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unit 3

fractions

KEYWORDS

Fraction Numerator Top number Denominator Bottom number Part of a whole

Quantity Proper fraction Improper fraction Mixed number Equivalent fractions To simplify

To cancel Cancellation Lowest terms Simplest form Common denominator Reciprocal

1. FRACTIONS You use a fraction to describe a part of a whole. A natural number divided by another natural number. The division symbol is a fraction bar (---) or sometimes a slash ( / ). In any fraction, the “top number” is called the numerator and the “bottom number” is called the denominator. The denominator of a fraction cannot be zero. (Division by zero is undefined.) The denominator tells us how many equal parts the whole is divided into and the numerator tells us how many of these parts there are. How do you read fractions?

You can see fractions in shops, on bills, in newspapers and recipes. It's a good idea to recognise fractions when they are written as words.

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fraction word plural

21 One (a) half halves

31 One third thirds

41 One quarter or a fourth quarters or fourths

51 One fifth fifths

61 One sixth sixths

71 One seventh sevenths

81 One eighth eighths

91 One ninth ninths

101 One tenth tenths

It's easy to make the words, apart from the first four. All the other fractions are like the numbers but with a 'th' sound added at the end. Even easier is making the plurals, as long as you are careful with the plural of

21 .

Just put an 's' on the end of the other fractions

Here are some more examples of fractions in words.

Words Figures Words Figures

two fifths 52 three eighths

83

four fifths 54 two thirds

32

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Can you see how it works? The first number goes on the top, the second number on the bottom.

If the number in the denominator is greater than 10 you have two forms to read

the fraction:

.

A fraction is a proper fraction if the numerator is smaller than the denominator. The fraction represents a number less than one.

Example: 5/9 (five ninths)

A fraction is an improper fraction if the numerator is larger than the denominator. The fraction represents a number greater than one.

Example: 15/2 (fifteen halves)

If the numerator is equal to the denominator, the fraction is equal to one.

An improper fraction can be convert to a mixed number (a natural number followed by a proper fraction) dividing the numerator by the denominator, the quotient is the natural number, the remainder is the new numerator and the denominator is the same.

Example:

(four and one fifth)

To convert a mixed number to an improper fraction: Multiply the natural number by the denominator. Add the numerator. This is the new numerator. Keep the same denominator.

2. EQUIVALENT FRACTIONS Two fractions are equivalent if they have the same value.

Examples:

;

;

HOW DO YOU OBTAIN EQUIVALENT FRACTIONS?

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1.- Reduce or simplify a fraction: Divide numerator and denominator by a common factor.

Example:

(we divide the numerator and the denominator by 2)

2.- Simplify to lowest terms: Reduce a fraction so the numerator and the denominator have no common factors, dividing by the highest common factor (Irreducible fraction)

Example:

3.- Amplify a fraction: Multiply numerator and denominator by the same number.

Example:

4.-Reduce to common denominator: Change the fractions to equivalent fractions with the same denominators.

a) Chose a common multiple to use as common denominator.

b) If you use the L.C.M. then you work with the smallest numbers.

c) Find equivalent fractions with this denominator.

Exercise 1:

Calculate an equivalent amplified fraction and another equivalent simplified fraction of the following:

Exercise 2:

Copy and complete:

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Exercise 3:

Reduce to common denominator the following fractions:

3. COMPARING FRACTIONS HOW DO YOU ORDER FRACTIONS?

1.- Fractions with the same numerator: If two fractions have the same numerator, the bigger fraction is the one with the smallest denominator.

Example:

2.-Fractions with the same denominator: If two fractions have the same denominator, the bigger fraction is the one with the biggest numerator.

Example:

3.-Fractions with different numerators and denominators: You have to reduce the fractions to common denominator.

Exercise 4:

Copy and complete with the sign < or >.

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Exercise 5:

Put the following fractions in increasing order:

4. OPERATIONS WITH FRACTIONS

ADDITION AND SUBTRACTION:

1.-To add or subtract fractions with the same denominators, add or subtract the numerators and keep the denominators.

Examples:

You must change mixed numbers to improper fractions before you add or subtract.

2.-To add or subtract fractions with different denominators, rewrite the fractions as equivalent fractions with common denominators. Then follow the rule for addition and subtraction of fractions with the same denominators.

Examples:

We use the L.C.M. of the

denominators

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MULTIPLICATION:

Multiply the numerators and multiply the denominators. You must change mixed numbers to improper fractions before you multiply.

Examples:

YOU MUST

SIMPLIFY !!!!

c)

DIVISION:

Multiply by the RECIPROCAL (the inverted form) of the divisor. Mixed numbers must be changed to improper fractions before they can be inverted.

Examples:

(Cross multiply)

Exercise 6:

Calculate and simplify the following operations:

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Exercise 7:

Calculate and reduce:

a)51 -

21 +

31

b)41 +

31 -

21

c) 1 + 21 -

43

d)32 +

21 - 3

e)31 - 2 +

61

f)21 +

31 - 2

Exercise 8:

Operate with fractions:

a) ¸¹·

¨©§

61 +

32 - 2 b) ¸

¹·

¨©§

¸¹·

¨©§

41 - 3 +

32 - 3

c) ¸¹·

¨©§

31 +

21 - 3 d) ¸

¹·

¨©§

¸¹·

¨©§

21 +

32 -

41 -

23

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Exercise 9:

Calculate and reduce:

a)41 +

31 -

21 : 2 ¸

¹·

¨©§ b) ¸

¹·

¨©§

¸¹·

¨©§

21 :

31 +

21 :

23

c) ¸¹·

¨©§

32

41 : 3 +

32 ·

5. A FRACTION OF AN AMOUNT

To calculate the fraction of an amount (a number) you divide the number by the denominator and multiply the result by the numerator.

Example:

Exercise 10:

Calculate the following fractions of amounts:

a) 2/3 of 15 b) 3/4 of 80 c) 3/20 of 400 d) 1/3 of 60

e) 2/7 of 21 f) 3/5 of 80 g) 5/4 of 16 h) 3/5 of 75

Exercise 11:

If Paul ran miles each day last week, how many miles did he run?

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Exercise 12:

Mary studied an hour on Monday and

an hour on Tuesday. How

much more did Mary study on Tuesday?

Exercise 13:

Peter wants to lose 10 kilograms. After two months he has lost of this amount. How

much weight has he lost?

Exercise 14:

Pat uses cups of sugar to make a pie. How many pies can be made from

15 cups of sugar?

Exercise 15:

On Friday, it snowed inches and on Saturday it snowed

inches. What was the

total snowfall for the two days?

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Exercise 16:

Which is greater; the product of or the quotient of

?

Exercise 17:

Jane spent 4/5 of her pocket money on a DVD player. Her sister spent 10/15 of her pocket money on clothing. Did they spend the same portion of their money?

Exercise 18:

Mike is 133 cm tall. If he is as tall as Jim, how tall is Jim?

Exercise 19:

What fraction of a day is 3 hours? How many minutes are in 65

of

an hour?

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Exercise 20:

Carlos spends 3 hours on homework each night. He spends ¼ of his time on Math, 1/3 on Social Studies, 1/6 on Science and 2/8 on English Language. Which two subjects does he spend an equal amount of time working on? How much time does Carlos spend working on each subject?

Exercise 21:

Twenty minutes before the school play, 4/9 of the auditorium was full. Ten minutes later, 7/8 of the auditorium seats were occupied. What fraction of the people arrived during those 10 minutes?

Exercise 22:

Henry made nine apple pies and five chocolate pies for the school festival. What fraction of the pies were apple pies?

Exercise 23:

If students spend 10 months in school, then what fraction of the year are students off?

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Exercise 24:

Sam gave Jesse 21

of her chocolate bar and

Jesse gave Abbey 32

of his piece. What fraction

of the original chocolate bar did Abbey get?

Exercise 25:

Jeremy gave 21

of his candy bar to 4 friends. If his

friends shared the piece equally, then what fraction of the original candy bar did each one get?

Exercise 26:

Justin gave 32

of his pizza to 4 friends who shared the pizza

equally. What fraction of the original pizza did each one get?

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Across 5. Write

as a mixed number

7. When adding fractions with ________ denominators, you must find the least common denominator

8. Reduce

to lowest terms

11. Equivalent fractions have the ________ value but different names 12. Reducing a fraction to lowest terms in _________step requires dividing its numerator and denominator by the greatest common factor 13. When comparing fractions with the same denominator, you only have to compare the ________

14. Reduce

to lowest terms

15. What is the reciprocal of ?

16. When adding or subtracting fractions, we ________ add or subtract denominators Down

1. To find an equivalent fraction for a certain fraction, you could _________ its numerator and denominator by the same number

2. Write 2 as an improper fraction

3. The numerator of an improper fraction is ________ than its denominator 4. Mixed numbers have a _________ number part and a fraction part 5. The least common denominator of 15 and 20

6. and

are _________ fractions

7. If a fraction is not in lowest terms you could ________ its numerator and denominator by the same number to get an equivalent fraction

9.

and are equivalent fractions. What is the value of x?

10. Any whole number could be written as a fraction with a _________ of one