fractions and multiples

Index Divisibility Fractions Simplest form Operations Exercises

Factors and Fractions

Matematicas 2o E.S.O.Alberto Pardo Milanes

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1 Divisibility

2 Fractions

3 Simplest form of a fraction

4 Operations

5 Exercises

Alberto Pardo Milanes Factors and Fractions


Divisibility



Divisibility

Factors and multiples

A factor of a number n, is a number d which divides n.Read ⇐⇒ if and only if.

d is a factor of n ⇐⇒d is a divisor of n ⇐⇒d divides n ⇐⇒n is divisible by d ⇐⇒n is a multiple of d.

Examples:−7 divides 14 ⇐⇒−7 is a factor of 14 ⇐⇒14 is divisible by −7 ⇐⇒14 is a multiple of −7.



Divisibility

Primes

A prime number is a positive number that has only two positivefactors 1 and the number itself (1 is not considered a primenumber as it only has one positive factor). A number with morethan two positive factors it is called composite number.

Examples: 3 is a prime number because has only two positive fac-tors (1 and 3). 6 is a composite number because has four positivefactors (1, 2, 3 and 6).

Two numbers are relatively prime if they have no common positivedivisors except 1.

Example: 6 and 25 are relatively prime because positive factorsof 6 are 1, 2, 3, 6 and positive factors of 25 are 1, 5, 25.



Divisibility

Prime decomposition

Prime decomposition is to find the set of prime factors of aninteger: To factorize a number you have to express the number asa product of its prime factors.To factorize negative numbers use also −1.

Examples:• 90 = 2 · 45 = 2 · 3 · 15 = 2 · 3 · 3 · 5 = 2 · 32 · 5.• −25 = −1 · 25 = −1 · 5 · 5 = −1 · 52.



Divisibility

GCD and LCM

The Greatest Common Divisor (GCD) is the highest number thatis a common factor of two or more numbers. It is clear that ifGCD(a, b) = 1, a and b are relatively prime.

Examples: GCD(42, 110) = 2, because positive factors of 42are 1, 2, 3, 6, 7, 14, 21, 42, and positive factors of 110 are1, 2, 5, 10, 11, 22, 55, 110.12 and 35 are relatively prime, because GCD(12, 35) = 1.

The Least Common Multiple (LCM) is the lowest positive numberthat is a common multiple of two or more numbers.

Example: LCM(6, 9) = 18, because positive multiples of 6 are6, 12, 18, 24, . . . and positive multiples of 9 are 9, 18, 27, . . .



Fractions



Fractions

What’s a fraction?

A fraction is a number that represents a part of something.

Fractions are written in the forma

b, where a and b are integers,

and the number b is not zero. Reada

ba over b.

The number a is called numerator, and the number b is calleddenominator.

Example:13

25, 13 is the numerator and 25 is the denominator.



Fractions

Proper fractions and improper fractionsA proper fraction is a fraction that is less than one. A fractiongreater than one is called an improper fraction.

Examples:12

17is a proper fraction.

23

15is an improper fraction.

Equivalent fractionsEquivalent fractions are different fractions that name the sameamount.

Example:3

7=

6

14

because they represent the same amount.



Fractions

Amplify and reduce fractions

To find equivalent fractions, multiply or divide the numerator andthe denominator by the same number. If you multiply by the samenumber, you amplify the fraction. If you divide by the same factor,you reduce the fraction.

Examples:

• To amplify6

9we

multiply the numerator and the denominator by 10:6

9=

60

90.

• To reduce6

9we

divide the numerator and the denominator by 3:6

9=

2

3.



Simplest form of a fraction




Lowest Terms Fraction

A lowest terms fraction is a fraction that can not be reducedanymore. To write a fraction in the simplest form find the lowestterms fraction. To reduce a fraction to the lowest terms fraction,we can use two methods:

• Divide the numerator and the denominator by the GreatestCommon Factor.

• Divide the numerator and the denominator by any commonfactor and keep dividing until they are relatively prime.




Examples:

• To obtain the lowest terms fraction of70

105we can divide

70 and 105 by GCD(70, 105) = 35, so70

105=

2

3.

• To reduce70

105to the lowest terms fraction we can di-

vide by 5 and keep dividing until there are no common factors

but one70

105=

14

21=

2

3, (note 2 and 3 are relatively prime ).



Operations



Operations

Add and subtract

To add or subtract fractions with like denominators, add thenumerators and keep the same denominator. To add or subtractfractions with unlike denominators, first amplify these fractions toobtain like denominators (using the LCM or any multiple).Simplify, if possible.

Examples:

• 4

15+

8

15=

12

15=

4

5.

• To add3

15+

1

6using the LCM:

3

15+

1

6=

6

30+

5

30=

11

30.

Because LCM(15, 6) = 30.• Using another common multiple of 15 and 6, for example 60:3

15+

1

6=

12

60+

10

60=

22

60=

11

30.



Operations

Multiply and divide

To multiply fractions, multiply the numerators and multiply thedenominators then simplify.

Example:12

5· 314

=12 · 35 · 14

=36

70=

18

35.

You can get the reciprocal of a fraction by switching its numeratorand denominator.

Example: The reciprocal of14

21is

21

14.

To divide by a fraction,multiply by its reciprocal and simplify,ifpossible.

Example:2

8:4

17=

2

8· 174

=2 · 178 · 4

=34

32=

17

16.



Exercises



Exercises

Exercise 1

There are 28 students in our class and we want to divide it intogroups with equal number of students. How many ways can theclass be divided into groups? What are they?



Exercises

Exercise 2

Mary wants to serve hot dogs for 48 people. Sausages come inpackages of 8 and hot dog buns come in packages of 12. She wantsto have enough to serve everyone and have none left over, howmany packages of sausages and hot dog buns should she purchase?



Exercises

Exercise 3

Peter works in a florist. Today He is making identical floralarrangements for a bridal party. He has 84 daisies, 66 lilies, and 30orchids. He wants each arrangement to have the same number ofeach flower. What is the greatest number of arrangements that hecan make if every flower is used?



Exercises

Exercise 4

Samantha loves the sea. She has kayaking lessons every fifth dayand diving lessons every seventh day. If she had a kayaking lessonand a diving lesson on June the sixth, when will be the next dateon which she has both kayaking and diving lessons?



Exercises

Exercise 5

There are two flashing neon lights. One blinks every 4 seconds andthe other blinks every 6 seconds. If they are turned on exactly atthe same time, how many times will they blink at the same time ina minute?



Exercises

Exercise 6

Peter sells books. He made 240e selling children’s books, 140efrom cookbooks, and 280e from paperback books. He gets exactlythe same benefit from each book. What is the most that Petercould get for each book? How many books could Peter have soldthen?



Exercises

Exercise 7

Complete:

A half=

5

2=

Two thirds=

1

3=

A quarter=

3

4=

Three fifths=

1

5=

Two quarters=

1

7=

An/one eighth=

3

6=

One and a half=

41

3=

Two and a quarter=

42

5=

Two and four tenths=

11

12=



Exercises

Exercise 8

Write a fraction in simplest form beside each sentence:There are forty-two girls out of seventy-two people:

The baby snake was just four and three quarters inches long:

Fifteen out of the twenty students has dogs as pets:

I walk five and a quarter miles a day:



Exercises

Exercise 9

An orchard has 72 peach trees 15 apple trees, and 23 lemon trees.In simplest form, what fraction of the trees are peach trees?



Exercises

Exercise 10

George swims seven eighths miles and Mat swims ten twelfthsmiles. Who swims farther?



Exercises

Exercise 11

Compare these fractions using the least common denominator:2

−6,3

4,

9

15,7

5,−1−3

,−210

,13

12.



Exercises

Exercise 12

Lance and Frank ate seven twelfths of a pizza. If Frank ate a thirdof the pizza, how much of the pizza did Lance eat?



Exercises

Exercise 13

Tangram means seven boards of skill.The Tangram is a Chinesepuzzle consisting of seven flatshapes (called tans) which areput together to form differentshapes. Study the tans at theright forming a square. Theside of the square is 1 cm, sothe area is 1 cm2. Find thefractional value of each piece.



Exercises

Exercise 14

Of the 20 students, a quarter of the students wear jeans, a fithwear shorts and a tenth wear a skirt. How many students wearsomething else?



Exercises

Exercise 15

My car´s gas tank was empty and now is full but I paid forty eurosfor it. To go to Marbella I need two fifths of the tank, how muchmoney is this fraction of the tank?



Exercises

Exercise 16

Peter cut six apples into quarters, how many pieces of apple did hehave?



Exercises

Exercise 17

Mike cleaned1

4of the 24 m2 yard . How many are left?



Exercises

Exercise 18

Thomas drunk3

5cup of milk, and now he has 24 cl left. How

many ml did the cup have originally?



Exercises

Exercise 19

Eric is reading a book with 195 pages. He read2

5of the pages

today. How many are left?



Exercises

Exercise 20

Of Paul’s stone collection1

5are white stones. He has total of 75

stones. How many of them are not white?



Exercises

Exercise 21

Mom baked cookies, and gave1

2of them to Beth, and of the

remaining ones, she gave2

3to Seth. The rest, which was 4

cookies, she ate herself. How many did she bake originally?



Exercises

Exercise 22

This time Mom baked again cookies, and gave1

2of them to Beth,

and of the remaining ones she gave1

3to Seth. There were 2

cookies left. How many did she bake originally?


fractions and multiples

Education

improper fractions

fractions matemticas

dierent fractions

prime number

number b

nd equivalent fractions

number n

number d