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Free Pre-Algebra Lesson 35 ! page 1

© 2010 Cheryl Wilcox

Lesson 35

Comparing Sizes

Comparing with Greater Than and Less Than One number is greater than another if it is farther to the right on a number line:

5 > 2 because 5 is further to the right than 2, but –2 is further to the right than –5, so –2 > –5.

Comparing Fractions If we know where fractions are located on the number line, it’s easy to see which is greater.

We can locate the fractions and see that

5

8>

9

16

and that

11

2>

5

8.

Many times it’s easy to see which of two fractions is greater without looking at a number line or doing too much calculation. For example, it’s obvious that 11/2 is greater than either 5/8 or 9/16, because 11/2 is more than one and the other two proper fractions are less than one. (A mixed number or improper fraction is always greater than a proper fraction.) But it’s not always easy to tell just by looking at the numbers themselves, and drawing is not always accurate.

For example, which is greater, 9/14 or 13/21? The fractions as written don’t give us much clue. We could make a very careful drawing:

You can see that

9

14>

13

21, but it’s not easy to draw a

picture so accurately.

If the fractions have a common denominator, though, we can just compare the numerators. This lets us just work with numbers instead of pictures.

9

14=

9

2 • 7•

3

3=

27

42

13

21=

13

3 • 7•

2

2=

26

42

Since

27

42>

26

42, we know that

9

14>

13

21.

Example: Compare the fractions with > , < or =.

Which is greater, 4/3 or 11/16?

4/3 is an improper fraction and is greater than one.

11/16 is a proper fraction and is less than one.

So 4/3 > 11/16

Which is greater, 2/3 or 11/16?

2

3=

2

3•

2 • 2 • 2 • 2

2 • 2 • 2 • 2=

32

48

11

16=

11

2 • 2 • 2 • 2•

3

3=

33

48

33

48>

32

48

11

16>

2

3



Comparing Decimals Since people in the U.S. don’t regularly use the metric system, we have a tougher time picturing decimals on the number line. Remember that decimal numbers have denominators that are powers of 10. Each unit is divided into tenths. Each tenth has ten divisions to make hundredths.

The key to comparing decimals is to match the same decimal places. If you look at the decimals 0.3 and 0.23 without looking at the number line or thinking about place value, you might think “3 < 23, so 0.3 < 0.23. (wrong!)” But if you find 0.3 on the number line above and remember that 0.3 is 3/10 = 30/100, and 0.23 is 23/100, you see the correct comparison is

0.3 > 0.23. In a way, arranging decimal numbers ls like alphabetizing. Align the decimal points of the numbers you want to compare so that place values are identical, then compare left to right. If numbers have the same number of decimal places you can drop the decimal point and compare directly. Repeating decimals can be written with more places, and you can add zeros at the end of any decimal without changing the value. For

example, which is greater, 2.3 or 2.3 ? Expand to 2.30 and 2.33 and you can see that 230 < 233, so 2.3 < 2.3

These words are arranged in alphabetical order. These decimals are arranged in increasing order.

The words that begin with “a” come before the words that begin with “b” and “g” below.

Within the words beginning with “a” those that begin with “ab” come before those that begin with “ac” or “at”, because “c” and “t” come later in the alphabet than “b”.

abstract

abuse

actual

attorney

batter

butter

good

0.15876

0.196

0.27

0.8

1.06

1.6

5

All the decimals with “0” in the ones place are less than those with “1” or “5” in the ones place.

Within the decimals with “0” in the ones place, those with “1” in the tenths place are less than those with “2” or “8” in the tenths place, because 2 and 8 are greater than 1.

Decimals with the same number of places can be compared directly. You know that 800 > 196 so the decimal

0.800 > 0.196

Example: Compare the decimals with > and <.

0.45 and 0.405

Method 1: Common denominator (same number of decimal places) 0.450 > 0.405 because 450 > 405.

Method 2: Alphabetical

0.45 The numbers match in the ones and tenths places. 0.405 Since 5 in the hundredths place is greater than 0 in the hundredths place 0.45 > 0.405.

0.45 and 0.45 and 0.45

Since some of the decimals are repeating, it’s easier to compare when they are written out in more places. You only need three places to find where the numbers are different.

0.4555… 5 in the thousandths place 0.4545… 4 in the thousandths place 0.4500… 0 in the thousandths place

0.45 < 0.45 < 0.45



Convert to Compare The ease of comparison for decimals gives an alternate method for comparing fractions. First convert the fractions to their decimal equivalents, then compare. This method also allows us to compare numbers written as fractions to numbers written as decimals.

Example: Compare the numbers with >, <, or =.

13

20 and

2

3

13

20= 13 ÷ 20 = 0.65

2

3= 2 ÷ 3 = 0.666...

The decimals differ in the thousandths place.

2

3>

13

20

0.2 and 4

19

4

19= 4 ÷ 19 = 0.210526...

0.2105

0.2000

The decimals differ in the hundredths place.

4

19> 0.2

Comparisons in Context Being able to tell which of two numbers is greater is a fundamental skill. To put the skill to work in interpreting data is the next step. Suppose you are writing a presentation for a nutrition class, comparing apples to chocolate bars. You have the information from the table below. If you are just comparing quantities, you might write:

medium apple 1 apple

chocolate bar 1 bar

serving size 182 g. 43 g. calories 95 210 calories from fat 3 110

The apple is bigger and weighs more than the chocolate bar and so will make you feel more full after you eat it. But it has fewer calories overall and fewer calories from fat than the chocolate bar.

Example: Write a paragraph comparing the information from the chart below.

California New York

Area (square miles) 163,707 54,475 Population (2010) 37,253,956 19,378,102 Median Household Income

(2008) $61,021 $56,033

California has a greater area than New York, and a greater population. The median household income is also higher.

These comparisons are accurate and helpful, but some further comparison tools can make your presentation really meaningful.



Comparing with Differences One way to compare two quantities is to find the difference between them by subtracting. You can take the absolute value of the difference to ensure a positive result.


chocolate bar 1 bar

Difference between apple and choc. bar

serving size 182 g. 43 g. |182 – 43| = 139 g. calories 95 210 |95 – 210| = 115 cal. calories from fat 3 110 | 3 – 110| = 107 cal.

The information about differences shows you that almost all the caloric difference between the apple and chocolate bar are due to the additional fat calories in the candy. Using the information about differences, you could write:

The apple weighs 139 g. more than the chocolate bar but has 115 fewer calories. The chocolate bar has 107 more calories from fat. In fact the extra fat calories in the chocolate account for almost all the calorie count over that of the apple.

Example: Find the differences and compare.

California New York Differences

Area (square miles) 163,707 54,475 | 163,707 – 54,475 | = 109,232 Population (2010) 37,253,956 19,378,102 | 37,253,956 – 19,378,102| =

17,875,854 Median Household Income (2008)

$61,021 $56,033 | $61,021 – $56,033| = $4988

California’s area is more than one hundred thousand square miles greater than New York’s. There are nearly 18 million more people in California, and the median income is nearly $5000 more.

Notice the use of rounding in the paragraph about the differences. Using your judgment about the level of accuracy required in the comparison is an important numeracy skill. Comparing with Quotients Probably the most useful numeric comparisons use quotients. A ratio compares quantities of the same type and units using division. It is often written as a fraction. The item mentioned first has its value in the numerator.


chocolate bar 1 bar

Ratio comparing apple and choc. bar

serving size 182 g. 43 g. 182 g. / 43 g. calories 95 210 95 cal / 210 cal calories from fat 3 110 3 cal / 110 cal

Right now the ratios don’t look useful, but that’s because they haven’t been simplified. Just as with fractions, both factors and units can be cancelled to simplify the result. Often the best comparisons come from a unit ratio – a ratio simplified by dividing so as to have 1 in the denominator. The division and related multiplication equations let us compare how many times greater one quantity is than another. For example, the ratio of weights for the apple and chocolate bar is 182 g / 43 g. In a ratio, the units should always match and cancel. We can say “The ratio of the weight of an apple to the weight of a chocolate bar is 182 to 43.” This sentence is accurate but not yet really helpful. If you divide, though, 182 ÷ 43 is about 4.2. This tells us that the apple weighs about 4.2 times as much as the chocolate bar, or that it would take more than 4 chocolate bars to make the same weight as an apple.



medium apple

1 apple chocolate bar

1 bar Ratio comparing

apple and choc. bar Ratio simplified

and rounded

serving size 182 g. 43 g. 182 g. / 43 g. 4.2 calories 95 210 95 cal / 210 cal 0.5 calories from fat 3 110 3 cal / 110 cal 0.03

We usually express the simplified ratio in terms of multiplication when writing about the comparisons. Sentences about ratios:

The ratio of the weight of the apple to the weight of the chocolate bar is 182 to 43 or about 4.2 to 1.

The ratio of the calories in an apple to the calories in a chocolate bar is 95 to 210 or about 0.5 to 1.

The ratio of the fat calories in an apple to the fat calories in a chocolate bar is 3 to 110 or about 0.03 to 1.

Sentences about multiplications

The apple weighs about 4 times as much as the chocolate bar.

The apple has about half as many calories as the chocolate bar.

The apple has only about three hundredths the fat calories of the chocolate bar.

The ratios can be written in either order, so long as you are clear which quantity comes first. If you switch the order, the new ratios will be the reciprocals of the previous ratios. For example, you could switch the calorie ratio around, and find the ratio of the calories in a chocolate bar to the calories in an apple is 210 cal to 95 cal, which simplifies to about 2.2. You could then write the sentence “The chocolate bar has more than twice as many calories as the apple.”

Example: Find the simplified ratios and compare.

California New York Ratios Simplified and Rounded Ratios

Area (square miles) 163,707 54,475 163707 / 54,475 3.0 Population (2010) 37,253,956 19,378,102 37253956 / 19378102 1.9 Median Household Income (2008)

$61,021 $56,033 61021 / 56033 1.1

Comparison: California’s area is three times that of New York, and the population is nearly twice as large. The median incomes for the two states are almost the same.

Notice that a ratio close to 1 means the two quantities are almost the same. Don’t you feel the paragraph above has a lot of punch? You often read these sorts of comparisons in news articles. A rate is like a ratio, except the quantities divided have different units. For example, your car’s speed is a rate, since it divides miles by hours, which are different units. Rates are usually described with the word “per” as in “miles per hour.” A rate gives us a way to compare using two quantities at once. Just as with ratios, if you do the division and simplify, you have a unit rate (denominator 1).



For example, you can divide miles driven by hours to find the rate. Simplify to find the unit rate.

140 miles

2 hours=

140 / 2 miles

2 / 2 hours=

70 miles

1 hour We would say the rate is “70 miles per hour.”

You might decide that the rate of calories per gram of food is important nutritionally, and figure out that rate for both the apple and the chocolate bar.


chocolate bar 1 bar

serving size 182 g. 43 g. calories 95 210 calories from fat 3 110 rate of calories per gram 95 cal / 182 g 210 cal / 43 g Simplified and rounded rate 0.5 cal. per gram 4.9 cal. per gram

Comparison: An apple has only 1/2 calorie per gram whereas a chocolate bar has nearly 5 calories per gram.

Example: Find the population per square mile and compare.

California New York

Area (square miles) 163,707 54,475 Population (2010) 37,253,956 19,378,102 Rate: Pop / Sq Mile 37253956 people / 163,707 sq mi 19378102 people / 54475 sq mi Simplified and rounded rate about 228 people per sq mile 356 people per sq mile

Comparison: Although California has a larger population than New York, the rate of people per square mile is only 228 compared to a rate of 356 people per square mile for New York.

If you are comparing quantities in larger tables of data, it’s easy to set up a spreadsheet to compute differences, ratios, and rates.

!



Lesson 35: Comparing Sizes

Worksheet Name _________________________________________

Use the graphic to locate and compare the quantities with >, <, or =.

Convert to common denominators or decimals to compare with >, <, or =.

4.

17

20 and

5

6

5. 1.6578 and 1.655

6.

0.89 and 8

9

7.

! and 355

113

8. Why is it obvious without finding a common denominator

that

19

23<

50

41

?

9. Why is it obvious with no conversions that

27

8< 3.0007 ?

1.

23

8 and 2

5

16

2.

7

12 and

3

5

3. Mark the number line with the decimals 0.7, 0.72, 0.77, and 0.8.

Use the number line to explain why, if we round to the nearest tenth, 0.72 rounds to 0.7 and 0.77 rounds to 0.8.



10. A car comparison website gives the information below:

Jeep Wrangler Ford Explorer

MSRP $22,045 $28,190

EPA City 15 17

EPA Highway 19 25

MSRP stands for “Manufacturer’s Suggested Retail Price”. The units are dollars.

EPA City and Highway are estimates of gas mileage under different driving conditions. The units are miles per gallon.

Write one or more sentences comparing the SUVs using simple comparisons. (Greater than, less than).

11. Compare the SUVs by finding the differences between the given values.

Jeep Wrangler Ford Explorer DIFFERENCE

MSRP $22,045 $28,190

EPA City 15 17

EPA Highway 19 25

Write one or more sentences comparing the SUVs using differences.

12. What is the ratio of the price of the Ford to the price of the Jeep?

Write a sentence about the price ratio.

13. The EPA gas mileage is a rate: “miles per gallon.” You can calculate using the formula

miles

gallon• gallons = miles or mpg • gallons = miles

If you drive 20,000 highway miles per year, according to the gas mileage given, how many gallons of gas will you use

a. In the Jeep?

b. In the Ford?

14. What is the difference in the amount of gas you will use? Write a sentence.

15. If gas is an average of $4 per gallon, how much money will you save on gas with the Ford each year?

16. How long will it take for the savings on gas to equal the difference in cost of the SUVs?




Homework 35A Name __________________________________

1. Write the decimals as fractions in lowest terms.

a. 0.08

b. 1.75

c. 0.625

2. Write the fractions as decimals.

a.

17

1000

b.

17

100

c.

100

17

3. Find the area of a rectangle with length 8.05 cm and width 3.99 cm. Round to the nearest hundredth.

4. Find the volume of a sphere with radius 2.5 cm. Round to the nearest tenth.

5. A rectangle has length 3.8 cm and perimeter 15.2 cm. What is the width?

6. A runner completed the 26.2 mile marathon in 4.08 hours. What was her speed, in miles per hour? Round to the nearest hundredth.

Convert 0.08 hours to minutes.

7. Solve the equation

2

3y = 72 .

8. Solve the equation 0.7x ! 0.6 = 1.5 .



9. Compare with >, <, or =. Show any conversion work.

a. 0.6 0.28

b. 7/8 0.875

c. 22/7 !

d. 1/4 4/15

Use the information below for #10 – 13.

Source: Diablo Valley College Fact Book 2009, information from DVC Office of Financial Aid.

10. Write one or more sentences with simple comparisons of the number and dollar amounts of fee waivers at DVC in the academic years 2000-1 and 2008-9.

11. What is the difference in the number of fee waivers in 2000-1 and the number of fee waivers in 2008-9?

What is the difference in the amount of Fee Waiver aid in dollars between 2000-1 and 2008-9?

Write one or more sentences comparing the differences in fee waivers.

12. Find the ratio of the number of students receiving fee waivers in 2008-2009 to the number of students receiving fee waivers in 2000-2001.

Write a sentence comparing the data.

Find the ratio of the amount of aid in fee waivers in 2008-9 compared with 2000-1.


13. Find the rate of dollars spent to students receiving aid for each of the two academic years 2000-1 and 2008-9.

What do you think accounts for the change in the rate of dollars spent per student?

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Homework 35A Answers


a.

0.08 =8

100=

2

25

b.

1.75 = 175

100= 1

3

4

c.

0.625 =625

1000=

5

8


a.

17

1000= 0.017

b.

17

100= 0.17

c.

100

17= 5.882352941...

(can’t see repeating pattern on calculator)

3. Find the area of a rectangle with length 8.05 cm and width 3.99 cm. Round to the nearest hundredth.

A =LW = 8.05 cm( ) 3.99 cm( )= 32.1195 cm

2! 32.12 cm

2

4. Find the volume of a sphere with radius 2.5 cm. Round to the nearest tenth.

V =4

3!r

3=

4

3! 2.5( )

3

" 65.4 cm3


P = 2L + 2W

15.2 = 2 3.8( ) + 2W

2W + 7.6 = 15.2 2W + 7.6 ! 7.6 = 15.2 ! 7.6

2W = 7.6 2W / 2 = 7.6 / 2

W = 3.8

The width is also 3.8 cm. The rectangle is square.


rate =distance

time

=26.2 miles

4.08 hours! 6.42 mph


0.08 hours

1•

60 minutes

1 hour

= 4.8 minutes


2

3y = 72 .

2

3y = 72

3

2•

2

3y = 72 •

3

2

y = 108

8. Solve the equation 0.7x ! 0.6 = 1.5 .

7x ! 6 = 15 7x ! 6 + 6 = 15 + 6

7x = 21 7x / 7 = 21/ 7

x = 3



9. Compare with >, <, or =. Show any conversion work.

a. 0.6 > 0.28 0.60 > 0.28

b. 7/8 = 0.875 7/8 = 0.875

c. 22/7 > ! 3.1428.. > 3.1415…

d. 1/4 < 4/15

1/4 = 15/60 1/4 = 0.25 4/15 = 16/60 4/15 = 0.266…

Use the information below for #10 – 13.

Source: Diablo Valley College Fact Book 2009, information from DVC Office of Financial Aid.

10. Write one or more sentences with simple comparisons of the number and dollar amounts of fee waivers at DVC in the academic years 2000-1 and 2008-9.

More students received financial aid in 2008-2009 than in 2000-2001. The amount

of aid in dollars is also higher in 2008-2009.

11. What is the difference in the number of fee waivers in 2000-1 and the number of fee waivers in 2008-9?

| 3,174 – 6,166 | = 2,992

What is the difference in the amount of Fee Waiver aid in dollars between 2000-1 and 2008-9?

| 599,164 – 2,103,728 | = 1,544,564

Write one or more sentences comparing the differences in fee waivers.

There were nearly 3000 more fee waivers in the 2008-2009 academic year than in the 2000-2001 academic year. The amount of

financial aid increased by over $1,500,000.

12. Find the ratio of the number of students receiving fee waivers in 2008-2009 to the number of students receiving fee waivers in 2000-2001.

6166 / 3174 = about 1.9


Nearly twice as many students received fee waivers in 2008-2009 as did in 2000-2001.

Find the ratio of the amount of aid in fee waivers in 2008-9 compared with 2000-1.

2,103,728 / 599,164 = about 3.5


The amount of aid distributed in 2008-9 is 3 and a half times what it was in 2000-1.

13. Find the rate of dollars spent to students receiving aid for each of the two academic years 2000-1 and 2008-9.

2000-2001: $599,164 / 3174 students = $188.77 per student

2008-2009: $2,103,728 / 6166 students = $341.18 per student

What do you think accounts for the change in the rate of dollars spent per student?

Fees have gone up, or students are taking more classes, or both.

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Homework 35B Name ________________________________________


a. 0.0005

b. 0.05

c. 0.55


a.

13

20

b.

13

200

c.

13

30

3. Find the area of a rectangle with length 19.2 cm and width 20.7 cm. Round to the nearest whole number.

4. Find the volume of a sphere with radius 5.1 cm. Round to the nearest whole number.





4

15w !

3

15=

10

15.

8. Solve the equation 7x + 0.06 = 0.9 .



9. Compare with <, >, or =. Show any conversion work.

a. 0.08543 0.08553

b. 1/6 0.16

c. 1/6 3/20

d. 1.57 !/2

Use the information below for #10 – 13. Source: Apple Investor Relations. 2010 Annual Report.

10. Write one or more sentences with simple comparisons of unit sales and net sales for the ipod and iphone in 2009 and 2010.

11. What is the difference in net sales for the ipod from 2009 to 2010?

What is the difference in unit sales from the ipod from 2009 to 2010?

Write one or more sentences comparing the differences in net sales and unit sales for the ipod.

12. Find the ratio of net sales 2010 to net sales 2009 for each item.

ipod:

iphone:

Write a sentence comparing the ratios.

13. Find the rate of net sales to unit sales for the ipod and iphone in 2010.

ipod:

iphone:

What are the units of the rate? What does this number represent, practically?

2009 unit sales

2009 net sales

2010 unit sales

2010 net sales

ipod 54,132,000 $8,091,000,000 50,312,000 $8,274,000,000

iphone 20,731,000 $13,033,000,000 39,989,000 $25,179,000,000

Information from Apple 2010 Annual Report

lesson 35 text - diablo valley collegevoyager.dvc.edu/~lmonth/prealg/lesson35student.pdf · free...

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