 + 41.653.82 + 41.6 size 12{"53" "." "82 "+" 41" "." 6} {} is about 96. In fact, 53.82 + 41.6 = 95.4253.82 + 41.6 = 95.42 size 12{"53" "." "82 "+" 41" "." "6 "=" 95" "." "42"} {}.

Practice Set E

Exercise 13

Estimate the sum: 61.02 + 26.861.02 + 26.8 size 12{"61" "." "02 "+" 26" "." 8} {}.

Solution

61.02 + 26.8 : 61 + 27 61.02+26.8:61+27. About 88. In fact, 87.82.

Exercise 14

Estimate the sum: 109.12 + 137.88109.12 + 137.88 size 12{"109" "." "12 "+" 137" "." "88"} {}.

Solution

109.12 + 137.88 : 110 + 138 109.12+137.88:110+138. About 248. In fact, 247. We could have estimated 137.88 with 140. Then 110 + 140 110+140 is an easy mental addition. We would conclude then that 109.12 + 137.88 109.12+137.88 is about 250.

Sample Set F

Example 8

Estimate the product: (31.28)(14.2)(31.28)(14.2) size 12{ \( "31" "." "28" \) \( "14" "." 2 \) } {}.

Notice that 31.28 is close to 30,︸one nonzerodigit30,︸one nonzerodigit and that 14.2 is close to 15.︸two nonzerodigits15.︸two nonzerodigits

The product can be estimated by 30 ⋅ 15 = 45030 ⋅ 15 = 450 size 12{"30 " cdot " 15 "=" 450"} {}. ( 3 ⋅ 15 = 453 ⋅ 15 = 45 size 12{"3 " cdot " 15 "=" 45"} {}, then affix one zero.)

Thus, (31.28)(14.2)(31.28)(14.2) size 12{ \( "31" "." "28" \) \( "14" "." 2 \) } {} is about 450. In fact, (31.28)(14.2)= 444.176(31.28)(14.2)= 444.176 size 12{ \( "31" "." "28" \) \( "14" "." 2 \) =" 444" "." "176"} {}.

Example 9

Estimate 21% of 5.42.

Notice that 21% =.2121% =.21 size 12{"21% "= "." "21"} {} as a decimal, and that .21 is close to .2.︸one nonzerodigit.2.︸one nonzerodigit

Notice also that 5.42 is close to 5.︸one nonzerodigit5.︸one nonzerodigit

Then, 21% of 5.42 can be estimated by (.2)(5)= 1(.2)(5)= 1 size 12{ \( "." 2 \) \( 5 \) =" 1"} {}.

Thus, 21% of 5.42 is about 1. In fact, 21% of 5.42 is 1.1382.

Practice Set F

Exercise 15

Estimate the product: (47.8)(21.1)(47.8)(21.1) size 12{ \( "47" "." 8 \) \( "21" "." 1 \) } {}.

Solution

( 47.8 ) ( 21.1 ) : ( 50 ) ( 20 ) (47.8)(21.1):(50)(20). About 1,000. In fact, 1,008.58.

Exercise 16

Estimate 32% of 14.88.

Solution

32% of 14.88 : ( .3 ) ( 15 )14.88:(.3)(15). About 4.5. In fact, 4.7616.

Exercises

Estimate each calculation using the method of rounding. After you have made an estimate, find the exact value and compare this to the estimated result to see if your estimated value is reasonable. Results may vary.

Exercise 17

1,402+2,1981,402+2,198 size 12{1,"402"+2,"198"} {}

Solution

about 3,600; in fact 3,600

Exercise 18

3,481+4,2163,481+4,216 size 12{"3,481"+"4,216"} {}

Exercise 19

921+796921+796 size 12{"921"+"796"} {}

Solution

about 1,700; in fact 1,717

Exercise 20

611+806611+806 size 12{"611"+"806"} {}

Exercise 21

4,681+9,3254,681+9,325 size 12{4,"681"+9,"325"} {}

Solution

about 14,000; in fact 14,006

Exercise 22

6,476+7,8146,476+7,814 size 12{6,"476"+7,"814"} {}

Exercise 23

7,805−4,2667,805−4,266 size 12{7,"805"-4,"266"} {}

Solution

about 3,500; in fact 3,539

Exercise 24

8,427−5,3428,427−5,342 size 12{8,"427"-5,"342"} {}

Exercise 25

14,106−8,41214,106−8,412 size 12{"14","106"-8,"412"} {}

Solution

about 5,700; in fact 5,694

Exercise 26

26,486−18,93126,486−18,931 size 12{"26","486"-"18","931"} {}

Exercise 27

32⋅5332⋅53 size 12{"32" cdot "53"} {}

Solution

about 1,500; in fact 1,696

Exercise 28

67⋅4267⋅42 size 12{"67" cdot "42"} {}

Exercise 29

628⋅891628⋅891 size 12{"628" cdot "891"} {}

Solution

about 540,000; in fact 559,548

Exercise 30

426⋅741426⋅741 size 12{"426" cdot "741"} {}

Exercise 31

18,012⋅32,41618,012⋅32,416 size 12{"18","012" cdot "32","416"} {}

Solution

about 583,200,000; in fact 583,876,992

Exercise 32

22,481⋅51,07622,481⋅51,076 size 12{"22","481" cdot "51","076"} {}

Exercise 33

287÷19287÷19 size 12{"287"÷"19"} {}

Solution

about 15; in fact 15.11

Exercise 34

884÷33884÷33 size 12{"884"÷"33"} {}

Exercise 35

1,254÷571,254÷57 size 12{1,"254"÷"57"} {}

Solution

about 20; in fact 22

Exercise 36

2,189÷422,189÷42 size 12{2,"189"÷"42"} {}

Exercise 37

8,092÷2398,092÷239 size 12{8,"092"÷"239"} {}

Solution

about 33; in fact 33.86

Exercise 38

2,688÷482,688÷48 size 12{2,"688"÷"48"} {}

Exercise 39

72.14+21.0872.14+21.08 size 12{"72" "." "14"+"21" "." "08"} {}

Solution

about 93.2; in fact 93.22

Exercise 40

43.016+47.5843.016+47.58 size 12{"43" "." "016"+"47" "." "58"} {}

Exercise 41

96.53−26.9196.53−26.91 size 12{"96" "." "53"-"26" "." "91"} {}

Solution

about 70; in fact 69.62

Exercise 42

115.0012−25.018115.0012−25.018 size 12{"115" "." "0012"-"25" "." "018"} {}

Exercise 43

206.19+142.38206.19+142.38 size 12{"206" "." "19"+"142" "." "38"} {}

Solution

about 348.6; in fact 348.57

Exercise 44

592.131+211.6592.131+211.6 size 12{"592" "." "131"+"211" "." 6} {}

Exercise 45

32.1248.732.1248.7 size 12{ left ("32" "." "12" right ) left ("48" "." 7 right )} {}

Solution

about 1,568.0; in fact 1,564.244

Exercise 46

87.01321.0787.01321.07 size 12{ left ("87" "." "013" right ) left ("21" "." "07" right )} {}

Exercise 47

3.00316.523.00316.52 size 12{ left (3 "." "003" right ) left ("16" "." "52" right )} {}

Solution

about 49.5; in fact 49.60956

Exercise 48

6.03214.0916.03214.091 size 12{ left (6 "." "032" right ) left ("14" "." "091" right )} {}

Exercise 49

114.06384.3114.06384.3 size 12{ left ("114" "." "06" right ) left ("384" "." 3 right )} {}

Solution

about 43,776; in fact 43,833.258

Exercise 50

5,137.118263.565,137.118263.56 size 12{ left (5,"137" "." "118" right ) left ("263" "." "56" right )} {}

Exercise 51

6.920.886.920.88 size 12{ left (6 "." "92" right ) left (0 "." "88" right )} {}

Solution

about 6.21; in fact 6.0896

Exercise 52

83.041.0383.041.03 size 12{ left ("83" "." "04" right ) left (1 "." "03" right )} {}

Exercise 53

17.31.00317.31.003 size 12{ left ("17" "." "31" right ) left ( "." "003" right )} {}

Solution

about 0.0519; in fact 0.05193

Exercise 54

14.016.01614.016.016 size 12{ left ("14" "." "016" right ) left ( "." "016" right )} {}

Exercise 55

93% of  7.0193% of  7.01 size 12{"93""% of "7 "." "01"} {}

Solution

about 6.3; in fact 6.5193

Exercise 56

107% of 12.6

Exercise 57

32% of 15.3

Solution

about 4.5; in fact 4.896

Exercise 58

74% of 21.93

Exercise 59

18% of 4.118

Solution

about 0.8; in fact 0.74124

Exercise 60

4% of .863

Exercise 61

2% of .0039

Solution

about 0.00008; in fact 0.000078

Exercises for Review

Exercise 62

((Reference)) Find the difference: 710−516.710−516. size 12{ { {7} over {"10"} } - { {5} over {"16"} } "." } {}

Exercise 63

((Reference)) Find the value 6−146+14.6−146+14. size 12{ { {6- { {1} over {4} } } over {6+ { {1} over {4} } } } "." } {}

Solution

23252325 size 12{ { {"23"} over {"25"} } } {}

Exercise 64

((Reference)) Convert the complex decimal 1.11141.1114 size 12{1 "." "11" { {1} over {4} } } {} to a decimal.

Exercise 65

((Reference)) A woman 5 foot tall casts an 8-foot shadow at a particular time of the day. How tall is a tree that casts a 96-foot shadow at the same time of the day?

Solution

60 feet tall

Exercise 66

((Reference)) 11.62 is 83% of what number?

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