# 8.1 Estimation by rounding

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## Sample set c

Estimate the product: $\text{73}\cdot \text{46}$ .

Notice that 73 is near $\underset{\text{digit}}{\underset{\text{one nonzero}}{\underbrace{70,}}}$ and that 46 is near $\underset{\text{digit}}{\underset{\text{one nonzero}}{\underbrace{50.}}}$

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

Estimate the product: $\text{87}\cdot \text{4,316}$ .

Notice that 87 is close to $\underset{\text{digit}}{\underset{\text{one nonzero}}{\underbrace{90,}}}$ and that 4,316 is close to $\underset{\text{digit}}{\underset{\text{one nonzero}}{\underbrace{4,000.}}}$

The product can be estimated by $\text{90}\cdot \text{4,000}=\text{360,000}$ .

Thus, $\text{87}\cdot \text{4,316}$ is about 360,000. In fact, $\text{87}\cdot \text{4,316}=\text{375,492}$ .

## Practice set c

Estimate the product: $\text{31}\cdot \text{87}$ .

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

Estimate the product: $\text{18}\cdot \text{42}$ .

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

Estimate the product: $\text{16}\cdot \text{94}$ .

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

## Sample set d

Estimate the quotient: $\text{153}÷\text{17}$ .

Notice that 153 is close to $\underset{\text{digits}}{\underset{\text{two nonzero}}{\underbrace{150,}}}$ and that 17 is close to $\underset{\text{digits}}{\underset{\text{two nonzero}}{\underbrace{15.}}}$

The quotient can be estimated by $\text{150}÷\text{15}=\text{10}$ .

Thus, $\text{153}÷\text{17}$ is about 10. In fact, $\text{153}÷\text{17}=\text{9}$ .

Estimate the quotient: $\text{742,000}÷\text{2,400}$ .

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

The quotient can be estimated by $\text{700,000}÷\text{2,000}=\text{350}$ .

Thus, $\text{742,000}÷\text{2,400}$ is about 350. In fact, $\text{742,000}÷\text{2,400}=\text{309}\text{.}1\overline{6}$ .

## Practice set d

Estimate the quotient: $\text{221}÷\text{18}$ .

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

Estimate the quotient: $\text{4,079}÷\text{381}$ .

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

Estimate the quotient: $\text{609,000}÷\text{16,000}$ .

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

## Sample set e

Estimate the sum: $\text{53}\text{.}\text{82}+\text{41}\text{.}6$ .

Notice that 53.82 is close to $\underset{\text{digits}}{\underset{\text{two nonzero}}{\underbrace{54,}}}$ and that 41.6 is close to $\underset{\text{digits}}{\underset{\text{two nonzero}}{\underbrace{42.}}}$

The sum can be estimated by $\text{54}+\text{42}=\text{96}$ .

Thus, $\text{53}\text{.}\text{82}+\text{41}\text{.}6$ is about 96. In fact, $\text{53}\text{.}\text{82}+\text{41}\text{.}\text{6}=\text{95}\text{.}\text{42}$ .

## Practice set e

Estimate the sum: $\text{61}\text{.}\text{02}+\text{26}\text{.}8$ .

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

Estimate the sum: $\text{109}\text{.}\text{12}+\text{137}\text{.}\text{88}$ .

$109.12+137.88:110+138$ . 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.

## Sample set f

Estimate the product: $\left(\text{31}\text{.}\text{28}\right)\left(\text{14}\text{.}2\right)$ .

Notice that 31.28 is close to $\underset{\text{digit}}{\underset{\text{one nonzero}}{\underbrace{30,}}}$ and that 14.2 is close to $\underset{\text{digits}}{\underset{\text{two nonzero}}{\underbrace{15.}}}$

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

Thus, $\left(\text{31}\text{.}\text{28}\right)\left(\text{14}\text{.}2\right)$ is about 450. In fact, $\left(\text{31}\text{.}\text{28}\right)\left(\text{14}\text{.}2\right)=\text{444}\text{.}\text{176}$ .

Estimate 21% of 5.42.

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

Notice also that 5.42 is close to $\underset{\text{digit}}{\underset{\text{one nonzero}}{\underbrace{5.}}}$

Then, 21% of 5.42 can be estimated by $\left(\text{.}2\right)\left(5\right)=\text{1}$ .

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

## Practice set f

Estimate the product: $\left(\text{47}\text{.}8\right)\left(\text{21}\text{.}1\right)$ .

$\left(47.8\right)\left(21.1\right):\left(50\right)\left(20\right)$ . About 1,000. In fact, 1,008.58.

Estimate 32% of 14.88.

32% of $14.88:\left(.3\right)\left(15\right)$ . 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.

$1,\text{402}+2,\text{198}$

$\text{3,481}+\text{4,216}$

$\text{921}+\text{796}$

$\text{611}+\text{806}$

$4,\text{681}+9,\text{325}$

$6,\text{476}+7,\text{814}$

$7,\text{805}-4,\text{266}$

$8,\text{427}-5,\text{342}$

$\text{14},\text{106}-8,\text{412}$

$\text{26},\text{486}-\text{18},\text{931}$

$\text{32}\cdot \text{53}$

$\text{67}\cdot \text{42}$

$\text{628}\cdot \text{891}$

$\text{426}\cdot \text{741}$

$\text{18},\text{012}\cdot \text{32},\text{416}$

$\text{22},\text{481}\cdot \text{51},\text{076}$

$\text{287}÷\text{19}$

$\text{884}÷\text{33}$

$1,\text{254}÷\text{57}$

$2,\text{189}÷\text{42}$

$8,\text{092}÷\text{239}$

$2,\text{688}÷\text{48}$

$\text{72}\text{.}\text{14}+\text{21}\text{.}\text{08}$

$\text{43}\text{.}\text{016}+\text{47}\text{.}\text{58}$

$\text{96}\text{.}\text{53}-\text{26}\text{.}\text{91}$

$\text{115}\text{.}\text{0012}-\text{25}\text{.}\text{018}$

$\text{206}\text{.}\text{19}+\text{142}\text{.}\text{38}$

$\text{592}\text{.}\text{131}+\text{211}\text{.}6$

$\left(\text{32}\text{.}\text{12}\right)\left(\text{48}\text{.}7\right)$

$\left(\text{87}\text{.}\text{013}\right)\left(\text{21}\text{.}\text{07}\right)$

$\left(3\text{.}\text{003}\right)\left(\text{16}\text{.}\text{52}\right)$

$\left(6\text{.}\text{032}\right)\left(\text{14}\text{.}\text{091}\right)$

$\left(\text{114}\text{.}\text{06}\right)\left(\text{384}\text{.}3\right)$

$\left(5,\text{137}\text{.}\text{118}\right)\left(\text{263}\text{.}\text{56}\right)$

$\left(6\text{.}\text{92}\right)\left(0\text{.}\text{88}\right)$

$\left(\text{83}\text{.}\text{04}\right)\left(1\text{.}\text{03}\right)$

$\left(\text{17}\text{.}\text{31}\right)\left(\text{.}\text{003}\right)$

$\left(\text{14}\text{.}\text{016}\right)\left(\text{.}\text{016}\right)$

107% of 12.6

32% of 15.3

74% of 21.93

18% of 4.118

4% of .863

2% of .0039

## Exercises for review

( [link] ) Find the difference: $\frac{7}{\text{10}}-\frac{5}{\text{16}}\text{.}$

( [link] ) Find the value $\frac{6-\frac{1}{4}}{6+\frac{1}{4}}\text{.}$

$\frac{\text{23}}{\text{25}}$

( [link] ) Convert the complex decimal $1\text{.}\text{11}\frac{1}{4}$ to a decimal.

( [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

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

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