Chapter 8 Student Textbook

8/7/2019 Chapter 8 Student Textbook

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chapter 8 ratons Sus and erenesratons Sus and erenes | 377

Chapter

8

Fraction Sums and Dierences

Chapter Objecties

Use prime factorization to simplify fractions. Find equivalent fractions to simplify fractions. Compare fractions, add fractions, and nd differences of fractions. Express fractions greater than a whole using mixed number notation. Use whole numbers, fractions and mixed numbers to solve problems and

estimate measurements.

Lessons

Lesson 39

Simplifying Fractions

Lesson 40

Common

Denominators

Lesson 41

Adding Fractions

Lesson 42

Mixed Numbers

Lesson 43

Fractional Differences

Are You reAdY?

Express each number in prime factored form using exponents.

1. 24

2. 50

3. The following number line shows equal jumps where three jumps equal a

whole. What fraction is represented by pointp?

4. Solve the following equation.

3__85__9 = k

5. Use division to compare the fractions by placing a , or = symbol to

make the statement true.

4__7 7__11


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378 | chapter 8 raton Sus and erenes

Lesson 39 Simpliying Fractions

Objecties

Use prime factorization to determine if fractions are equivalent.

Simplify fractions by using prime factorization.

Simplify expressions involving exponents with a common base.

Apply fraction simplication to solving problems involving rates.

Concepts and Skills

RN.7 Determine if two fractions are equivalent.

RO.6 Use prime factorization to simplify fractions, generate equivalent

fractions and nd a common denominator for a pair of fractions.

ER.1 Use exponents to represent repeated multiplication.

ER.6 Multiply, divide and simplify exponential expressions involvingexponents with a common base.

Remember rom Beore

What are equivalent fractions?

What is a simplied fraction?

What is the multiplicative property of one?

Get Your Brain in Gear

1. Use mental math to nd the value of each expression.

a. 24 b. 9

2

c. 33 d. 7

0 3

2 2

2

e. 23 5

1 f. 2

3 5

3

Vocabulary

equvalentratons

splfed raton


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Lesson 39 Splyng ratonsSplyng ratons | 379

Lesson 39 Simpliying Fractions

Conepts and Sklls: R.7, RO.6,

ER.1, ER.6

In the previous few lessons we learned how to describe the composition of whole

numbers using prime factors. Here we will use prime factorization to gain a better

understanding of equivalent fractions.

EquialentFractions

Lets rst review the meaning ofequivalent fractions. When two fractions describe

the same point on the number line it means they equal the same number. We call

such fractions equivalent fractions. For example10__15and 2_3are equivalent fractions.

Lets use the number line to see why this is true.

Here is10__15 on the number line:on the number line:on the number line:

And here is2__3 expressed on the other side of the number line:expressed on the other side of the number line:expressed on the other side of the number line:

Since both fractions represent the same point, they are equal:

10

__

15 =2

__

3Prime factorization of a whole number tells us the basic building blocks that

compose that whole number. We can also use this building-block view to see

why two fractions are equivalent. For example, lets use this approach to see why10__15

equals 2_3.Here are the prime factorizations of 10 and 15:

10 = 2 5

15 = 3 5

We can use these representations to express the fraction

10__15

:

10__15 = 2 5___3 5

From what we know about multiplying fractions, we can rewrite25___35 as follows:as follows:as follows:

10__15 = 2__35__5


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380 | chapter 8 raton Sus and erenesraton Sus and erenes

Since5_5 equals 1, we get:

10__15 = 2__3 1

Finally, lets remove the 1 part because the multiplicative property of 1 tells us

that multiplying by 1 doesnt change the product:

10__15

=

2__3

We just used prime factorization to show that10__15 andandand 2_3 are equivalent fractions.are equivalent fractions.are equivalent fractions.

Check orUnderstanding

1. Use prime factorization to find equivalent fractions for the following

numbers:

a.6__10 b. 15__9 c. 4__14 d. 14__7

SimplifedFractions

Sometimes we run across fractions that look complicated, such as90___360. By usingprime factorization, we can often nd an equivalent fraction that is simpler. Lets

try this.

Here is the prime factorization for 90:

And here is the prime factorization for 360:

This means we can express90___360 like this:

Immediately we see that this fraction has many multiplications by 1, such as3_3, 2_2

and5_5:


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Since multiplication by 1 doesnt change the product, we can eliminate these from

the expression:

This fraction is a lot simpler, but we can simplify it even further by using what we

know about the properties of multiplication:

Since 2 2 equals 4, we conclude that1_4 andandand 90___360 are equivalent fractions:are equivalent fractions:are equivalent fractions:

90___360 === 1__4

Obviously1_4 is a much simpler fraction to work with. When a fraction is simplied

as much as possible, we call it a simpliedfraction.

Instead of writing out all of the steps, we can use a shorthand notation for thissimplication process. We can simply cross out matching prime factors in the

numerator and the denominator like this:


2. Find the simplified fraction for each of the following numbers:

a.25___100 b. 270___360 c. 30__75 d. 9__20

SimpliyingExpressions

with Exponents

In Lesson 38 we learned how to use exponents to represent the repeated factors in

a prime factorization. Using this knowledge, lets simplify the following fraction:

Here the numerator and denominator are already in prime factorization form. Notice

that we are using exponents to represent the repeated factors. Here 23

means we

have three factors of 2 in the numerator:


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The 72

means that there are two factors of 7 in the numerator:

In the denominator we have 2

2

, which means there are two factors of 2:

Finally, the 71

means there is just one factor of 7:

Now we can simplify this fraction as we learned earlier:

From this we see that the original complicated looking fraction simplies to the

whole number 14.

Check or

Understanding

3. Simplify the expression.

Simpliyingwith Rates

We can also use prime factorization to help us solve problems involving rates. As

an example, consider the following situation:

We bought 5 pies for $20. At this rate, how much does1_2of a pie cost?

The problem asks us to convert from pies to dollars, so here is the rate:

To nd how much1_2of a pie costs, we multiply by the above rate:


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To make the problem less cluttered, lets remove the units for the moment. We just

need to remember that the value of the expression will have units of dollars:

Lets multiply the fractions:

Now we simplify using prime factorization. Since 20 equals 225, we rewrite the

fraction as:

We can cancel out a 2_2 and a 5_5because the multiplicative property of 1 tells usthat multiplying by

n_n doesn't change the value of an expression:

From this we conclude that 1_2 of a pie costs 2 dollars. We can summarize this resultof a pie costs 2 dollars. We can summarize this resultof a pie costs 2 dollars. We can summarize this resultwith the following equation:

Its easy to visualize why $2 for 1_2 of a pie is the same rate as $20 for 5 pies:of a pie is the same rate as $20 for 5 pies:of a pie is the same rate as $20 for 5 pies:

Here we have 10 half pies coming together to form 5 whole pies. Since each half

pie costs $2, the 5 whole pies cost $20.

Check or

Understanding

4. If 8 batteries get used up every 6 hours, how many batteries are used after

15 hours?

5. Seven out of 10 people at the company are vegetarians. If there are 56

vegetarians at the company, what is the total number of people at the

company?

6. If you drive 63 miles per hour, how long will it take to drive 210 miles?


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Problem Set Write the numerator and denominator of each fraction in prime factored form.

Example:12__30 Solution: 12__30= 2 2 3_______2 3 5

1.8__12 2. 40__25 3. 210___270

Write the numerator and denominator of each fraction in prime factored form.

Use this to simplify the fraction.

4.2__8 5. 16__18 6. 45__36

7.63__14 8. 12__30

Express the exponents as repeated multiplication, then simplify the expression.

9. 10. 11.

Whichofthefractionsbelowarealreadysimplied?Fortheotherfractions,

writetheequivalentsimpliedfraction.

12.

14__28

13.

4__15

14.

6__36

15.

48__30 16. 21__16 17. 24__81

Findthevalueoftheexpression.Writetheresultasasimpliedfraction.

18.5__63__5 19. 7__124__3 20. 54__9 15__3 21. 25__6 4__15

Usefractionmultiplicationtondthevalueoftheexpression.Dontforgetthe

units of your answer.

22. 23.

24.


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Multiple Choice Practice

1. Which is a simplied fraction?

14__49 14__21 14__8 14__25

2. Here is where3__6is located on the number line:

After simplifying3__6, the new location on the number line will be:

closer to zero. closer to one.

in the exact same location as3__6. on the opposite side of zero.

Math Journal Questions

1. Using the fraction30__24, write clear and logical step by step directions that show

how to simplify the fraction. For each step, explain what mathematical properties

are being used.

2. Explain how you can tell if a fraction is as simplied as possible.

3. Explain how you would simplify the expression below without having to write

out all of the factors? What is the value of this expression?

Find the Errors A student made a mistake below. Find and correct the mistake.

looking bAck

Vocabulary: Equvalent ratons, pre atoraton, sply,

splfed raton

Student Sel Assessment: o I get t?

1. How do I deterne ratons are equvalent?

2. How do I sply ratons usng pre atoraton?

3. Whh propertes are used to sply ratons n ths way?

4. How s splyng ratons useul when solvng rate probles?


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Lesson 40 Coon enonators | 387

Lesson 40 Common Denominators

Objecties

Use prime factorization to form common denominators.

Compare fractions by rst forming equivalent fractions with a common

denominator.

Apply fraction comparison to compare rates such as nding the better price or

the faster speed.

Concepts and Skills

RN.9 Find a common denominator for a pair of fractions.



RO.7 Compare fractions by rst writing them as equivalent fractions with a

common denominator.SN.8 Solve word problems involving rates.

Remember rom Beore

What are equivalent fractions?

What are some ways weve learned to compare fractions?

What is the multiplicative property of one?


1. Estimate the value n on the number line below.

2. Estimate the value m on this next number line.

3. Name four fractions equivalent to 4_3.4. Find equivalent fractions that have denominators of 10.

a.3__5 b. 7__2 c. 10__20

Vocabulary

oondenonator

equvalent

ratons

splfed raton


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Lesson 40 Common Denominators


RO.7, S.8

In the previous lesson, we generated equivalent fractions using prime factorization.

This technique is useful not only for putting fractions in simplied form but its

also helpful when getting two fractions to agree on a common denominator. Well

discuss this second situation here.

ComparingFractions

Lets start the lesson with a real world example:

Brothers John and Bill had a contest to see who could eat the most pie in

6 minutes. John and Bill sliced their pie in different ways:

When the time was up, John had eaten 3 slices whereas Bill had eaten 5

slices. Who won?

Bill ate more slices, but Bills slices were smaller than Johns. It would have

been easier to compare if they had sliced the pies in the same way in the beginning.

But thats okay because we can use mathematics to cut the pies into a common

number of slices, even though some of the slices are being digested in John and

Bills stomachs. Lets see how this works.

John ate 3 out of 4 slices, so thats3_4 of a pie. Bill ate 5 out of 8 slices, totaling 5_8

of a pie:

Lets now express these fractions using prime factorization:

Since Bills fraction has one more factor of 2 in the denominator, let's multiply

Johns fraction by 2_2 to get an equivalent fraction:


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Lesson 40 Coon enonatorsCoon enonators | 389

The multiplicative property of 1 tells us that we can multiply by2_2without changing

the value of the fraction. Now the fractions have a common denominator:

Since 3 2 equals 6, and 2 2 2 equals 8, we end up with the following two

fractions:

We have just used mathematics to slice Johns pie in the same way as Bills. This

shows us that John won the competition by eating 6_8 of a pie, which is more thanthe

5_8 of a pie that Bill ate. We can write this as an inequality:6__8 >>> 5__8

Since

6_8is equivalent to

3_4, we can also summarize the contest results as:

3__4 >>> 5__8

Before we move on, lets look at the above inequality on the number line:

The number line gives us a good picture of how much more John ate to win the

pie-eating competition.


1. Place a , or = sign in each circle to make the statement true.

a.5__7 4__7 b. 5__9 8__27 c. 11__10 6__5 d. 2__5 9__25

Rewriting Both

Fractions

Lets look at a more sophisticated example by nding which is bigger, 3__10 ororor 2_6.

The blue parts below show 3__10 of a strip of paper on the left, and 2_6 of a strip ofpaper on the right:

Visually, these two fractions look almost equal. Its hard to tell them apart, but with


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prime factorization, we can nd out very easily which is bigger. Here are the two

fractions expressed using prime factors:

The fraction on the left has a 5 in the denominator, but the one on the right doesnt.

So lets multiply the right fraction by5_5:

The denominators arent common yet. The fraction on the right has a 3 in the

denominator, but the one on the left doesnt. So we multiply the left fraction by3_3:

We haven't changed the values of these fractions, but now they share a common

denominator, which makes it easy to compare them. The numerator on the left is

3 3 = 9, and the numerator on the right is 2 5=10:

This shows us that the fraction on the left is smaller, and we can summarize this

with an inequality:

3__10


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ComparingRates

We can compare rates just like we compare fractions. Comparing rates is very useful

in the real world such as when searching for the best prices. For example:

The following two stores have different deals for the same brand of soda.

Which store has the better deal?

First of all, what do we mean by a better deal? Heres one way to think of it: We

have a certain amount of money for buying soda, and we want to buy as much soda

as possible. Which store will give us the most soda for our money?

Store A is selling soda at the rate of 2 bottles for 3 dollars. We want to turn dollars

into bottles of soda, so here is the rate:

Store B is selling 5 bottles of soda for 7 dollars, so here is Store Bs rate:

We now want to compare these two rates to see which gives us more soda. As we

did earlier, we can do this by creating a common denominator.

Store As rate has a 3 in the denominator, but Store Bs rate doesnt. So we multiplyStore Bs rate by

3_3:

Store Bs rate has a 7 in the denominator, but Store As rate doesnt, so we multiply

by7_7 like this:

Now the two rates have a common denominator of 3 7 = 21.

Since 2 7 = 14, and 5 3 = 15, we get the following rates:

From this we see that $21 will only buy 14 bottles of soda at Store A. But $21 will

buy 15 bottles at Store B. This means Store B has the better deal.


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To express this mathematically, we say that Store As rate is less than Store Bs

rate. We can summarize this with the following inequality:

Check orUnderstanding 3. An 11 pound bag of ice costs $2. A 39 pound bag of ice costs $7. Which isthe better deal (assuming you will use all the ice you buy)?

4. In 2 minutes Jo walked 9 miles. Delicia walked 13 miles in 3 minutes.

Who had the faster average walking speed?

5. Haz wrote 200 lines of code in 8 hours. It took Mike 12 hours to write 300

lines of code. Which computer programmer is the faster coder?

Problem Set 1. For each number nd an equivalent fraction that has a denominator of 12.

Example: 1__6 Solution: 1__62__2 = 2__12a.

5__3 b. 3__4 c. 7_2 d. 5__62. For each number nd an equivalent fraction that has a denominator of 10.

a.4__5 b. 3__2 c. 1__5 d. 4__20

3. For each number nd an equivalent fraction that has a denominator of 18.

a.3__2 b. 5__3 c. 7__9 d. 5__6

4. For each number nd an equivalent fraction that has a denominator of 15.

a.2__3 b. 7__5 c. 6__30

Compare the fractions using the , or = sign.

5. Compare by rst forming a common denominator of 12.

a.7__4 5__3 b. 5__6 3__4 c. 2__3 3__4 d. 7__12 2__3


a.7__5 3__2 b. 2__5 3__10 c. 6__10 3__5


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a.4__9 2__6 b. 15__18 5__6 c. 5__6 8__9 d. 11__18 2__3


a.2__3 3__5 b. 2__3 4__5 c. 11__15 2__3 d. 10__15 2__3

Compare by forming a common denominator.

9.5__9 2__3 10. 7__24 3__8 11. 5__4 19__16 12. 13__6 7__3

Find an equivalent expression with a denominator of 2 3 5.

13. 14.

Find equivalent expressions that are easier to compare by forming a common

denominator. Compare the expressions.

15. 16.

17. 18.

For each pair, nd equivalent fractions that have a common denominator.

Compare the fractions.

19.5__6 4__5 20. 3__4 5__7 21. 5__12 7__16 22. 7__15 9__20

23.4__15 5__12 24. 5__18 7__24 25. 9__20 7__24


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26. Use these store ads to answer the questions below.

The Good Price Store The 400 Cents Store

Erasers...........24 for $7 Erasers...........15 for $4

Pencils...........12 for $5 Pencils............9 for $4

Rulers.............6 for $5 Rulers.............5 for $4

Markers............2 for $1 Markers............9 for $4

a. Which store has the better deal on pencils?

b. Which store sells markers for the better price?

c. Which store has the more expensive erasers?

d. Which store has the cheaper rulers?

27. Julian walked 30 feet in 7 seconds. Mia walked 50 feet in 13 seconds. Who

walked faster?

28. On the rst test Hang got 28 out of 35 problems correct. On the second test hegot 52 out of 65 problems correct. On which test did he get a greater fraction

of the problems correct?

29. Abby made 15 out of 20 shots in the basketball game. Miko made 22 out of 35

shots. Who made a greater fraction of their shots?

30. From the garden hose, 13 gallons of water ow in 2 minutes. From the kitchen

faucet, 40 gallons of water ow in 7 minutes. Which has the higher ow rate?

In other words, which has water ow out at a faster rate?

Challenge Problems

1. Here is a similar question to problem 30 in the problem set, but this time someunit conversion is needed:

From the garage faucet, 14 gallons of water ow in 3 minutes. From the

upstairs shower, 4 gallons ow every 45 seconds. Which has the higher ow

rate?

Here is one approach: Convert the 3 minutes to seconds rst. Then compare the

rates when they are both in gallons per second.

2. Either point a or b on the number line below has the value4__21. Figure out

which point it must be based on where5__28is located. Justify your answer using

mathematics.

3. Compare the following fractions.

a.25__14 13__8 b. 10__9 13__12 c. 21__25 14__15 d. 11__30 7__24


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1.2__33__3=

6__3 2__9 2__3 5__6

Math Journal Questions1. Describe how to make two fractions share a common denominator. When you

do this, do the values of the fractions change? Use mathematics to justify your

answer.

2. Can you nd a fraction equivalent to1__3 that has a denominator of 10? If you nd

one, explain how you found it. If you cant nd one, explain why you cant nd

one.

3. Explain the different ways you know how to compare fractions. Weve discussed

at least four different ways in this book so far.

4. Consider this comparison of rates:

From this inequality, which rate represents the better deal for buying apples?

Explain your reasoning.

Find the ErrorsA student made a mistake comparing the fractions below. Identify and correct

the mistake.

looking bAck

Vocabulary: nequalty, equvalent ratons, pre atoraton,

ultplatve dentty

Student Sel Assessment: o I get t?1. How do I fnd a oon denonator?

2. How do I rewrte ratons wth a oon denonator?

3. How do I opare ratons wth a oon denonator?

4. How do I opare ratons that dont have a oondenonator?

5. How an I opare rates?


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Lesson 41 Adding Fractions

Objecties

Understand the meaning of addition on the number line and know that the

meaning is the same whether adding whole numbers or adding fractions. Add fractions with like and unlike denominators.

Add whole numbers to fractions and express as a mixed number.

Concepts and Skills




RO.8 Add fractions with like denominators.

RO.10 Add and nd the difference of fractions with unlike denominators by rst

rewriting them as equivalent fractions with a common denominator.

Remember rom Beore

How is addition dened on the number line?

How is factoring used to nd a common denominator?


1. Find equivalent fractions that have denominators of 12.

a.7__6 b. 1__4 c. 4__3

2. Find equivalent fractions that have denominators of 18.

a.7__9 b. 11__3 c. 5__6

Vocabulary

oondenonator

xed nuber


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Lesson 41 Addng ratonsAddng ratons | 397

Lesson 41 Adding Fractions


RO.8, RO.10

In the previous lesson we compared fractions by nding common denominators.

Here well nd common denominators when adding fractions.

AddingFractions

The denition of addition on the number line is to place the start of one expression

at the end of another expression. We learned this in Lesson 1, and it still works for

fractions. For example, here are expressions for1_2 and 1_3:

To add these two expressions, we simply make one start where the other ends:

We labeled the nal point as1_2+ 1_3. To express this number as a single fraction, we

rst nd a common denominator for1_2 and 1_3:

Weve expressed1

_

2+1

_

3 as3

_

6+2

_

6. Now that we have a common denominator, letsshow this equivalent expression on the number line:

Here we have 3 jumps of + 1_6plus 2 jumps of + 1_6which equals a total of 5 jumps of+ 1_6. This is simply the fraction 5_6:

In symbols we write this as:

3__6 + 2__6 = 5__6


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Since3_6+ 2_6 is equivalent to 1_2+ 1_3, we arrive at the following conclusion:

1__2 + 1__3 = 5__6

Once fractions have a common denominator, adding them is easy. We just add the

numerators. We can generalize this result with the following identity:


1. Find the value of the expression.

a.3__5 + 7__5 b. 1__7 + 1__5 c. 2__9 + 4__3

d.1__5 + 7__10 e. 3__5 + 5__2

AddingFractions o

Area

Fractions describe quantities where units are broken into equal parts. But sometimes

we run across things in the real world that arent broken into equal parts. For

example, lets determine what fraction of the following unit square is shaded gray:

Since this unit square is not broken into equal parts, we can treat it as adding

fractions with different denominators. The above gure can be formed by adding

the following areas:

We can express this addition in symbols as follows:

1__2 + 1__4

To nd this value, we rst create a common denominator:

Now we add the fractions:

2__4 + 1__4 = 3__4

From this we conclude that the shaded area is3_4 of a unit square.

We can visualize that the area is 3_4 of a unit square by imagining what it would look


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like if we rearranged the shaded regions like this:

Lets look at another example.What fraction of the following unit square is shaded gray?

We can form the above gure by adding fractions together:

In symbols we write this as:

1__3 + 1__6

By forming a common denominator, we can easily add the fractions:

This means the area is3_6of a unit square. If we simplify 3_6, we see that its equivalent

to 1_2:

Again, by imagining the shaded areas rearranged, we can visualize that the gure

is1_2 shaded gray:


2. What fraction addition reasonably describes how much of each unit square

is shaded gray? Add the fractions. Check the reasonableness of the result

by comparing it to the original picture.


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Mixed Numbers Here is an area formed by adding a fraction to a whole number of unit squares:

We can describe this as 2 unit squares plus 1_4 of a unit square:2 +

1__4To express this as a single fraction, we need to remember that 2 equals the

fraction2_1:

2__1 + 1__4

Now we form a common denominator and add:

From this we conclude that the above area is9_4 of a unit square.

Some people prefer to express9_4 as 2+ 1_4. Its convenient because 2+ 1_4shows us

right away that theres a little more than 2 whole unit squares. Its more difcult to

see this from looking at9_4.

Adding fractions to whole numbers is done so often that people started getting lazy

and began writing 2+ 1_4 without the + sign:

When one puts a whole number right next to a fraction like this, its called a mixed

number. Well learn more about mixed numbers in the next lesson.

Its easy to visualize that 2+ 1_4 is the same as 9_4. All we have to do is break eachwhole unit square into 4 equal parts:

Now if we count up all of the quarters, we will nd that there are 9 of them.


3. Write each mixed number as the addition of a whole number and a fraction.

Then add to get a single fraction.

a. 41__3 b. 51__2 c. 7 3__10 d. 5 9__10


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4. For each group of unit squares, what fraction most reasonably describes

the total shaded area?

Problem Set Writethenumberlineexpression insymbolsandthenndthevalueoftheexpression.

1.

2.

3.

4. Foreachfraction,ndanequivalentfractionthathasadenominatorof12.

a.2__3 b. 1__6 c. 3__4 d. 5__2


a.1__5 b. 1__2 c. 2__5 d. 4__20


a.2__9 b. 5__6 c. 7__3 d. 3__2


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Addbyrstformingacommondenominatorof6.

7.2__3 + 1__2 8. 4__3 + 5__6


9. 1__3 + 3__4 10. 1__4 + 5__6 11. 5__12 + 1__2Addbyrstformingacommondenominatorof14.

12.1__2 + 3__7 13. 3__7 + 9__14


14.4__3 + 2__5 15. 4__15 + 4__5

Addthefractions.Useyourknowledgeofprimefactorizationtondacommon

denominator.

16.2__7 + 3__7 17. 3__4 + 1__8 18. 4__3 + 1__9

19.3__8 + 1__6 20. 3__5 + 2__3 21. 5__9 + 7__6

22.5__12 + 11__12 23. 7__12 + 5__8 24. 4__15 + 4__25

Add the whole number to the fraction. Express the value as a single fraction.

25. 3 +7__8 26. 1__15 + 1 27. 4 + 5__9

Write each mixed number as the addition of a whole number and a fraction.

Then, express each value as a single fraction.

28. 41__2 29. 7 2__3 30. 6 8__15

Challenge Problems

What fraction addition reasonably describes how much of each unit square is

shaded green? Add tondhowmucheachunitsquareisshaded.

1. 2. 3.


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Find the value of the following expressions.

4.1__6 + 1__4 + 3__8 5. 5__14 + 3__8 + 1__2

Here are some two-step word problems.

6. The vanpool traveled at a constant rate of 40 miles per hour. They traveled for1__2

of an hour. Then they traveled another 3__5 of an hour. How many miles did theytravel?

7. Lisa bought3__4 of a yard of owered fabric, 5__8 of a yard of dotted fabric, and 1__2

of a yard of purple fabric. If the fabric costs $2 for every 3 yards, how much did

the fabric cost (before taxes)?

Here is a tricky one. Read it carefully.

8. The cake was cut into 6 equal slices. Jared and Mickey each took one of the

slices. Jared didnt want all of his, so he cut his slice into 5 equal pieces and gave

one of the pieces to Mickey. What fraction of the total cake did Mickey get?


1. Estimate where1__2 + 7__8 is located on the number line.

2. Estimate where1__5 + 2__9 is located on the number line.

3. What is the value of point kon the number line below?

4__6 4__2 3__8 10__8


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1. Here is the addition of two generic fractions where the numerators and the

denominators are unknown:

a__b + c__

d

Explain how you would form a common denominator for

a__band

c__d. After you form

a common denominator, add the fractions. Use this result to add the fractions2_3+ 4_5.

2. A tangram is an ancient Chinese puzzle where a square is broken into parts as

shown here:

Can you gure out what fraction of the puzzle each piece represents? For starters,

the big piece at the very top is1__4of the puzzle.

After you nd all the fractions, verify your result by adding all the values

together. They should add to make a whole.

Find the Errors A student made 2 mistakes below. Identify and correct each mistake.

1. 2. 3.

looking bAck

Vocabulary: xed nuber, ratons, oon denonator


1. How do I add ratons wth oon denonators?

2. How do I add ratons wth unoon denonators?3. How do I add whole nubers to ratons?

4. What are xed nubers? What do they look lke?


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Lesson 42 Mxed ubers | 405

Lesson 42 Mixed Numbers

Objecties

Write mixed numbers using proper notation.

Express mixed numbers as fractions.

Express fractions greater than a whole as mixed numbers using long division.

Concepts and Skills

RN.11 Understand the meaning of mixed number notation.

RN.12 Locate mixed numbers on a number line.

RN.13 Express mixed numbers as improper fractions. Express improper

fractions as mixed numbers.

Remember rom Beore How is addition dened on the number line?

How is factoring helpful when adding fractions?


1. Use mental math to nd equivalent fractions that have denominators of 15.

a.7__3 b. 4__1

c.12__5 d. 78__

78

2. Use mental math to nd equivalent fractions that have denominators of 6.

a.21__2 b. 27__18

c.11__3 d. 32__24

Vocabulary

xed nuber


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Lesson 42 Mixed Numbers

Conepts and Sklls: R.11,

R.12, R.13

In the previous lesson we learned how to add rational numbers expressed as

fractions. We also learned about mixed numbers, which are fractions added to

whole numbers. In this lesson, well discuss mixed numbers in more depth.

Mixed NumberNotation

As we learned in the previous lesson, we can write expressions such as 2 + 1_3in ashorthand notation by removing the + sign:

2 +1__3 = 2 1_3

When we express the addition of a whole number and a fraction in this way, we

call it a mixed number because it mixes together two different ways of notating

numbers.

The following are some rules about the mixed number notation:

1. The whole number part cannot be 0. In other words, 0 + 1_2 should not bewritten as 0

1_2. We simply write 0 + 1_2 as 1_2.2. The fractional part must be greater than 0 and less than 1. This means

that 3 +5_2 should not be written as 3 5_2because 5_2 is greater than 1.

3. When writing a mixed number, the fractional part should come right after

the whole number part. For example, we should never write 2 1_3 as 1_32.4. When reading mixed numbers out loud, we separate the whole number

part from the fractional part using the word and. For example, 23__5 is

read as 2 and 3 fths.


1. Which of the following are not in correct mixed number notation? Explain

your reasoning.

a. 02__3 b. 1 9__10 c. 7 9__4 d. 7__82 e. 4 0__3

2. Rewrite the following expressions using mixed number notation when

appropriate:

a. 12 +3__4 b. 2__3 + 8 c. 0 + 4__5 d. Five and two-thirds

Mixed Numbersas Fractions

Mixed number notation is useful because it lets us instantly see how close the

number is to a whole number. For example, 3 1_4 tells us that the value is + 1_4 morethan 3. To see this, here is 3

1_4 on the number line:


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Fractions asMixed Numbers

When a fraction is greater than a whole, we can express it as a mixed number. We

use the same method that we used several lessons ago to determine if a fraction is a

whole number. Lets review that briey.

A fraction is a whole number when the numerator is divisible by the denominator.

For example, to test if 12__4 is a whole number, we attempt to use 12 unit squares tocreate a rectangle of width 4:

The resulting rectangle is 3 units tall. From this we conclude that 12__4 equals thewhole number 3:

12__4 = 3

We use this same process for mixed numbers. For example, lets express thefollowing fraction as a mixed number:

7__2

We rst try to use 7 unit squares to form a rectangle that is 2 units wide:

Because 7 is not divisible by 2, we have a remainder. If we break the remaining

1 unit square into 2 equal parts, we can add those parts to the top of our rectangle

like this:

Now we have a rectangle that is 2 units wide and 3 +1

_

2 tall. This means that7

_

2equals 3 + 1_2:7__2 = 3 + 1__2

We can write 3 + 1_2 as the mixed number 3 1_2:7__2 = 3 1__2


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Lesson 42 Mxed ubersMxed ubers | 409

Lets go over another example. Consider the following fraction:

8__3

To express this as a mixed number, we attempt to use 8 unit squares to form a

rectangle that is 3 units wide. We can use long division to help us with this by

remembering that8_3 is the same as 8 3:

From this we see that 8 3 is 2 with a remainder of 2. This means our rectangle will

have a height of 2 units with 2 unit squares left over:

To add the remainder to our rectangle, lets break each of the left-over squares into

3 equal parts:

Now we add the rst left-over unit square like this:

Finally we add the second remainder in a similar way:

The rectangle now has a height of 2 + 2_3, which we can express as the mixed number2 2_3. From this we conclude that 8_3 is equal to 2 2_3:

8__3 = 2 2__3


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We can easily visualize this on the number line by making 8 jumps of + 1_3:

Its clear that this is just two jumps of + 1_3more than 2.


5. Express the following fractions as mixed numbers:

a.3__2 b. 7__4 c. 4__3 d. 9__2

Using LongDiision

As we saw earlier, long division makes it easy to express fractions as mixed numbers.

Lets consider a more sophisticated fraction such as:

98__5

Since this is the same as 98 5, we simply carry out the following division:

From this we see that 98 5 is 19 with a remainder of 3. To add the remainders

to our rectangle of height 19, we rst divide the 3 remainders by 5. Since 3 5 is3_5, we add 3_5 to 19:

98__5 = 19 + 3__5

Finally, we write this in mixed number notation:

98__5 = 19 3__5

People typically express the result of long division using mixed number notation:

A Closer Look Earlier we showed that 98__5 is equal to 98 5. Then we showed that 98 5 equals19 with a remainder of 3. In mixed number notation this means that 98 5 is 19

3__5.Lets examine how this all works in more detail.


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When we say that 98 5 equals 19 with a remainder of 3, we mean that when

we multiply 5 19, we need to add 3 to get 98. We write this with the following

equation:

5 19 + 3 = 98

This means that 5 19 equals 95, which gives us:

95 + 3 = 98

Lets verify that 5 19 equals 95:

This also shows us that 95 5 equals 19. We can verify this with long division:

Now lets look back at our original fraction:

98__5We know that 98 equals 95+3, so we can write this as:

(95+3)______5Our knowledge of fraction addition tells us that this is equal to:

95__5 + 3__5We just showed that 95__5 equals 95 5 which equals 19. This gives us:19 +

3__5In mixed number notation 19 +

3__5 equals 19 3__5:19

3__5This is the same result we got before.

Check or

Understanding

6. Express the results of the following long division problems as mixed

numbers:

a. b. c. d.


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7. Rewrite the following fractions as mixed numbers:

a.10__3 b.360___7 c.123___8 d.156___155 e.23__10

Problem SetWrite the mixed numbers using words.

1. 53__5 2. 1 1__2 3. 17 8__10 4. 340 7__9

Express as a fraction or as a mixed number.

5. Seventeen forty-fths 6. Six and six eighths

7. Five sevenths 8. One hundred three and three fths

Find the value of the number line expression. Write the value as both a mixed

number and as a single fraction.

9.

10.

11.

12.

Draw a rectangle that has the given width using the given number of unit

squares. Express the height as a mixed number.

Example: A width of 7 out of 17 unit squares.

Answer:

13. A width of 4 out of 10 unit squares.




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Express each mixed number as the addition of a whole number and a fraction.

Thenndthevalueoftheexpression as a single fraction.

16. 21__3 17. 4 1__2 18. 6 3__7

19. 104__5 20. 7 5__12 21. 8 3__32

Express the result of each long division problem as a mixed number.

22. 23. 24.

Express the fractions as mixed numbers.

25. 13__5 26. 23__2 27. 123___428.

156___8 29. 277___9 30. 7__6Challenge Problems

For each group of unit squares, nd the mixed number that most reasonably

represents the total green shaded area.

1.

2.

3.

4.

Add the fractions, then write in mixed number notation.

5.3__4 + 1__2 6. 7__9 + 2__6 7. 2__3 + 4__5

8.13__9 + 5__8 9. 1__4 + 1__2 + 1__3 10. 7__2 + 5__4 + 3__8


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1. Which number is greater than a whole?

17__19 9__10 8__8 15__14

2. Which number is less than a whole?

23__21 37__39 5__3 16__1

3. Estimate where9__4 is located on the number line.

4. Which mixed number is equal to

83__79

?

83__79 1 4__79 83 1__79 8 7__9


1. The number11__4 and the number 2 3__4are the same value but in different notations.

Explain why both types of notation are useful. Give two examples of situations

where one notation is more convenient than the other.

2. Explain how you nd where a mixed number is located on the number line. As

your example, show where the mixed number 4 2__5 is located on a number line.Make your drawing as precise and as accurate as you can.

Find the Errors

A student made 3 mistakes below. Identify and correct each mistake.

1. 2.

3. 4.

A student wrote 4 of the numbers below in incorrect mixed number notation.Find and correct the mistakes.

5. 6. 7. 8. 9.

10. 11. 12. 13. 14.


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looking bAck

Vocabulary: xed nubers, whole nubers, ratons, addton


1. What s a xed nuber?

2. What ust I reeber when wrtng a xed nuber?3. How do I rewrte a xed nuber as a sngle raton?

4. How do I rewrte a raton as a xed nuber?

5. How an I tell a raton s greater than, less than, or equal to one?


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Lesson 43 Fractional Dierences

Objecties

Find the difference between two fractions.

Solve problems involving fraction arithmetic arising from concrete situations.

Estimate fractional values and measure the accuracy of an estimate.

Concepts and Skills


RN.13 Express mixed numbers as improper fractions. Express improper

fractions as mixed numbers.



RO.8 Add fractions with like denominators.

RO.9 Find the difference between two fractions with a common denominator.

RO.10 Add and nd the difference of fractions with unlike denominators by rst

rewriting them as equivalent fractions with a common denominator.

Remember rom Beore

How do you nd the difference between two whole numbers?

What is a mixed number?

What is estimation?


1. Use mental math to solve each equation.

a. 5 + a = 12

b. 18 = b + 3

c. 14 = 8 + c

d. 15 + d = 26

2. Estimate the fraction n_5on the number line below.

Vocabulary

aurayderene

dstane

estate


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Lesson 43 ratonal erenesratonal erenes | 417

Lesson 43 Fractional Dierences

Conepts and Sklls: R.9,

R.13, RO.6, RO.8, RO.9, RO.10

In the previous lesson we learned about mixed number notation, which provides a

fast way to see how much more a value is from a whole number. In this lesson well

discuss how to nd the distance between any two rational numbers on the number

line by calculating the difference.

Consider the following two strips of paper. The blue portions represents 1_3on thetop and 1_2on the bottom:

What is the difference between1_2 and 1_3? In other words, how much does d

represent in the picture below?

We can describe this question with the following equation:

1__3 + d = 1__2

Here the value drepresents the difference between1_3 and 1_2. An easy way to nd

this difference is to form a common denominator between1_3 and 1_2. To do this we

multiply 1_3by 2_2 to create the equivalent fraction 2_6:

Now we multiply 1_2by 3_3 to create the equivalent fraction 3_6:

Now we can use these equivalent fractions in our equation:

2__6

+ d = 3__6

In the context of our strips of paper, this equation means:


42/50


Its clear that we need to add1__6 to 2__6 to make 3__6

2__6 + 1__6 = 3__6This means d=

1__6is the solution, and therefore 1__6 is the difference between 1__3and 1__2:1__3 + 1__6 = 1__2


1. Find the difference by solving for the variable d.

a.2__10 + d = 7__10 b. 4__9 + d = 17__9

c.3__4 + d = 11__12 d. 6__10 + d = 5__6

2. How much more is3_4 than 7__10? Write an equation to describe the result.

3. How much more is1_3 than 3__10? Write an equation to describe the result.

Distance Differences are very useful for solving problems in the real world such as ndinghow far apart things are. For example, consider the following situation:

Its3_4of a mile to travel from home to school along the road shown above.

Using the same road, its 2 7_8 miles to travel from home to the movietheater. How far is it to travel from school to the movie theater?

Lets visualize this problem on the number line:

The variable drepresents the distance between school and the movie theater, which

is the value we are trying to nd. Here is the above equation written with symbols:

3__4 + d = 2 7_8

To solve this, lets rst express 27_8 as a single fraction by remembering that 2 7_8

equals 2 +7_8:


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This means 27_8equals 16__8 + 7_8. Lets add 16 + 7:

Now we can write 27_8 as a single fraction:

Since 27_8 is equivalent to 23__8, we can rewrite our equation as follows:

3__4 + d = 23__8

Now lets form a common denominator between3_4and 23__8:

This gives us the following equation:

6__8 + d = 23__8

The difference between6_8 and 23__8 is found by calculating the difference between 6

and 23:

From this we see that the solution is d=

17__8:

6__8 + 17__8 = 23__8

This means that its17__8 of a mile to travel from school to the movies.

Lets express17__8 as a mixed number using long division:

The mixed number 2 1_8 is easier to interpret than 17__8. It shows us that its just a littlemore than 2 miles to travel from school to the movie theater.


4. A full can of juice holds 31_2 liters. Maria opened the can and drank most of

it. If there is only 1_5of a liter left, how much did Maria drink? Illustrate thequestion on the number line. Write an equation. Then solve.


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5. Kevin is 6 1_6 feet tall. Vick is 5 1__2 feet tall. How much taller is Kevincompared to Vick? How much shorter is Vick compared to Kevin?

Accuracy o anEstimate

Differences are useful when evaluating the accuracy of an estimate. The difference

between an estimate and the actual value tells us how close the estimate was.

The following strip of paper shows an unspecied fraction colored blue:

When three people were asked to estimate the value of the above fraction, they

came up with three different estimates:

If the actual value is2_3, how accurate was the closest estimate?

Person As estimate was 2_9. Since 2_9 is closer to zero, we know this is not a very goodestimate. The actual value is closer to a whole.

Person Bs estimate was 4__11, which is less than a half. The actual value is more thana half, so this is probably not a very good estimate.

Person Cs estimate was6__10 which is a little more than half. This seems like a more

accurate estimate than the others. In order to get a measure of the accuracy, lets ndthe difference between

6__10and 2_3.The rst step is to form a common denominator:

Now that the fractions have a common denominator, we get the difference by

solving fordin the following equation:

18__30 + d = 20__30

The solution is d=2__30:

18__30 + 2__30 = 20__30

From this we conclude that Person Cs estimate was merely2__30away from the actual

value. Thats reasonably accurate.


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3.

4.

Solve for the variable in each equation.

5.5__8 + m = 11__8 6. 1__2 + p = 35__4

7.7__15 + g = 4__5 8. n + 2__6 = 8__15

Express the mixed number as a single fraction.

9. 11__9 10. 3 7__10 11. 6 5__7

Express the fraction as a mixed number.

12.23__7 13. 65__5 14. 165___8

Foreachnumberline,ndthedistancebetweenthetwopointsshown.Write

an equation then solve for the difference.

15.

16.

17.

18.

Solve for the variable in each equation.

19.5__6 + d = 2 1__3 20. 1 1__5 + h = 3 1__3

21. 21__2 + z = 4 2__5 22. 5__7 + b = 5 4__5


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Denevariables,writeanequation,thensolvetoanswerthequestion.

23. How much more is4__5 compared to 7__9?

24. Ned spent1__3 of his $57 allowance. How much is left?

25. The blue bucket was9__16 lled with sand. The red bucket was 7__12 lled with

sand. If the buckets are the same size, which one has more sand in it? How

much more?

26. Hugo walked 47__10 miles. Marie walked 24__5 miles. Who walked farther? How

much farther did that person walk?

27. Last years art class painted a mural 101__3 feet tall. This years class is designing

a mural that is 113__4 feet tall. How much taller will the mural be this year?

28. It was a 3-hour-long movie. The rst1__4hour was interesting. The next 1__2hour

was exciting. The next3__5of an hour was funny. The rest of the movie was dull.

How many hours of the movie were dull?

Find the closest estimate for the fraction of the rectangle that is blue.

29. If the actual value is1__8, whose estimate is the most accurate? Find the difference

between the estimate and the actual value.

30. If the actual value is6__7, whose estimate is the most accurate? Find the difference

between the estimate and the actual value.

Challenge Problems

Find the value of the variable in each equation.

1. (5__7+ 1__2) + d = 20__14 2. 1 1__3 + k = (2__3+ 5__6)3. p + (1__8+ 3__4) = 1 1__4 4. 15__2 + 8__4 = 2 3__4 + b


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Denevariables,writeanequation,andthensolvetoanswerthequestion.

5. Kim and Myra recorded how many hours they read for each day during the

week. The information is shown below in their nightly reading log. Who read

more hours? How many more hours?

Name Monday Tuesday Wednesday Thursday Friday

Kim 1__2 3__4 2 11__2 13__4Myra 2

1__4 11__2 3__4 11__4 2 3__46. After the team won the 5 games, the coach bought 3 pizzas to celebrate. One was

pepperoni, one was cheese, and the other was vegetarian. After 5 minutes,3__4 of

the vegetarian was gone,5__6of the cheese pizza was eaten, and 1__2of the pepperoni

was devoured. How much pizza was left?


1. Estimate the distance between pointj and point kon the number line below.

14__5 11__5 7__5 4__52. Which is the best estimate of the fraction of the circle that is colored green?

4

__

95

__

82__3 1__5


1. Describe the technique you use to visually estimate fractions. For example,

explain how you would estimate what fraction of the square is colored green for

each square below. Describe how you arrived at your estimates.

2. Explain what it means to nd the difference between two points on the number

line.

3. Measure how tall you are. Measure how tall your friend is. Record the heights

in units of feet using mixed number notation (each inch is1__12of a foot). What is

the difference in height between you and your friend?


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Find the Errors A student made mistakes in all 3 problems below. Identify and correct eachmistake.

1. Find the distance between3__4 and 9__4.

2.

3.

looking bAck

Vocabulary: auray, estate, derene, dstane, raton, oon

denonator, xed nuber


1. How do I fnd the derene between two ratons?

2. How do I wrte equatons to represent stuatons nvolvng

derenes?

3. How do I deterne whh estate s losest?4. How do I fnd the derene between the estate and the atual

raton?


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Summary and Reiew

Chapter 8: Fraction Sums andDierences

ChapterAccomplishments

We know how to use prime factorization to simplify fractions. We know how to nd equivalent fractions so that two fractions share a

common denominator.

We know how to compare fractions, add fractions, and nd differences offractions.

We understand mixed number notation. We know how to use whole numbers, fractions and mixed numbers to solve

problems and estimate measurements.

Vocabulary romthe Chapter

accuracy

common denominator

difference

distance

equivalent fractions

estimate

mixed number

simplied fraction

Concepts and

Skills Check

1. Simplify the fraction6__15. 2. Simplify the fraction 20__45.

3. Write90__7 as a mixed number. 4. Add the fractions 3__8 + 2__4.

5. Add the fractions2__3 + 1__5. 6. Simplify the expression:

7. Write 73__4as a single fraction. 8. Add the mixed numbers 2 1__4 + 12__5.

9. Simplify

1 million_______1 billion. 10. Solve the equation 2__5 + d = 5__6.Whats Next? Next well learn how to use decimal notation to express fractions. Decimal notation

Documents

Chapter 8 Student Textbook