FRACTION AND MIXED NUMBERS

THE FUNDAMENTAL PROPERTY OF FRACTIONS

A fraction describes a part of a whole. The whole must be divided into equal parts.

Fractions are number that represent one or more equal parts of a whole. FRACTIONS: 1/2, 2/3, 3/5.

Fractions can be also be used to name a part of a collection or a set of objects or a group.

Example,

Each pencil is 1 out of 4 in the group, i.e. ¼ of the group.

Fractions can be presented with diagrams.

Example,

The shaded parts in the diagram represent 8 parts out 9 equal parts.

i.e. 8/9 of the diagram.

In a fraction, 1/6

The top number shows how many parts you have.

The bottom number shows how many parts there are altogether.

The top number is the numerator.

The bottom number is the denominator.

EQUIVALENT FRACTIONS

Equivalent fractions are fractions that represent the same value although they have different numerators and denominators.

An equivalent fraction for a given fraction can be obtained by multiplying or dividing both the numerator and denominator of the given fraction by the same number.

EQUIVALENT FRACTIONS: 1/2 = 2/4= 4/8.

5/8=10/16=15/24.

MIXED NUMBERS

A mixed number consists of a whole number and a fraction.

Example mixed numbers:

1 ½, 3 1/4, 154 5/8.

A mixed number can be represented by using a diagram and a number line can be use to arrange the mixed numbers in order.

PROPER FRACTION

Proper fractions are fractions in which the numerator (top) is less than the denominator (bottom).

Example,

1/7, 2/5, 4/9.

The value of a proper fraction is less than 1.

IMPROPER FRACTION

Improper fractions are fractions in which the numerator is equal to or greater than the denominator.

Example,

7/ 2, 11/7, 25/4.

The value of an improper fraction is equal to or greater than 1.

MULTIPLYING FRACTIONS

Multiplication of Fractions.

The multiplication of a whole number by a fraction or a mixed number is the repeated addition of the fraction or the mixed number.

Example

4 x 2/3 = 2/3 + 2/3 + 2/3 + 2/3

= 8/3

= 2 2/3.

When multiplying a fraction by a whole number:

a)

b) Multiply the numerator of the fraction by that whole number.

c) Simplify the answer to the lowest terms whenever possible.

Example,

Evaluate 8 x 4/5

Solution

8 x4/5 = 8x 4 / 5

= 32 /5

=6 2/5

When multiplying a fraction with another fraction:

a) Multiply the numerator by the numerator.

b) Multiply the denominator by the denominator.

c) Simplify the answer to the lowest terms whenever possible.

Example,

1/5 x 6/11

=1x 6/ 5x 11

=6/55.

DIVIDING FRACTIONS

Division of Fractions

A fraction is also a division between the numerator and denominator.

The division of a whole number by a fraction is the process of finding the number of times the fraction is contained in that whole number.

The division of a fraction by a whole number is actually the equal sharing of a fraction.

When dividing a fraction by a whole number:

a) Multiply the fraction with the reciprocal of the whole number.

b) Simplify the answer to the lowest terms whenever possible.

Example

2/3 ÷ 6=

2/3 ÷ 6=2/3 x 1/6

=1/9

When dividing a fraction by another fraction:

The division is done by multiplying the first fraction by the reciprocal of the divisor.

Solve 5/16 ÷25/256

Solution

5/16 ÷ 25/256

=5/16 x256/25

=16/5

=3 1/5.

When dividing a fraction by a mixed number:

a) Change the mixed number into an improper fraction.

b) Then, multiply the first fraction by the reciprocal of the divisor.

Example,

1 5/6 ÷ 3 2/3.

1 5/6 ÷ 3 2/3

=11/6 ÷11/3

=11/6 x 3/11

= 1/2

Example,

¾ x 7/18 ÷ 49/27

= ¾ x 7/18 x 27/49

=9/ 56

ADDING AND SUBTRACTING FRACTIONS

Addition of Fractions

Addition of fractions is the process of finding the sum of two or more fractions.

To add fractions having the same denominator:

a) Retain the denominator.

b) Add the numerators only.

c) Simplify the answer to the lowest terms whenever possible.

Example,

15/25 + 5/25

=15+5 /25

=20/ 25

=4/5

SUBTRACTION OF FRACTIONS

Subtraction of fractions is the process of finding the different between fractions.

To subtract fractions having the same denominator:

a) Retain the denominator.

b) Subtract only the numerators

Example

8/15 – 2/15

=8-2 /15

=6 /15

=2/5

To subtract fractions that has different denominators:

Find the LCM of the denominators to get the common denominator for fractions.

Example,

7/9 -2/5

=7X5 / 9X5 - 2X9/ 5X9

=35-18 /45

=17/45

MULTIPLYING AND DIVIDING MIXED NUMBERS

When multiplying a mixed numbers and a whole number:

a) Change the mixed number into an improper fraction.

b) Then, multiply the numerator with the whole number.

c) When multiplying two mixed numbers : Change both the mixed numbers into improper fractions before multiplying them.

d) Simplify the answer to the lowest terms whenever possible.

Example, 7 x 3 1/14

Solution

7x 3 1/14 =7 x 43/14

=7x 43 /14

=43/2

=21 ½.

Solve

4 2/5 x 3 4/7

= 22/5 x 25/7

=110 /7

=15 5/7

When multiplying a mixed numbers and a whole number:

e) Change the mixed number into an improper fraction.

f) Then, multiply the numerator with the whole number.

g) Simplify the answer to the lowest terms whenever possible.

Example, 7 x 3 1/14

Solution

7x 3 1/14 =7 x 43/14

=7x 43 /14

=43/2

=21 ½.

When dividing a fraction by a mixed number:

c) Change the mixed number into an improper fraction.

d) Then, multiply the first fraction by the reciprocal of the divisor.

Example,

1 5/6 ÷ 3 2/3.

1 5/6 ÷ 3 2/3

=11/6 ÷11/3

=11/6 x 3/11

= 1/2

ADDING AND SUBTRACTING MIXED NUMBERS

To add fractions and mixed numbers

a) Add the fractions and the whole numbers separately.

b) Add the fractions before adding the whole numbers.

Example,

3 3/5 + 13/25

=3 + 3/5 + 13/25

=3 + 3x5 /5x5 + 13/25

=3 + 15+13/25

=3 +28/25

=3+ 1 3/25

=4 3/25.

To add a mixed number to another mixed number:

Change the mixed numbers into improper fractions first, and then add them up.

Example, solve 5 1/3 + 4 2/5

=16/3 +22/5

= (16 x5) + (22 x 3) / 15 =80+66 /15

=146/15

=9 11/ 15

To subtract a mixed number from another mixed number that is greater

Change the mixed numbers into improper fractions first and then subtract them.

Example,

20 7/8 -18 2/7

=167/8 - 128/7

=1169 -1024 / 56

=145 /56

=2 33/56.

Solve 13 2/3 + 7 – ¼

= (13 +7) + (2/3 – ¼)

=20 + 8-3/12

=20 + 5/12

=20 5 /12.

DECIMALS

Decimals are fractions whose denominator is a multiple of 10, that is 10, 100,

1 000 and so on.

Example, the shade areas represent 8 of 10 parts, that is 8 /10 parts.

1 of 10 parts = 1/ 10

= 0.1

Since: 1/ 10 =0.1

Therefore: 8/ 10 = 0.1 x 8 = 0.8

Hence, decimal and fraction are interchangeable.

Decimal numbers can be represented on a number line as shown below. In this way, the values of any two decimals can be compared easily.

Decimals can be arranged in ascending or descending order.

Place value and value of a digit in decimals

Each digit in a decimal has a specific place value which determines the value of the digit.

For the number 37.156, the table below shows the place value of each digit and the value of a each digit.

PLACE VALUE

DECIMAL

VALUE OF THE DIGIT

Tens (10)

3

30

Units (1)

7

7

Decimal point

Tenths (1/ 10)

1

0.1

Hundredths (1 / 100)

5

0.05

Thousandths (1 / 1000)

6

0.006

Each digit has only one place value and that place determines the value of the digit. For example , the place value of the digit 5 is hundredths and therefore, the value of the digit is 0.05.

37.156 =37 + 0.1 + 0.05 +0.006

= 37 +1/10 + 5/100 + 6 /1 000

ROUNDING OFF DECIMALS

The decimal place ( d.p) of a decimal is the number of digits to the right of the decimal point.

A decimal can be rounded off correct to a certain number of decimal places.The method of rounding off a decimal is similar to the method of rounding off a whole number.The following are steps to round off a decimal to a specific number of decimal places.

Example

Round off 87.4592 to

a) the nearest whole number

b) 1 decimal place

c) 2 decimal places

Solution

a)87.4592 = 87

b)87.4592 =87.5

c)87.4592 =87.46

ADDITION AND SUBTRACTION WITH DECIMALS

ADDITION OF DECIMALS

Addition of decimals is the process of finding the sum of two or more decimals.

To add two or more decimals:

a) Arrange the digits of the decimals according to their place values.

b) Then, add the digits from right to left.

Example

Solve 3.5 + 7.029 +18.953 =

3.5

+ 7.029

+ 18.953

=29.482

SUBTRACTION OF DECIMAL

Subtraction of decimals is the process of finding the difference between two decimals.

To subtract two or more decimals

a) Arrange the digits of the decimals according to their place values. Line up the decimal points.

b) Subtract the digits from right to left.

Example,

Solve 49.81 – 10.19 -3.54

= 49.81

- 10.19

39.62

- 3.54

36.08

DIVISION AND MULTIPLICATION WITH DECIMALS

DIVIDING DECIMALS

To divide a decimal by a whole number:

Use the long division method and place the decimal point exactly above the decimal point of the number to be divided.

Example

Evaluate 75.38 ÷ 5

=75.38 / 5

=15.076

To divide a decimal by 10, 100, or 1 000:

Move the decimal point 1, 2, or 3 places to the left according to the number of zeros in 10, 100, 1000 respectively.

To divide a decimal by 0.1, 0.01, or 0.001:

Move the decimal point 1, 2 or 3 places to the right according to the number of decimal places in 0.1, 0.01, and 0.001 respectively.

Example,

Divide the following

a) 79.88 ÷100

=0.7988

b) 22.128 ÷ 0.001

=22 128

MULTIPLYING TWO OR MORE DECIMALS

The multiplication of a whole number and a decimal is a process of repeated addition of that decimal.

To multiply two or more decimals:

a) Multiply as for whole numbers from right to left.

b) Find the total number of decimal places in the product which is the sum of decimal places in each number involved.

c) Place the decimal point according to the total number of decimal places in the product.

Example

a) 18.13 x 5

= 18.13

X 5

90.65

To multiply a decimal by 10, 100 and 1 000 :

Move the decimal pints 1, 2 or 3 places to the right of the decimals according to the number of zeros in 10, 100 and 1 000 respectively.

Example

a) 0.054 x 10 =0.54

b) 1.796 x 100 = 179.6

c) 48.38 X 1 000 = 48 380

FRACTIONS AND DECIMALS

Representing fractions with denominators 10,100, and 1 000 can be expressed in decimals.

Since 1/10 = 0.1 and 1/100 =0.01 and 1/ 1000 =0.001

- 81/ 100 =0.81

- 2 73/1 000 = 2.073

Changing fractions to decimals and vice versa

When changing a fraction to a decimal:

Divide the numerator by its denominator.

Example,

15/8

=15 ÷8

= 1.875.

When changing a decimal to a fraction

a) Count the number of digits in the decimal.

b) Then, write a fraction with a denominator that is a multiple of 10.

c) Simplify the answer to the lowest terms whenever possible.

Example,

Convert the decimals into fractions

a) 0.02

= 2/100

b) 0.075

=75/1 000

PERCENTAGES

PERCENTAGE

A fraction with 100 as the denominator is called a percentage. In this case, the numerator represents the number of parts in every 100.

The symbol % is used o represent percentage.

Example 80 / 100 is stated as 80 % and as 14 % stated as 14 /100.

RELATION BETWEEN PERCENTS, DECIMALS AND FRACTIONS

Decimals, Fractions and Percentages are just different ways of showing the same value:

A Half can be written…

As a fraction: ½ As a decimal: 0.5 As a percentage: 50%

A Quarter can be written …

As a fraction: ¼

As a decimal: 0.25

As a percentage: 25%

Converting Between Percentage and Decimal

Percentage means "per 100", so 50% means 50 per 100, or simply 50/100. If you divide 50 by 100 you get 0.5 (a decimal number). So:

To convert from percentage to decimal: divide by 100 (and remove the "%" sign).

To convert from decimal to percentage: multiply by 100 (and add a "%" sign).

The easiest way to multiply (or divide) by 100 is to move the decimal point 2 places. So:

From Decimal- 0.125 0 1 2 5 To percent -12.5%

Move the decimal point 2 places to the right, and add the % sign.

From Percent -75% 0 7 5 To decimal -0.75

Move the decimal point 2 places to the left, and remove the % sign. Converting Between Fractions and Decimals

The easiest way to convert a fraction to a decimal is to divide the top number by the bottom number (divide the numerator by the denominator in mathematical language)

Example: Convert 2/5 to a decimal

Divide 2 by 5: 2 ÷ 5 = 0.4

Answer: 2/5 = 0.4

To convert a decimal to a fraction needs a little more work:

· Write down the decimal “over” the number 1. E.g 0.75/1

· Then multiply top and bottom by 10 for every number after the decimal point. E.g 0.75x100 / 1x 100. This makes it a correctly formed fraction =75/100.

· Then, simplify the fraction =3/4.

Converting Between Percentages and Fractions

The easiest way to convert a fraction to a percentage is to divide the top number by the bottom number. then multiply the result by 100 (and add the "%" sign)

Example: Convert 3/8 to a percentage

First divide 3 by 8: 3 ÷ 8 = 0.375,Then multiply by 100: 0.375 x 100 = 37.5Add the "%" sign: 37.5%

Answer: 3/8 = 37.5%

Changing fractions and decimals to percentages and vice versa.

A fraction or a decimal number can be changed into a percentage by

a) Changing the denominator ( of fractions ) to 100 or

b) Multiplying the fraction or decimal number by 100 %

Example,

2/5

= 2/5 x 100 %

= 40 %

0.3

= 0.3 x 100 %

=30 %

A percentage can be changed into a fraction or a decimal number by dividing it with 100.

Example

Change each of the following into

i) a fraction ii) a decimal number -for 65 % and 18%

65 %

i) 65 % = 65/100 = 13/20

ii) 65 % = 65/100 = 0.65

18 %

i) 18% =18/100 = 9/50

ii) 18% = 18/100 = 0.18

IMPORTANT FORMULAE

Percentage increase = increase in value x 100 %

Original value

Percentage decrease = decrease in value x 100%

Original value

Profit = selling price – cost price

Loss = cost price – selling price

Discount =New selling price – Original selling price

Percentage discount = Discount x 100%

Original selling price

Interest = Interest x deposit

Interest rate = Interest x 100 %

Deposit

Commission = Rate of commission x selling price

Rate of commission = Commission x 100%

Selling price

TAS

Examples for daily life problem

APPLICATION OF FRACTION AND MIXED NUMBER

1. An oil tank contains 30 2/5 litres of oil. It is then filled with 37 2/3 litres of oil. Later 56 1/9 litres of oil is pumped out from the tank. What is the volume of oil left in the tank now, in litres?

Solution,

30 2/5 litres + 37 2/3 litres – 56 1/9 litres

=30 + 2/5 + 37 + 2/3 – (56 + 1/9)

=( 30 + 37 + - 56 ) + ( 2/5 + 2/3 – 1/9)

= 11 + ( 18/45 + 30/45 – 5/45)

=11 43 /45 litres.

It contains 11 43/45 litres of oil now.

2. Lin completed 1/6 of her project in 3 ½ days. How long would she take to complete the whole project?

Solution

1/6 of the project takes 3 ½ days = 7/2

The whole project, 1 = 7/2 ÷1/6

= 7/2 x 6/1

=21 days.

Therefore, the time Lin would tale to complete the whole project = 21 days.

3. Hooi cuts a 36 2/5 m long rope into pieces, each measuring 6 1/15 m. Calculate the number of pieces cut.

Solution 36 2/5 ÷ 6 1/15 = 182/5 ÷91 /15

=182/5 x 15/91

=6.

The number of pieces cut is 6.

DECIMALS

1) Swee Yee and her mother made 12.5 litres and 59.7 litres of soya milk respectively for sale. At the end of the day, how much soya milk was sold if they are left with 3.45 litres?

Solution 12.5 L + 59.7 L – 3.45 L

=72.2 L – 3.45 L

=68.75 L

Quantity of soya milk sold = 68.75 L

2) 7.2 kg of durians costs RM 28.08. How much would 15.8 kg of the same kind of durians cost ?

Solution

Price of 7.2 kg of durians = Rm 28.08

Price of 1 kg durians =Rm 28.08

7.2

Price of 15.8 kg of durians = Rm 28.08 X 15.8

7.2

=RM 61.62

3) Idham bought a piece of wire measuring 53 m long. He used 13 .9 m of it and sold the remainder to Samuel at 50 sen metre. How much did Samuel pay for the wire?

Solution

Length of wire bought by Samuel

=(53 – 13.9 ) m

= 39.1 m

Price of wire per metre= Rm 0.50

Price of 39.1 m of wire = Rm 0.50 x 39.1

=Rm19.55

Therefore, the amount Samuel paid

=Rm 19.55.

PERCENT

1) A handbag which is normally sold for Rm180, is sold for Rm108 during a sale. What is the percentage discount?

Solution

Discount = Rm180 –Rm 108

=Rm 72.

Percentage discount = Rm72 x 100%

Rm180

= Rm 40%

2) Wen is paid a commission of 7 % for every house she sells. If she sells a house for Rm 80 000, what is her commission?

Solution

Commission = 7% of Rm80 000

= 7 x Rm80 000

100

=Rm 5 600

3) Jason bought a new car for Rm 50 000 and sold it for Rm45 000 after one year. Calculate the percentage loss.

Solution

Loss =Rm 50 000 –Rm 45 000

=Rm 5 000

Percentage loss

=Rm 5 000 x 100%

Rm 50 000

=10%

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