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Fractions Packet, Study Guides, Projects, Research of Mathematics

Ateneo de Davao University (ADDU)Mathematics

The denominators are different numbers. Therefore, change to equivalent fractions. See page 25. Simplifying and reducing completes addition and.

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Fractions Packet

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Fractions

Packet

Contents

Intro to Fractions…………………………………….. page 2

Reducing Fractions………………………………….. page 13

Ordering Fractions…………………………………… page 16

Multiplication and Division of Fractions………… page 18

Addition and Subtraction of Fractions………….. page 26

Answer Keys………………………………………….. page 39

Note to the Student: This packet is a supplement to your textbook

Partial preview of the text

Download Fractions Packet and more Study Guides, Projects, Research Mathematics in PDF only on Docsity!

Fractions Packet

Fractions

Packet

Intro to Fractions…………………………………….. page 2

Reducing Fractions………………………………….. page 13

Ordering Fractions…………………………………… page 16

Multiplication and Division of Fractions………… page 18

Addition and Subtraction of Fractions………….. page 26

Answer Keys………………………………………….. page 39

Note to the Student: This packet is a supplement to your textbook

Fractions Packet

Intro to Fractions

Reading Fractions

Fractions are parts. We use them to write and work with amounts that are less

than a whole number (one) but more than zero. The form of a fraction is one

number over another, separated by a fraction (divide) line.

i.e.

, and 4

These are fractions. Each of the two numbers tells certain information about

the fraction (partial number). The bottom number (denominator) tells how many

parts the whole (one) was divided into. The top number (numerator) tells how

many of the parts to count.

says, “Count one of two equal ports.”

says, “Count three of four equal parts.”

says, “Count five of nine equal parts.”

Fractions can be used to stand for information about wholes and their parts:

EX. A class of 20 students had 6 people absent one day. 6 absentees are

part of a whole class of 20 people.

represents the fraction of people

absent.

EX. A “Goodbar” candy breaks up into 16 small sections. If someone ate 5

of those sections, that person ate

of the “Goodbar”.

Fractions Packet

Complementary Fractions

Fractions tell us how many parts are in a whole and how many parts to count.

The form also tells us how many parts have not been counted (the complement).

The complement completes the whole and gives opposite information that can

be very useful.

says, “Count 3 of 4 equal parts.” That means 1 of the 4 was not counted and

is somehow different from the original 3.

implies another

(its complement). Together,

make 4

and 4

, the whole

thing.

says, “Count 5 of 8 equal parts.” That means 3 of the 8 parts have not been

counted, which implies another

, the complement. Together,

and 8

make

which is equal to one.

Complementary Situations

It’s 8 miles to town, We have driven 5 miles. That’s

of the way, but we still

have 3 miles to go to get there or

of the way.

= 8

= 1 (1 is all the way to town).

A pizza was cut into 12 pieces. 7 were eaten

. That means there are 5 slices

left or

of the pizza.

= 12

= 1 (the whole pizza).

Mary had 10 dollars. She spent 5 dollars on gas, 1 dollar on parking, and 3

dollars on lunch. In fraction form, how much money does she have left?

Gas =

, parking =

, lunch =

= 10

is the complement (the leftover money)

Altogether it totals

or all of the money.

Fractions Packet

Exercise 2 (answers on page 39)

Write the complements to answer the following questions:

1. A cake had 16 slices. 5 were eaten. What fraction of the cake was

left?

2. There are 20 people in our class. 11 are women. What part of the class

are men?

3. It is 25 miles to grandma’s house. We have driven 11 miles already.

What fraction of the way do we have left to go?

4. There are 36 cookies in the jar. 10 are Oreos. What fraction of the

cookies are not Oreos?

Reducing Fractions

If I had 20 dollars and spent 10 dollars on a CD, it’s easy to see I’ve spent half

of my money. It must be that

. Whenever the number of the part (top)

and the number of the whole (bottom) have the same relationship between

them that a pair of smaller numbers have, you should always give the smaller

pair answer. 2 is half of 4. 5 is half of 10.

is the reduced form of

and

and many other fractions.

A fraction should be reduced any time both the top and bottom number can be

divided by the same smaller number. This way you can be sure the fraction is as

simple as it can be.

both 5 and 10 can be divided by 5

describes the same number relationship that

did, but with smaller

numbers.

is the reduced form of

both 6 and 8 can be divided by 2.

Fractions Packet

Hint 3

If the 2 numbers of the fraction end in 0 and/or 5, you can divide by

Hint 4

If both numbers end in zeros, you can cancel the zeros in pairs, one from the

top and one from the bottom. This is the same as dividing them by

for each

cancelled pair.

Hint 5

If you have tried to cut the fraction by

, 3

, 5

and gotten nowhere, you

should try to see if the top number divides into the bottom one evenly. For

, none of the other hints help here, but 69 23 = 3. This means you can

reduce by

. 3

For more help on reducing fractions, see page 13

Exercise 4 (answers on page 39)

Directions: Reduce these fractions to lowest terms

100

5000

Fractions Packet

Higher Equivalents

There are good reasons for knowing how to build fractions up to a larger form.

It is exactly the opposite of what we do in reducing. If reducing is done by

division, it makes sense that building up should be done by multiplication.

A fraction can be built up to an equivalent form by multiplying by any form of

one, any number over itself.

All are forms of 3

; all will reduce to 3

Comparing Fractions

Sometimes it is necessary to compare the size of fractions to see which is

larger or smaller, or if the two are equal. Sometimes several fractions must be

placed in order of size. Unless fractions have the same bottom number

(denominator) and thus parts of the same size, you can’t know for certain which

is larger or if they are equal.

Which is larger

or 6

? Who knows? A ruler might help, but rulers aren’t

usually graduated in thirds or sixths. Did you notice that if 3 were doubled, it

would be 6?

Fractions Packet

Mixed Numbers

A “mixed” number is one that is part whole number and part fraction.

3 are samples of mixed numbers. Mixed numbers have to be

written as fractions only if you’re going to multiply or divide them or use them

as multipliers or divisors in fraction problems. This change of form is easy to

do. Think about

3. That’s 3 whole things and half another. Each of the 3

wholes has 2 halves ( 1

). The number 3 is 1+1+1 or.

That’s

and, with the original

, there’s a total of

. You don’t have to think of

every one this way; just figure the whole number times the denominator

(bottom) and add the numerator (top).

Exercise 6 (answers on page 39)

Change these mixed numbers to “top heavy” fractions:

These “top heavy” forms are “work” forms, but they are not usually acceptable

answers. If the answer to a calculation comes out a top heavy fraction, it will

have to be changed to a mixed number. This can be done by reversing the times

and plus to divide and minus.

3 became

by

can go back to

3 by dividing 7 and 2.

Fractions Packet

The answer is the whole number 3. The remainder 1 is the top number of

the fraction and the divider 2 is the denominator (bottom fraction number).

Exercise 7 (answers on page 39)

Reduce these top heavy fractions to mixed numbers.

Top heavy fractions may contain common factors as well. They will need to be

divided out either before or after the top heavy fraction is changed to a mixed

number.

but

can be divided by

. Then 8

= 4

If you had noticed that both 26 and 8 are even, you could divide out

right

away and then go for the mixed number. Either way, the mixed number is the

same.

Exercise 8 (answers on page 39)

Fractions Packet

Reducing Fractions

Divide by 2 if…

The top AND bottom numbers are EVEN numbers

Like:

32 , 26

14 , 4

Divide by 3 if …

The sum of the top numbers can be divided by 3 AND the sum of the

bottom numbers can be divided by 3

Like:

15 canbedividedevenlyby 3

12 canbedivided evenlyby 3

7 6 2 15

5 6 1 12

762

561

Divide by 5 if…

The top AND bottom numbers end in 0 or 5

Like:

460

255

60 , 15

5 ,

Divide by 10 if…

The top AND bottom numbers end in 0.

Like:

440

320 , 260

140 , 40

Divide by 25 if…

The top AND bottom numbers end in 25 or 50 or 75 or 100

Like:

4500

3275 , 275

150 , 400

225

Fractions Packet

Divisibility RULES!

Dividing by 3

Add up the digits: if the sum is divisible by three, then the number

divides by three.

Ex.

9 3

6 0 3

2 0 7

603

207 therefore 603

207 divides by 3

Dividing by 4

Look at the last two digits. If they are divisible by four, then the number

divides by four.

Ex.

36 4

24 4

136

124 therefore 136

124 divides by 4

Dividing by 6

If the digits can be divided by two and three, then the number divides by

six

Ex.

903

306

1806

612

1806

612

And therefore 1806

612 divides by 6

602

204

1806

612

1806

612

Dividing by 7

Take the last digit, double it, and then subtract it from the other

numbers. If the answer is divisible by seven, then the number divides by

seven.

Ex.

31 - 10

28 - 14

315

287

7 therefore 315

287 divides by 7

Dividing by 8

If the last three digits are divisible by eight then the number divides by

eight.

Ex.

160 8

104 8

3160

2104 therefore 3160

2104 divides by 8

Fractions Packet

ORDERING

Fractions

Being able to place numbers in order (smallest to largest or largest to smallest)

is fundamental to the understanding of mathematics. In these exercises we will

learn how to order fractions.

Ordering Fractions

There are several ways to order fractions. One way is to use common sense.

This method can be simple but requires some general knowledge. If nothing

else, it can be used as a starting point to finding the necessary order.

Take a look at the following examples:

Ex. Place the following fractions from smallest to largest order

The larger the number on the bottom of a fraction (assuming the numerator is

the same for all the fractions), the smaller the fraction is. In the above

example,

is the smallest fraction because the 5 is the largest denominator.

Next in order would be the

because the 3 is the next largest denominator.

This leaves the

, which has the smallest denominator. Therefore, the order

for these fractions is:

Ex. Place the following fractions from smallest to largest

The larger bottom number here is the 6 in

. But the student should ask, “Is this the

smallest fraction?” By inspection, it does not seem to be. But with fractions of this sort

(different numerators), students run into the most problems when ordering.

Fractions Packet

Another way to order fractions is to find common denominators for all the

fractions; build up the fractions; then compare the top numbers (numerators)

of all the fractions.

Look at the following example:

Ex. Order the following fractions from smallest to largest

Fractions Packet, Study Guides, Projects, Research of Mathematics

Related documents

Partial preview of the text

Download Fractions Packet and more Study Guides, Projects, Research Mathematics in PDF only on Docsity!

Fractions

Packet

Contents

Intro to Fractions…………………………………….. page 2

Reducing Fractions………………………………….. page 13

Ordering Fractions…………………………………… page 16

Multiplication and Division of Fractions………… page 18

Addition and Subtraction of Fractions………….. page 26

Answer Keys………………………………………….. page 39

Reading Fractions

Fractions are parts. We use them to write and work with amounts that are less

than a whole number (one) but more than zero. The form of a fraction is one

number over another, separated by a fraction (divide) line.

i.e.

These are fractions. Each of the two numbers tells certain information about

the fraction (partial number). The bottom number (denominator) tells how many

parts the whole (one) was divided into. The top number (numerator) tells how

many of the parts to count.

says, “Count one of two equal ports.”

says, “Count three of four equal parts.”

says, “Count five of nine equal parts.”

Fractions can be used to stand for information about wholes and their parts:

EX. A class of 20 students had 6 people absent one day. 6 absentees are

part of a whole class of 20 people.

represents the fraction of people

absent.

EX. A “Goodbar” candy breaks up into 16 small sections. If someone ate 5

of those sections, that person ate

of the “Goodbar”.

Complementary Fractions

Fractions tell us how many parts are in a whole and how many parts to count.

The form also tells us how many parts have not been counted (the complement).

The complement completes the whole and gives opposite information that can

be very useful.

says, “Count 3 of 4 equal parts.” That means 1 of the 4 was not counted and

is somehow different from the original 3.

implies another

(its complement). Together,

, the whole

thing.

says, “Count 5 of 8 equal parts.” That means 3 of the 8 parts have not been

counted, which implies another

, the complement. Together,

make

which is equal to one.

Complementary Situations

It’s 8 miles to town, We have driven 5 miles. That’s

of the way, but we still

have 3 miles to go to get there or

of the way.

= 1 (1 is all the way to town).

A pizza was cut into 12 pieces. 7 were eaten

. That means there are 5 slices

left or

of the pizza.

= 1 (the whole pizza).

Mary had 10 dollars. She spent 5 dollars on gas, 1 dollar on parking, and 3

dollars on lunch. In fraction form, how much money does she have left?

Gas =

, parking =

, lunch =

is the complement (the leftover money)

Altogether it totals

or all of the money.

Exercise 2 (answers on page 39)

Write the complements to answer the following questions:

1. A cake had 16 slices. 5 were eaten. What fraction of the cake was

left?

2. There are 20 people in our class. 11 are women. What part of the class

are men?

3. It is 25 miles to grandma’s house. We have driven 11 miles already.

What fraction of the way do we have left to go?

4. There are 36 cookies in the jar. 10 are Oreos. What fraction of the

cookies are not Oreos?

Reducing Fractions

If I had 20 dollars and spent 10 dollars on a CD, it’s easy to see I’ve spent half