8.1: Estimation by Rounding

Learning Objectives

• understand the reason for estimation
• be able to estimate the result of an addition, multiplication, subtraction, or division using the rounding technique

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 [link] (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 \(80 \div 26\) 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.

Definition: 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

Estimate the sum: \(2,357 + 6,106\).

Solution

Notice that 2,357 is near \(\underbrace{2,400}_{\text{two nonzero digits}}\) and that 6,106 is near \(\underbrace{6,100}_{\text{two nonzero digits}}\)

The sum can be estimated by \(2,400 + 6,100 = 8,500\). (It is quick and easy to add 24 and 61.)

Thus, \(2,357 + 6,106\) is about 8,400. In fact, \(2,357 + 6,106 = 8,463\).

Practice Set A

Estimate the sum: \(4,216 + 3,942\).

Answer

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

Practice Set A

Estimate the sum: \(812 + 514\).

Answer

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

Practice Set A

Estimate the sum: \(43,892 + 92,106\).

Answer

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

Sample Set B

Estimate the difference: \(5,203 - 3,015\).

Solution

Notice that 5,203 is near \(\underbrace{5,200}_{\text{two nonzero digits}}\) and that 3,015 is near \(\underbrace{3,000}_{\text{one nonzero digit}}\)

The difference can be estimated by \(5,200 - 3,000 = 2,200\).

Thus, \(5,203 - 3,015\) is about 2,200. In fact, \(5,203 - 3,015 = 2,188\).

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 \(5,000 - 3,000 =2,000\). and conclude that \(5,203 - 3,015\) is about 2,000. This is why we say "answers may vary."

Practice Set B

Estimate the difference: \(628 - 413\).

Answer

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

Practice Set B

Estimate the difference: \(7,842 - 5,209\).

Answer

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

Practice Set B

Estimate the difference: \(73,812 - 28,492\).

Answer

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

Sample Set C

Estimate the product: \(73 \cdot 46\).

Solution

Notice that 73 is near \(\underbrace{70}_{\text{one nonzero digit}}\) and that 46 is near \(\underbrace{50}_{\text{one nonzero digit}}\).

The product can be estimated by \(70 \cdot 50 = 3,500\). (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, \(73 \cdot 46\) is about 3,500. In fact, \(73 \cdot 46 = 3,358\).

Sample Set C

Estimate the product: \(87 \cdot 4,316\).

Solution

Notice that 87 is near \(\underbrace{90}_{\text{one nonzero digit}}\) and that 4,316 is near \(\underbrace{4,000}_{\text{one nonzero digit}}\).

The product can be estimated by \(90 \cdot 4,000 = 360,000\).

Thus, \(87 \cdot 4,316\) is about 360,000. In fact, \(87 \cdot 4,316 = 375,492\).

Practice Set C

Estimate the product: \(31 \cdot 87\).

Answer

\(31 \cdot 87 : 30 \cdot 90\). About 2,700. In fact, 2,697.

Practice Set C

Estimate the product: \(18 \cdot 42\).

Answer

\(18 \cdot 42 : 20 \cdot 40\). About 800. In fact, 756.

Practice Set C

Estimate the product: \(16 \cdot 94\).

Answer

\(16 \cdot 94 : 15 \cdot 100\). About 1,500. In fact, 1,504.

Sample Set D

Estimate the quotient: \(153 \div 17\).

Solution

Notice that 153 is close to \(\underbrace{150}_{\text{two nonzero digits}}\) and that 17 is close to \(\underbrace{15}_{\text{two nonzero digits}}\).

The quotient can be estimated by \(150 \div 15 = 10\).

Thus, \(153 \div 17\) is about 10. In fact, \(153 \div 17 = 9\).

Sample Set D

Estimate the quotient: \(742,000 \div 2,400\).

Solution

Notice that 742,000 is close to \(\underbrace{700,000}_{\text{one nonzero digit}}\) and that 2,400 is close to \(\underbrace{2,000}_{\text{one nonzero digit}}\).

The quotient can be estimated by \(700,000 \div 2,000 = 350\).

Thus, \(742,000 \div 2,400\) is about 350. In fact, \(742,000 \div 2,400 = 309.1\overline{6}\).

Practice Set D

Estimate the quotient: \(221 \div 18\).

Answer

\(221 \div 18: 200 \div 20\). About 10. In fact, 12.27.

Practice Set D

Estimate the quotient: \(4,079 \div 381\).

Answer

\(4,079 \div 381: 4,000 \div 400\). About 10. In fact, 10.70603675...

Practice Set D

Estimate the quotient: \(609,000 \div 16,000\).

Answer

\(609,000 \div 16,000: 600,000 \div 15,000\). About 10. In fact, 38.0625.

Sample Set E

Estimate the sum: \(53.82 \div 41.6\).

Solution

Notice that 53.82 is close to \(\underbrace{54}_{\text{two nonzero digits}}\) and that 41.6 is close to \(\underbrace{42}_{\text{two nonzero digits}}\).

The quotient can be estimated by \(54 + 42 = 96\).

Thus, \(53.82 + 41.6\) is about 96. In fact, \(53.82 + 41.6 = 95.42\).

Practice Set E

Estimate the sum: \(61.02 + 26.8\).

Answer

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

Practice Set E

Estimate the sum: \(109.12 + 137.88\).

Answer

\(109.12 + 137.88 : 110 + 138\). 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.

Sample Set F

Estimate the product: (31.28)(14.2).

Solution

Notice that 31.28 is close to \(\underbrace{30}_{\text{one nonzero digit}}\) and that 14.2 is close to \(\underbrace{15}_{\text{two nonzero digits}}\).

The product can be estimated by \(30 \cdot 15 = 450\). (\(3 \cdot 15 = 45\), then affix one zero.)

Thus, (31.28)(14.2) is about 450. In fact, \(31.28)(14.2) = 444.176\).

Sample Set F

Estimate 21% of 5.42.

Solution

Notice that \(21\% = .21\) as a decimal, and that .21 is close to \(\underbrace{.2}_{\text{one nonzero digit}}\)

Notice also that 5.42 is close to \(\underbrace{5.}_{\text{one nonzero digit}}\).

Then, 21% of 5.42 can be estimated by \((.2)(5) = 1\).

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

Practice Set F

Estimate the product: (47.8)(21.1).

Answer

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

Practice Set F

Estimate 32% of 14.88.

Answer

32% of 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 \(\PageIndex{1}\)

\(1,402 + 2,198\)

Answer

about 3,600; in fact 3,600

Exercise \(\PageIndex{2}\)

\(3,481 + 4,216\)

Exercise \(\PageIndex{3}\)

\(921 + 796\)

Answer

about 1,700; in fact 1,717

Exercise \(\PageIndex{4}\)

\(611 + 806\)

Exercise \(\PageIndex{5}\)

\(4,681 + 9,325\)

Answer

about 14,000; in fact 14,006

Exercise \(\PageIndex{6}\)

\(6,476 + 7,814\)

Exercise \(\PageIndex{7}\)

\(7,805 - 4,266\)

Answer

about 3,500; in fact 3,539

Exercise \(\PageIndex{8}\)

\(8,427 - 5,342\)

Exercise \(\PageIndex{9}\)

\(14,106 - 8,412\)

Answer

about 5,700; in fact 5,694

Exercise \(\PageIndex{10}\)

\(26,486 - 18,931\)

Exercise \(\PageIndex{11}\)

\(32 \cdot 53\)

Answer

about 1,500; in fact 1,696

Exercise \(\PageIndex{12}\)

\(67 \cdot 42\)

Exercise \(\PageIndex{13}\)

\(628 \cdot 891\)

Answer

about 540,000; in fact 559,548

Exercise \(\PageIndex{14}\)

\(426 \cdot 741\)

Exercise \(\PageIndex{15}\)

\(18,012 \cdot 32,416\)

Answer

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

Exercise \(\PageIndex{16}\)

\(22,481 \cdot 51,076\)

Exercise \(\PageIndex{17}\)

\(287 \div 19\)

Answer

about 15; in fact 15.11

Exercise \(\PageIndex{18}\)

\(884 \div 33\)

Exercise \(\PageIndex{19}\)

\(1,254 \div 57\)

Answer

about 20; in fact 22

Exercise \(\PageIndex{20}\)

\(2,189 \div 42\)

Exercise \(\PageIndex{21}\)

\(8,092 \div 239\)

Answer

about 33; in fact 33.86

Exercise \(\PageIndex{22}\)

\(2,688 \div 48\)

Exercise \(\PageIndex{23}\)

\(72.14 + 21.08\)

Answer

about 93.2; in fact 93.22

Exercise \(\PageIndex{24}\)

\(43.016 + 47.58\)

Exercise \(\PageIndex{25}\)

\(96.53 - 26.91\)

Answer

about 70; in fact 69.62

Exercise \(\PageIndex{26}\)

\(115.0012 - 25.018\)

Exercise \(\PageIndex{27}\)

\(206.19 + 142.38\)

Answer

about 348.6; in fact 348.57

Exercise \(\PageIndex{28}\)

\(592.131 + 211.6\)

Exercise \(\PageIndex{29}\)

\((32.12)(48.7)\)

Answer

about 1,568.0; in fact 1,564.244

Exercise \(\PageIndex{30}\)

\((87.013)(21.07)\)

Exercise \(\PageIndex{31}\)

\((3.003)(16.52)\)

Answer

about 49.5; in fact 49.60956

Exercise \(\PageIndex{32}\)

\((6.032)(14.091)\)

Exercise \(\PageIndex{33}\)

\((114.06)(384.3)\)

Answer

about 43,776; in fact 43,833.258

Exercise \(\PageIndex{34}\)

\((5,137.118)(263.56)\)

Exercise \(\PageIndex{35}\)

\((6.92)(0.88)\)

Answer

about 6.21; in fact 6.0896

Exercise \(\PageIndex{36}\)

\((83.04)(1.03)\)

Exercise \(\PageIndex{37}\)

\((17.31)(.003)\)

Answer

about 0.0519; in fact 0.05193

Exercise \(\PageIndex{38}\)

\((14.016)(.016)\)

Exercise \(\PageIndex{39}\)

93% of 7.01

Answer

about 6.3; in fact 6.5193

Exercise \(\PageIndex{40}\)

107% of 12.6

Exercise \(\PageIndex{41}\)

32% of 15.3

Answer

about 4.5; in fact 4.896

Exercise \(\PageIndex{42}\)

74% of 21.93

Exercise \(\PageIndex{43}\)

18% of 4.118

Answer

about 0.8; in fact 0.74124

Exercise \(\PageIndex{44}\)

4% of .863

Exercise \(\PageIndex{45}\)

2% of .0039

Answer

about 0.00008; in fact 0.000078

Exercises for Review

Exercise \(\PageIndex{46}\)

Find the difference: \(\dfrac{7}{10} - \dfrac{5}{16}\).

Exercise \(\PageIndex{47}\)

Find the value \(\dfrac{6 - \dfrac{1}{4}}{6 + \dfrac{1}{4}}\).

Answer

\(\dfrac{23}{25}\)

Exercise \(\PageIndex{48}\)

Convert the complex decimal \(1.11\dfrac{1}{4}\) to a decimal.

Exercise \(\PageIndex{49}\)

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?

Answer

60 feet tall

Exercise \(\PageIndex{50}\)

11.62 is 83% of what number?