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Sample set c

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

Notice that 73 is near 70, one nonzero digit and that 46 is near 50. one nonzero digit

The product can be estimated by 70 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 [link] for a review of this technique.)

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

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

Notice that 87 is close to 90, one nonzero digit and that 4,316 is close to 4,000. one nonzero digit

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

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

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Practice set c

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

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

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

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

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

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

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Sample set d

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

Notice that 153 is close to 150, two nonzero digits and that 17 is close to 15. two nonzero digits

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

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

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

Notice that 742,000 is close to 700,000 one nonzero digit , and that 2,400 is close to 2,000. one nonzero digit

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

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

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Practice set d

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

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

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

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

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

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

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Sample set e

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

Notice that 53.82 is close to 54, two nonzero digits and that 41.6 is close to 42. two nonzero digits

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

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

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Practice set e

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

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

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

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

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Sample set f

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

Notice that 31.28 is close to 30, one nonzero digit and that 14.2 is close to 15. two nonzero digits

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

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

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Estimate 21% of 5.42.

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

Notice also that 5.42 is close to 5. one nonzero digit

Then, 21% of 5.42 can be estimated by ( . 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.

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Practice set f

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

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

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Estimate 32% of 14.88.

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

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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.

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

about 3,600; in fact 3,600

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

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

about 1,700; in fact 1,717

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

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

about 14,000; in fact 14,006

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

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

about 3,500; in fact 3,539

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

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

about 5,700; in fact 5,694

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

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32 53 size 12{"32" cdot "53"} {}

about 1,500; in fact 1,696

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67 42 size 12{"67" cdot "42"} {}

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

about 540,000; in fact 559,548

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

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

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

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

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287 ÷ 19 size 12{"287"÷"19"} {}

about 15; in fact 15.11

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884 ÷ 33 size 12{"884"÷"33"} {}

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1, 254 ÷ 57 size 12{1,"254"÷"57"} {}

about 20; in fact 22

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2, 189 ÷ 42 size 12{2,"189"÷"42"} {}

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

about 33; in fact 33.86

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2, 688 ÷ 48 size 12{2,"688"÷"48"} {}

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72 . 14 + 21 . 08 size 12{"72" "." "14"+"21" "." "08"} {}

about 93.2; in fact 93.22

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43 . 016 + 47 . 58 size 12{"43" "." "016"+"47" "." "58"} {}

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96 . 53 26 . 91 size 12{"96" "." "53"-"26" "." "91"} {}

about 70; in fact 69.62

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115 . 0012 25 . 018 size 12{"115" "." "0012"-"25" "." "018"} {}

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206 . 19 + 142 . 38 size 12{"206" "." "19"+"142" "." "38"} {}

about 348.6; in fact 348.57

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592 . 131 + 211 . 6 size 12{"592" "." "131"+"211" "." 6} {}

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

about 1,568.0; in fact 1,564.244

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

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

about 49.5; in fact 49.60956

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

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

about 43,776; in fact 43,833.258

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

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

about 6.21; in fact 6.0896

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

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

about 0.0519; in fact 0.05193

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

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

about 6.3; in fact 6.5193

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32% of 15.3

about 4.5; in fact 4.896

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18% of 4.118

about 0.8; in fact 0.74124

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2% of .0039

about 0.00008; in fact 0.000078

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Exercises for review

( [link] ) Find the difference: 7 10 5 16 . size 12{ { {7} over {"10"} } - { {5} over {"16"} } "." } {}

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( [link] ) Find the value 6 1 4 6 + 1 4 . size 12{ { {6- { {1} over {4} } } over {6+ { {1} over {4} } } } "." } {}

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

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( [link] ) Convert the complex decimal 1 . 11 1 4 size 12{1 "." "11" { {1} over {4} } } {} to a decimal.

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( [link] ) 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?

60 feet tall

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( [link] ) 11.62 is 83% of what number?

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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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