rounding and estimating

Rounding and EstimatingRounding and EstimatingPRE-ALGEBRA LESSON 3-1PRE-ALGEBRA LESSON 3-1

Round each number to the underlined place value.

a. 89.63 b. 579,122 c. 0.83476

90 600,000 0.8348

(For help, go to Skills Handbook page 727.)

Use the number 27.3865. Write the value of the given digit.

1. 2 2. 3

3. 8 4. 6

Check Skills You’ll Need

Solutions

1. 27.3865; 2 tens 2. 27.3865; 3 tenths

3. 27.3865; 8 hundredths 4. 27.3865; 6 thousandths

a. Round 8.7398 to the nearest tenth.

8.7398

tenths place

less than 5

Round down to 7.

(continued)

b. Round 8.7398 to the nearest integer.

8.7398

nearest integer is ones place

5 or greater

Round up to 9.

Quick Check

Estimate to find whether each answer is reasonable.

a. Calculation

+$ 59.98

$ 83.21

$115.67

$258.86+$ 60

Estimate

The answer is close to the estimate. It is reasonable.

(continued)

b. Calculation

–$ 39.34

$176.48

$147.14

Estimate

–$ 40

The answer is not close to the estimate. It is not reasonable.

Quick Check

Add thefront-end digits.

You are buying some fruit. The bananas cost $1.32,

the apples cost $2.19, and the avocados cost $1.63. Use

front-end estimation to estimate the total cost of the fruit.

The total cost is about $5.10.

Estimate byrounding.

Quick Check

Estimate the total electricity charge: March: $81.75; April: $79.56; May: $80.89.

The total electricity charge is about $240.00.

The values cluster around $80. 80

3 months

Quick Check

Round to the underlined place.

1. 6.557 2. 3.0448

Estimate.

3. $4.95 + $.89 4. 4.589 + 5.098 + 5.179

about $6.00

6.6 3.04

about 15

Estimating Decimal Products and QuotientsEstimating Decimal Products and QuotientsPRE-ALGEBRA LESSON 3-2PRE-ALGEBRA LESSON 3-2

Round to the nearest whole number and to the nearest tenth.

a. 4.07 b. 0.708 c. 364.825

4, 4.1 1, 0.7 365, 364.8

(For help, go to Lesson 3-1.)

Round to the nearest one.

1. 145.89 2. 199.27

3. 101.06 4. 28.45

Solutions

nearest integer nearest integer 1. 145.89 2. 199.27

5 or greater; round up to 6 less than 5; do not change

146 199

nearest integer nearest integer3. 101.06 4. 28.45

less than 5; do not change less than 5; do not change

101 28

Estimate 6.43 • 4.7.

Multiply.6 • 5 = 30

6.43 6 4.7 5 Round to the nearest integer.

6.43 • 4.7 30

Quick Check

Joshua bought 3 yd of fabric to make a flag. The fabric

cost $5.35/yd. The clerk said his total was $14.95 before tax.

Did the clerk make a mistake? Explain.

Multiply 5 times 3, the number ofyards of fabric.

5 • 3 = 15

5.35 5 Round to the nearest dollar.

The sales clerk made a mistake. Since 5.35 > 5, the actual cost should be more than the estimate. The clerk should have charged Joshua more than $15.00 before tax.

Quick Check

The cost to ship one yearbook is $3.12. The total cost

for a shipment was $62.40. Estimate how many books were in

the shipment.

3.12 3 Round the divisor.

The shipment is made up of about 20 books.

62.40 60 Round the dividend to a multiple of 3 that is close to 62.40.

60 ÷ 3 = 20 Divide.

Quick Check

Is 3.29 a reasonable quotient for 31.423 ÷ 5.94?

Since 3.29 is not close to 5, it is not reasonable.

5.94 6 Round the divisor.

31.423 30 Round the dividend to a multiple of 6 that is close to 31.423.

30 ÷ 6 = 5 Divide.

Quick Check

Estimate each product or quotient.

1. $4.78 ÷ 0.891 2. 983.24 • 2.41

3. –5.89 ÷ (–2.7) 4. 24.69 • 0.7

5. 20.498 • 4.908

about 100

about $5 about 2,000

about 2 about 25

Mean, Median, and ModeMean, Median, and ModePRE-ALGEBRA LESSON 3-3PRE-ALGEBRA LESSON 3-3

Order from greatest to least: 0.677, 0.855, 0.760, 0.078, 0.541

0.855, 0.760, 0.677, 0.541, 0.078

(For help, go Skills Handbook page 723.)

Write the numbers from least to greatest.

1. 8, 6, 4, 9, 3, 5, 6 2. 72, 68, 69, 71, 72

3. 112, 101, 98, 120, 101 4. 3.74, 3, 3.7, 3.3, 37

Solutions

1. 3, 4, 5, 6, 6, 8, 9 2. 68, 69, 71, 72, 72

3. 98, 101, 101, 112, 120 4. 3, 3.3, 3.7, 3.74, 37

Six elementary students are participating in a one-week

Readathon to raise money for a good cause. Use the graph. Find

the (a) mean, (b) median, and (c) mode of the data if you leave

out Latana’s pages.

a. Mean:

40 + 45 + 48 + 50 + 505=

The mean is 46.6.

sum of data valuesnumber of data values

(continued)

b. Median: 40 45 48 50 50 Write the data in order.

The median is the middle number, or 48.

c. Mode: Find the data value that occurs most often.

The mode is 50.

Quick Check

a. $1.10 $1.25 $2.00 $2.10 $2.20 $3.50

No values are the same, so there is no mode.

b. 1 3 4 6 7 7 8 9 10 12 12 13

How many modes, if any, does each have? Name them.

c. tomato, tomato, grape, orange, cherry, cherry, melon, cherry, grape

There is one mode.

Both 7 and 12 appear more than the other data values.

Cherry appears most often.

Since they appear the same number of times, there are two modes.

Quick Check

a. Which data value is an outlier?

Use the data: 7%, 4%, 10%, 33%, 11%, 12%.

The data value 33% is an outlier.

b. How does the outlier affect the mean?

The outlier raises the mean by about 5.5 points.

12.8 – 7.3 = 5.5

It is an outlier because it is 21% away from the closest data value.

Find the mean with the outlier.776 12.8

Find the mean without the outlier.446

Quick Check

a. the monthly amount of rain for a year

Which measure of central tendency best describes

each situation? Explain.

since the average monthly amount of rain for a year is not likely to have an outlier, mean is the appropriate measure.

b. most popular color of shirt

When the data have no outliers, use the mean.

When determining the most frequently chosen item, or when the data are not numerical, use the mode.

since the data are not numerical, the mode is the appropriate measure.

c. times school buses arrive at school

(continued)

since one bus may have to travel much farther than other buses, the median is the appropriate measure.

Median;

When an outlier may significantly influence the mean, use the median.

Quick Check

Which measure of central tendency best describes each situation?

1. numbers of legs on the animals in a zoo

2. favorite digits (from 0 to 9) of the students in a class

3. numbers of days-per-student that students are absent from school

4. test scores

median

Using FormulasUsing FormulasPRE-ALGEBRA LESSON 3-4PRE-ALGEBRA LESSON 3-4

Evaluate each expression for r = 4, s = 6, t = 2.

a. 4r b. 18 ÷ s + t c. (r + s) • 2

16 5 20

Evaluate each expression for x = 3 and y = 4.

1. 2x + 2y 2. 2x + y

3. 2(x + y) 4.x + y

Solutions

1. 2x + 2y = 2(3) + 2(4) 2. 2x + y = 2(3) + (4)

= 6 + 8 = 6 + 4

= 14 = 10

3. 2(x + y) = 2(3 + 4) 4. =

= 2(7)

Suppose you ride your bike 18 miles in 3 hours.

Use the formula d = r t to find your average speed.

d = r t Write the formula.

18 = (r )(3) Substitute 18 for d and 3 for t.

Divide each side by 3.183 =

Simplify.6 = r

Your average speed is 6 mi/h.

Quick Check

Use the formula F = + 37, where n is the number

of chirps a cricket makes in one minute, and F is the

temperature in degrees Fahrenheit. Estimate the temperature

when a cricket chirps 76 times in a minute.

The temperature is about 56°F.

F = + 37n4 Write the formula.

F = + 37764 Replace n with 76.

F = 19 + 37 Divide.

F = 56 Add.

Quick Check

Find the perimeter of a rectangular tabletop with a

length of 14.5 in. and width of 8.5 in. Use the formula for the

perimeter of a rectangle, P = 2 + 2w.

The perimeter of the tabletop is 46 in.

P = 2 + 2w Write the formula.

P = 46 Add.

P = 2(14.5) + 2(8.5) Replace with 14.5 and w with 8.5.

P = 29 + 17 Multiply.

Quick Check

Use the formula P = 4s to find the perimeter of a square with side s.

1. s = 5.6 m 2. s = 9.3 in.

3. Find the side of a square with a perimeter of 164 yd.

22.4 m 37.2 in.

Solving Equations by Adding or Subtracting DecimalsSolving Equations by Adding or Subtracting DecimalsPRE-ALGEBRA LESSON 3-5PRE-ALGEBRA LESSON 3-5

Separate the digits 1 through 9 into three groups that have the same sum.

1, 5, 9; 2, 6, 7; 3, 4, 8

Simplify.

1. 2.8 + 7.06 2. 0.65 + 1.8

3. 4.52 – 2.48 4. 3.7 – 0.62

Solutions

1. 2.80 2. 0.65 + 7.06 + 1.80

9.86 2.45

3. 4.52 4. 3.70– 2.48 – 0.62

2.04 3.08

Solve 6.8 + p = –9.7.

6.8 + p = –9.7

6.8 – 6.8 + p = –9.7 – 6.8 Subtract 6.8 from each side.

p = –16.5 Simplify.

Check 6.8 + p = –9.7

6.8 + (–16.5) –9.7 Replace p with –16.5.

–9.7 = –9.7

Quick Check

Ping has a board that is 14.5 ft long. She saws off a

piece that is 8.75 ft long. Use the diagram below to find the

length of the piece that is left.

The length of the piece that is left is 5.75 ft.

x = 5.75 Simplify.

x + 8.75 – 8.75 = 14.5 – 8.75 Subtract 8.75 from each side.

x + 8.75 = 14.5

Quick Check

Solve –23.34 = q – 16.99.

–23.34 = q – 16.99

–23.34 + 16.99 = q – 16.99 + 16.99 Add 16.99 to each side.

–6.35 = q Simplify.

Quick Check

Alejandro wrote a check for $49.98. His new account

balance is $169.45. What was his previous balance?

Alejandro had $219.43 in his account before he wrote the check.

Simplify.p = 219.43

Add 49.98 to each side.p – 49.98 + 49.98 = 169.45 + 49.98

p – 49.98 = 169.45

Equation p – $49.98 = $169.45

Let p = previous balance.

Words previous balance minus check is new balance

Quick Check

Solve each equation.

1. a + 10 = 7.9 2. –1.01 = c – 9

3. s – (–2.6) = 1.6 4. 3.02 + d = 2.91

5. –23.7 = 13.3 + g

– 37

– 2.1 7.99

– 1 – 0.11

Solving Equations by Multiplying or Dividing DecimalsSolving Equations by Multiplying or Dividing DecimalsPRE-ALGEBRA LESSON 3-6PRE-ALGEBRA LESSON 3-6

Lucinda buys 12 gal of gasoline at $1.95 per gallon. How much does the gas cost?

$23.40

Find each product.

1. 2.6(4.5) 2. 3.2(0.15)

3. 11.03(0.6) 4. 8.003(0.6)

Solutions

1. 2.6 2. 0.15

4.5 3.2 130 30

+ 1040 + 45011.70 = 11.7 0.480 = 0.48

3. 11.03 4. 8.003

0.6 0.66.618 4.8018

Solve –6.4 = 0.8b.

–6.4 = 0.8b

–6.40.8

0.8b0.8= Divide each side by 0.8.

–8 = b Simplify.

Check –6.4 = 0.8b

–6.4 0.8(–8) Replace b with –8.

–6.4 = –6.4

Quick Check

Every day the school cafeteria uses about 85.8

gallons of milk. About how many days will it take for the

cafeteria to use the 250 gallons in the refrigerator?

Let x = number of days.

numberof days

Words times is 250 gallonsdaily milk

consumption

Equation • =85.8 x 250

(continued)

85.8x = 250

The school will take about 3 days to use 250 gallons of milk.

x = 2.914 . . . Simplify.

x 3 Round to the nearest whole number.

85.8x85.8 =

25085.8 Divide each side by 85.8.

Quick Check

Solve –37.5 = .

c–1.2

–37.5 =c

–1.2

Multiply each side by –1.2.–37.5(–1.2) = (–1.2)c

–1.2

45 = c Simplify.

Quick Check

A little league player was at bat 15 times and had

a batting average of 0.133 (rounded to the nearest thousandth). The

batting average formula is a = , where a is the batting average, h is

the number of hits, and n is the number of times at bat. Use the

formula to find the number of hits she made.

0.133 =h

15 Replace a with 0.133 and n with 15.

(continued)

0.133(15) = (15)h

15 Multiply each side by 15.

Simplify.1.995 = h

2 h Since h (hits) represents an integer, round to the nearest integer.

The little league player made 2 hits.

Quick Check

Solve each equation.

1. 9b = –30.6 2. –10.8 =

3. 2.45 = –0.7k 4. = 240

5. y ÷ (–0.3) = 146.7

– 44.01

–3.4 27

p–2.5

–3.5

Using the Metric SystemUsing the Metric SystemPRE-ALGEBRA LESSON 3-7PRE-ALGEBRA LESSON 3-7

Use mental math to find each product or quotient.

a. 8.75 × 100

b. 9.6 × 100

c. 4.6 ÷ 10

d. 0.94 × 100

e. 4.06 ÷ 100

0.0406

Find each product or quotient.

1. 5 100 2. 14.06 ÷ 1,000

3. 0.294 10 4. 0.9 ÷ 100

Solutions

1. 5 • 100 2. 14.06 ÷ 1000Move the decimal point Move the decimal point2 places to the right. 500. 3 places to the left. 0.01406

5 • 100 = 500 14.06 ÷ 1000 = 0.01406

3. 0.294 • 10 4. 0.9 ÷ 100Move the decimal point Move the decimal point 1 place to the right. 02.94 2 places to the left. 0.009

0.294 • 10 = 2.94 0.9 ÷ 100 = 0.009

Choose an appropriate metric unit. Explain your choice.

a. the width of a textbook

b. the mass of a pair of glasses

c. the capacity of a thimble

Milliliter; a thimble will hold only a small amount of water.

Gram; glasses have about the same mass as many paperclips, but less than this textbook.

Centimeter; the width of a textbook is about two hands, or ten thumb widths, wide.

Quick Check

Choose a reasonable estimate. Explain your choice.

a. capacity of a drinking glass: 500 L or 500 mL

b. length of a hair clip: 5 m or 5 cm

c. mass of a pair of hiking boots: 1 kg or 1 g

1 kg; the mass is about one half the mass of your math book.

5 cm; the length of a hair clip would be about 5 widths of a thumbnail.

500 mL; a drinking glass holds less than a quart of milk.

Quick Check

Complete each statement.

a. 7,603 mL = L

7,603 ÷ 1,000 = 7.603 To convert from milliliters to liters, divide by 1,000.

7,603 mL = 7.603 L

4.57 m = 457 cm

To convert meters to centimeters, multiply by 100.

4.57 100 = 457 cm

b. 4.57 m = cm

Quick Check

A blue whale caught in 1931 was about 2,900 cm

long. What was its length in meters?

Wordscentimeters per meter

length in centimeters ÷

length in meters=

2,900Equation 100 29÷ =

The whale was about 29 m long.

Quick Check

PRE-ALGEBRA LESSON 3-7PRE-ALGEBRA LESSON 3-7

Write the metric unit that makes each statement true.

1. 23 kg = 23,000 2. 970 cm = 9.7

Complete each statement.

3. g = 42 mg 4. km = 5,000 m

0.042 5

Using the Metric SystemUsing the Metric System

Problem Solving Strategy: Act It OutProblem Solving Strategy: Act It Out

Describe the pattern. Then give the next three terms.3, 6, 12, 24, 48, . . .

Starting with 3, each number is two times the number before it: 96, 192, 384.

Write a rule for each number pattern. Find the next three numbers inthe pattern.

1. 0, 6, 12, 18,… 2. –18, –9, 0, 9,…

3. 0, 2, 1, 3, 2, 4, 3,… 4. 7, 6, 8, 7, 9, 8, 10,…

Solutions

1. 0, 6, 12, 18,…

Start with 0 and add 6 repeatedly.

0, 6, 12, 18, 24, 30, 36+6 +6 +6 +6 +6 +6

2. –18, –9, 0, 9,…

Start with –18 and add 9 repeatedly.

–18, –9, 0, 9, 18, 27, 36

+9 +9 +9 +9 +9 +9

3. 0, 2, 1, 3, 2, 4, 3,…

Start with 0. Alternately add 2 and subtract 1.

0, 2, 1, 3, 2, 4, 3, 5, 4, 6

+2 –1 +2 –1 +2 –1 +2 –1 +2

Solutions (continued)

4. 7, 6, 8, 7, 9, 8, 10,...

Start with 7. Alternately subtract 1 and add 2.

7, 6, 8, 7, 9, 8, 10, 9, 11, 10

–1 +2 –1 +2 –1 +2 –1 +2 –1

Marta gives her sister one penny on the first day of October,

two pennies on the second day, and four pennies on the third day. She

continues to double the number of pennies each day. On what date will

Marta give her sister $10.24 in pennies?

Problem Solving Strategy: Act It OutProblem Solving Strategy: Act It OutPRE-ALGEBRA LESSON 3-8PRE-ALGEBRA LESSON 3-8

1 2 2 • 2 = 4 4 • 2 = 8 8 • 2 = 1616 • 2 = 32

Number ofpennies

Days after the first

012345

Amount

$0.01$0.02$0.04$0.08 $0.16$0.32

(continued)

Problem Solving Strategy: Act It OutProblem Solving Strategy: Act It OutPRE-ALGEBRA LESSON 3-8PRE-ALGEBRA LESSON 3-8

2 • 2 • 2 • 2 • 2 • 2 • 2 • 2 • 2 • 2 = 1024

You can tell from the pattern in the chart that you just need to count the number of 2’s multiplied until you reach 1,024, which is $10.24 in pennies.

Marta will give her sister $10.24 in pennies on October 11.

10 twos = 10 days after the first penny is given

Quick Check

Solve using any strategy.

1. On Monday, Jon reads page 45 of his book and continues toread until he finishes page 89. How many page does he readon Monday?

2. Marion put one penny in a bank. Each day after that she put inthe bank double the number of pennies from the previous dayuntil the bank was full. If the bank was full on the 8th day, whenwas the bank only half-full?

89 – 45 + 1, or 45 pages

7th day

rounding and estimating

estimatingprealgebra

quotientsprealgebra

thousandthsprealgebra

quick check3

total cost

decimal products

nearest integer

fabric cost

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