6 percentages, conversion between fractions, decimals and percentages

$: 6 percentages, conversion between fractions, decimals and percentages$
Percentages

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PercentagesA percentage specified “how many out of per 100” and it’s written as #% or . #

100

PercentagesA percentage specified “how many out of per 100” and it’s written as #% or .

1% = 1 out of 1001 percent = 1/100

#100


1% = 1 out of 1001 percent = 1/100

#100

It’s useful think of % as the ratio of pennies to 1$, e.g. 1¢ is 1% of $1 (100 ¢).


1% = 1 out of 1001 percent = 1/100

#100

5% = 5 out of 1005 percent = 5/100 = 1/20


1% = 1 out of 1001 percent = 1/100

#100

5% = 5 out of 1005 percent = 5/100 = 1/20

10% = 10 out of 10010 percent = 10/100 = 1/10


1% = 1 out of 1001 percent = 1/100

#100

5% = 5 out of 1005 percent = 5/100 = 1/20

10% = 10 out of 10010 percent = 10/100 = 1/10

25% = 25 out of 10025 percent = 25/100 = 1/4


1% = 1 out of 1001 percent = 1/100

#100

5% = 5 out of 1005 percent = 5/100 = 1/20

50% = 50 out of 100 = 50 percent = 50/100 = 1/2

10% = 10 out of 10010 percent = 10/100 = 1/10

25% = 25 out of 10025 percent = 25/100 = 1/4


1% = 1 out of 1001 percent = 1/100

#100

5% = 5 out of 1005 percent = 5/100 = 1/20

50% = 50 out of 100 = 50 percent = 50/100 = 1/2

10% = 10 out of 10010 percent = 10/100 = 1/10

25% = 25 out of 10025 percent = 25/100 = 1/4

100% = 100 out of 100100 percent = 100/100 = 1.

Example A. What is ¾ of $100?

The expression “a fractional of the total..” means to divide the total into equal parts indicated by the denominator, then retrieve the number of parts indicated by the numerator.

Percentages



34 Divide $100 into

4 equal parts.

Percentages



34 Divide $100 into

4 equal parts.

100 ÷ 4 = 25 so each part is 25,

Percentages



34 Divide $100 into

4 equal parts.

Take 3 parts. 100 ÷ 4 = 25 so each part is 25,

hence 3 parts is 3 x $25 = $75.

Percentages



34 Divide $100 into

4 equal parts.


So ¾ of $100 is $75.


Percentages


The expression “a fractional of the total..” means to divide the total into equal parts indicated by the denominator, then retrieve the number of parts indicated by the numerator. We record this process with a division then a multiplication.

34 Divide $100 into

4 equal parts.


So ¾ of $100 is $75.


Percentages



34 Divide $100 into

4 equal parts.


So ¾ of $100 is $75. In symbols, 34 * 100


Percentages



34 Divide $100 into

4 equal parts.


So ¾ of $100 is $75. In symbols, 34 * 100

25hence 3 parts is 3 x $25 = $75.

Percentages



34 Divide $100 into

4 equal parts.


So ¾ of $100 is $75. In symbols, 34 * 100 = 75.

25hence 3 parts is 3 x $25 = $75.

Percentages



34 Divide $100 into

4 equal parts.


So ¾ of $100 is $75. In symbols,

The same steps of calculation apply for calculating “the #% of a total” andin such a problem, simplify the percent to a reduced fraction first.

34 * 100 = 75.

25hence 3 parts is 3 x $25 = $75.

Percentages



34 Divide $100 into

4 equal parts.




34 * 100 = 75.

25hence 3 parts is 3 x $25 = $75.

Percentages

Example B. 45% of 60 pieces of candy are chocolates, how many is that?



34 Divide $100 into

4 equal parts.




34 * 100 = 75.

25

4510045% is


Percentages




34 Divide $100 into

4 equal parts.




34 * 100 = 75.

25

4510045% is = 9

20


Percentages

÷5

÷5




34 Divide $100 into

4 equal parts.




34 * 100 = 75.

25


4510045% is = 9

20 so “45% of 60” is 920

* 60


Percentages

÷5

÷5



34 Divide $100 into

4 equal parts.




34 * 100 = 75.

25

45100

345% is = 9

20 so “45% of 60” is 920

* 60


Percentages

÷5

÷5


divide 60 pieces into 20 groups so each group consists of 3 pieces



34 Divide $100 into

4 equal parts.




34 * 100 = 75.

25

divide 60 pieces into 20 groups so each group consists of 3 pieces and 9 groups make 27 pieces

45100

345% is = 9

20 so “45% of 60” is 920

* 60 = 27


Percentages

÷5

÷5




34 Divide $100 into

4 equal parts.




34 * 100 = 75.

25

So 27 pieces are chocolates.

45100

345% is = 9

20 so “45% of 60” is 920

* 60 = 27


Percentages

÷5

÷5


divide 60 pieces into 20 groups so each group consists of 3 pieces and 9 groups make 27 pieces

Following is a list of important simplified percentage, they are familiar because they are related to our coins: nickels, dimes, quarters, and the 50–cent pieces.

Percentages


51005% =

Percentages

= 120


51005% =

Percentages

= 120

: one nickel is 1/20 of a dollar and 20 nickels is $1.


51005% =

Percentages

= 120


1010010% = = 1

10: one dime is 1/10 of a dollar and 10 dimes is $1.


51005% =

Percentages

= 120


1010010% = = 1


2510025% = = 1

4: one quarter is 1/4 of a dollar and 4 quarters is $1.


51005% =

Percentages

= 120


1010010% = = 1


2510025% = = 1


5010050% = = 1

2: one 50–cent piece is 1/2 of a dollar and 2 of them is $1,


51005% =

Percentages

= 120


1010010% = = 1


2510025% = = 1


5010050% = = 1


100100and 100% = = 1, 200

100200% = = 2, etc..


It’s useful to think of percentages of multiples of 5 as counting nickels where one nickel is $1/20.

51005% =

Percentages

= 120


1010010% = = 1


2510025% = = 1


5010050% = = 1


100100and 100% = = 1, 200

100200% = = 2, etc..


It’s useful to think of percentages of multiples of 5 as counting nickels where one nickel is $1/20. For example, 35% = 7/20 because there are 7 nickels in 35 cents, or that 85% = 17/20 because there are 17 nickels in 85 cents.

51005% =

Percentages

= 120


1010010% = = 1


2510025% = = 1


5010050% = = 1


100100and 100% = = 1, 200

100200% = = 2, etc..


It’s useful to think of percentages of multiples of 5 as counting nickels where one nickel is $1/20. For example, 35% = 7/20 because there are 7 nickels in 35 cents, or that 85% = 17/20 because there are 17 nickels in 85 cents.

51005% =

Percentages

= 120


1010010% = = 1


2510025% = = 1


5010050% = = 1


100100and 100% = = 1, 200

100200% = = 2, etc..

Other useful approximate percentages in fractions are33% ≈ 1/3 and that 66% ≈ 2/3.

PercentagesWe may use percentages to indicate the “concentration” of specific groups in a larger population.


Example C. 120 people who watched the movie “As the Paint Dries” are surveyed to see if they enjoyed it. a. 60% of 120 people enjoyed the movie, how many people is that?



60% = 60100 = 3

5



60% = x 120 =35

60100 = 3

5 , so 60% of 120 people is



60% = x 120 = 72.35, so 60% of 120 people is60

100 = 35

24



60% = x 120 = 72.35, so 60% of 120 people is60

100 = 35

24

Hence 72 people like the movie.


Example C. 120 people who watched the movie “As the Paint Dries” are surveyed to see if they enjoyed it.

b. Of the people enjoyed it, 75% are men, how many men enjoyed the movie?

a. 60% of 120 people enjoyed the movie, how many people is that?

60% = x 120 = 72.35, so 60% of 120 people is60

100 = 35

24






60% = x 120 = 72.35, so 60% of 120 people is60

100 = 35

24


There are 72 people that like the movie.





60% = x 120 = 72.35, so 60% of 120 people is60

100 = 35

24

75% = x 7234, so 75% of 72 people is 75

100 = 34







60% = x 120 = 72.35, so 60% of 120 people is60

100 = 35

24

75% = x 7234, so 75% of 72 people is 75

100 = 34

18= 54.







60% = x 120 = 72.35, so 60% of 120 people is60

100 = 35

24

75% = x 7234, so 75% of 72 people is 75

100 = 34

18= 54.



Therefore there are 54 men who enjoyed the movie “As the Paint Dries”.





“The amount of adjustments” are often given as percentages such as the discount rates or tax rates etc..

60% = x 120 = 72.35, so 60% of 120 people is60

100 = 35

24

75% = x 7234, so 75% of 72 people is 75

100 = 34

18= 54.



Therefore there are 54 men who enjoyed the movie “As the Paint Dries”.

PercentagesExample D. A $45 nose–ring is on sale at a 15% discount rate. How much is the discounted price?


15% = 15100 = 3

20


15% =

x 45320

, so the amount of discount “15% of $45” is15100 = 3

20


15% =

x 45 =320


20

4

9274


15% =

x 45 =320


20

4

9274

= 6 34 = $6.75


15% =

x 45 =320


20

4

Hence the marked–down price of the nose–ring is 45 – 6.75 = $38.25.

9274

= 6 34 = $6.75

Conversion Between Decimals, Fractions and PercentagesFractions, decimals and percentages are three different ways to express answers to the question “How much?”.

Conversion Between Decimals, Fractions and PercentagesFractions, decimals and percentages are three different ways to express answers to the question “How much?”.A fraction gives the instruction for obtaining a quantity,

Conversion Between Decimals, Fractions and PercentagesFractions, decimals and percentages are three different ways to express answers to the question “How much?”.A fraction gives the instruction for obtaining a quantity, for example

34


34

Divide the unit (1), i.e. the whole, into 4 equal parts


34


then take 3 parts.


34


then take 3 parts.

Fractions are easy to understand conceptually so it’s important for mathematics which emphasis the study of relations among numbers.


34


then take 3 parts.

Fractions are easy to understand conceptually so it’s important for mathematics which emphasis the study of relations among numbers. However, it’s cumbersome to add or subtract fractions because searching for the least common denominator is difficult for a lengthy list of fractions.


34


then take 3 parts.

Fractions are easy to understand conceptually so it’s important for mathematics which emphasis the study of relations among numbers. However, it’s cumbersome to add or subtract fractions because searching for the least common denominator is difficult for a lengthy list of fractions.One way to over come the difficulty of adding or subtracting fractions is to standardize the denominators to powers of 10, which leads to the decimal system.


34


then take 3 parts.

Fractions are easy to understand conceptually so it’s important for mathematics which emphasis the study of relations among numbers. However, it’s cumbersome to add or subtract fractions because searching for the least common denominator is difficult for a lengthy list of fractions.One way to over come the difficulty of adding or subtracting fractions is to standardize the denominators to powers of 10, which leads to the decimal system. For example, the decimal number 0.75

means


34


then take 3 parts.

Fractions are easy to understand conceptually so it’s important for mathematics which emphasis the study of relations among numbers. However, it’s cumbersome to add or subtract fractions because searching for the least common denominator is difficult for a lengthy list of fractions.One way to over come the difficulty of adding or subtracting fractions is to standardize the denominators to powers of 10, which leads to the decimal system. For example, the decimal number 0.75

means 710

5100+ denominators are powers of 10.

To convert fractions to decimals, we attach 0’s after the decimal point and perform long division.

Conversion Between Decimals, Fractions and Percentages

Example E. Convert the fractions into a decimal.81



Example E. Convert the fractions into a decimal.81

)8 1.Perform long division,



attach 0’s

0 0 0

Example E. Convert the fractions into a decimal.

4 0

81

)8 1.Perform long division,

.0 0 0 1

8 2 0

2 5

1 6


4 00

0


attach 0’s


4 0

81

)8 1.Perform long division, we obtain that

. 1

8 2 0

2 5

1 6


4 00

0

81 = 0.125.


0 0 0attach 0’s


4 0

81


. 1

8 2 0

2 5

1 6


4 00

0

81 = 0.125.

=21

Here is a list of common fractions and their decimal expansions:

0.50 =41 0.25 =5

1 0.20 =101 0.10

=201 0.05 =25

1 0.04 =501 0.02 =100

1 0.01

81 = 0.125 8

2 = 0.250 83 = 0.375 8

4 = 0.500= 41 = 2

1

85 = 0.625 8

6 = 0.750 = 43

87 = 0.875


0 0 0attach 0’s


4 0

81


. 1

8 2 0

2 5

1 6


4 00

0

81 = 0.125.

=21

Here is a list of common fractions and their decimal expansions:

0.50 =41 0.25 =5

1 0.20 =101 0.10

=201 0.05 =25

1 0.04 =501 0.02 =100

1 0.01

81 = 0.125 8

2 = 0.250 83 = 0.375 8

4 = 0.500= 41 = 2

1

85 = 0.625 8

6 = 0.750 = 43

87 = 0.875

It’s easy to add or subtract decimals numbers–we don’t need to look for common denominators as the case with fractions.


0 0 0attach 0’s

Example F. Calculate the 0.375 x 1600 using fractions.

Conversion Between Decimals, Fractions and PercentagesSome problems are easier to do using fractions.



83

0.375 =

Some problems are easier to do using fractions.


83so 0.375 x 1600 =


83

0.375 = x 1600 =



83so 0.375 x 1600 =


83

0.375 = x 1600 = 600


200



Fact About Shifting the Decimal Points in a Fraction Given a fraction, if the decimal points of the numerator and the denominator are shifted in the same direction with the same number of spaces, the resulting fraction is an equivalent fraction, i.e. it’s the same.

83so 0.375 x 1600 = 8

30.375 = x 1600 = 600

200




To change a decimal number of the form 0 . # # # # to a fraction:

1. Put “1.” in the denominator and line up the decimal points.

83so 0.375 x 1600 = 8

30.375 = x 1600 = 600

200






0 . # # # #1 .

83so 0.375 x 1600 = 8

30.375 = x 1600 = 600

200






0 . # # # #1 .

83so 0.375 x 1600 = 8

30.375 = x 1600 = 600

200

2. Slide the decimal point of the numerator to end of the number.






0 . # # # #1 .

0 . # # # #1 .

. =

Drag the decimal point to the end of the number 2. Slide the decimal point of the

numerator to end of the number.

83so 0.375 x 1600 = 8

30.375 = x 1600 = 600

200






0 . # # # #1 .

0 . # # # #1 .

. =

Drag the decimal point to the end of the number 2. Slide the decimal point of the

numerator to end of the number.

3. Pack a “0” for each move to the right.

83so 0.375 x 1600 = 8

30.375 = x 1600 = 600

200






0 . # # # #1 .

0 . # # # #1 .

.

. 0 0 0 0=

Drag the decimal point to the end of the number

then fill in a “0” for each move.

2. Slide the decimal point of the numerator to end of the number.

3. Pack a “0” for each move to the right.

83so 0.375 x 1600 = 8

30.375 = x 1600 = 600

200

Example G. Convert the following decimals to fractions.

a. 0.023



a. 0.0231. Insert “1.” in the denominator and line up the decimal points.



a. 0.023

0 . 0 2 31 .

1. Insert “1.” in the denominator and line up the decimal points.



a. 0.023

0 . 0 2 31 .

1. Insert “1.” in the denominator and line up the decimal points.2. Slide the pair of points in to the back of the last non-zero digit in the numerator, and pack 0’s in the skipped slots in the denominator.




a. 0.023

0 . 0 2 31 .0 0 0

0 . 0 2 31 .

= . .




a. 0.023

0 . 0 2 31 .0 0 0 1000

23=

0 . 0 2 31 .

= . .




a. 0.023

0 . 0 2 31 .0 0 0 1000

23=

0 . 0 2 31 .

= . .

b. 37. 25




a. 0.023

0 . 0 2 31 .0 0 0 1000

23=

0 . 0 2 31 .

= . .

b. 37. 25

we only need to convert the decimal 0.25 to a fraction.

Since 37.25 = 37 + 0.25




a. 0.023

0 . 0 2 31 .0 0 0 1000

23=

0 . 0 2 31 .

= . .

b. 37. 25

0 . 2 51 .


Since 37.25 = 37 + 0.25

0. 2 5 =




a. 0.023

0 . 0 2 31 .0 0 0 1000

23=

0 . 0 2 31 .

= . .

b. 37. 25

0 . 2 51 .


Since 37.25 = 37 + 0.25

0. 2 5 =0 . 2 51 . 0 0

=..




a. 0.023

0 . 0 2 31 .0 0 0 1000

23=

0 . 0 2 31 .

= . .

b. 37. 25

0 . 2 51 .


Since 37.25 = 37 + 0.25

0. 2 5 =0 . 2 51 . 0 0

=.. 100

25=




a. 0.023

0 . 0 2 31 .0 0 0 1000

23=

0 . 0 2 31 .

= . .

b. 37. 25

0 . 2 51 .


Since 37.25 = 37 + 0.25

0. 2 5 =0 . 2 51 . 0 0

=.. 100

25= =


41



a. 0.023

0 . 0 2 31 .0 0 0 1000

23=

0 . 0 2 31 .

= . .

b. 37. 25

0 . 2 51 .


Since 37.25 = 37 + 0.25

0. 2 5 =0 . 2 51 . 0 0

=.. 100

25= = . Therefore 37.25 = 37


41

41



a. 0.023

0 . 0 2 31 .0 0 0 1000

23=

0 . 0 2 31 .

= . .

b. 37. 25

0 . 2 51 .


Since 37.25 = 37 + 0.25

0. 2 5 =0 . 2 51 . 0 0

=.. 100

25= = . Therefore 37.25 = 3741


Percentages are fractions with 100 as the denominator. It’s useful to think of 1% as 1¢ = $0.01 and that 30% as 30 ¢ = $0.30, etc..

41

The conversion between decimal numbers and percentages is done by shifting decimal points.




To change a decimal number into a #% # . # # #



1. insert “1.” as the denominator,

To change a decimal number into a #% # . # # # 1 .




2. move the decimal points two places to the right to make the denominator “100.”, write the result using the % notation.







# . # # # 1 .

. . 0 0=

move right, expand denom. to “100”





Example H. a. Convert 4.5 to %. b. Convert 0.045 to %.


# . # # # 1 .

. . 0 0=


4.5 =





Example H. a. Convert 4.5 to %. b. Convert 0.045 to %.4.51.


# . # # # 1 .

. . 0 0=


4.5 =






=4.50.1.00.


# . # # # 1 .

. . 0 0=


move right, expand to “100”

4.5 =






=4.50.1.00. = 450%

Hence 4.50 = 450.%.


# . # # # 1 .

. . 0 0=



4.5 =






=4.50.1.00. = 450% 0.045 = 1.

0.045

Hence 4.50 = 450.%.


# . # # # 1 .

. . 0 0=



4.5 =






=4.50.1.00. = 450% 0.045 = 1. =

0.04.501.00. = 4.50%

0.045

Hence 4.50 = 450.%. Hence 0.045 = 4.5%.


# . # # # 1 .

. . 0 0=



4.5 =






=4.50.1.00. = 450% 0.045 = 1. =

0.04.501.00. = 4.50%

0.045

Hence 4.50 = 450.%. Hence 0.045 = 4.5%.

To change a decimal number into a #%

To change a #% into a decimal number

# . # # # 1 .

# . # # # 1 .

. . 0 0=



4.5 =






=4.50.1.00. = 450% 0.045 = 1. =

0.04.501.00. = 4.50%

0.045

Hence 4.50 = 450.%. Hence 0.045 = 4.5%.


1. write the #% asTo change a #% into a decimal number

#100

# . # # # 1 .

# . # # # 1 .

. . 0 0=


# # #. #1 0 0.


4.5 =






=4.50.1.00. = 450% 0.045 = 1. =

0.04.501.00. = 4.50%

0.045

Hence 4.50 = 450.%. Hence 0.045 = 4.5%.


1. write the #% as

2. move the two decimal points two places to the left so the denominator is 1 and the numerator is the answer.

To change a #% into a decimal number#

100

# . # # # 1 .

# . # # # 1 .

. . 0 0=



# # #. #1 0 0.

4.5 =






=4.50.1.00. = 450% 0.045 = 1. =

0.04.501.00. = 4.50%

0.045

Hence 4.50 = 450.%. Hence 0.045 = 4.5%.


1. write the #% as

2. move the two decimal points two places to the left so the denominator is 1 and the numerator is the answer.

To change a #% into a decimal number#

100

# . # # # 1 .

# . # # # 1 .

. . 0 0=


# . # # #1 .

# # #. #1 . .

0 0. =

move left, reduce denom. to “1”


Conversion Between Decimals, Fractions and PercentagesExample I. a. Convert 0.35% to a decimal number. b. Convert 3,000 % to a decimal number.


= 0.351 0 0.

0.35%move left to reduce to “1”


= 0.351 0 0.

0.35%

Hence 0.35% = 0.0035.

move left to reduce to “1”

= 0.00.35


= 0.351 0 0.

0.35%

Hence 0.35% = 0.0035.


= 0.00.35

=3 0 0 0.

1 0 0.3,000%


= 0.351 0 0.

0.35%

Hence 0.35% = 0.0035.


= 0.00.35

=3 0 0 0.

1 0 0.3,000%

= 30.00.

Hence 3,000% = 30.00 or 30.


= 0.351 0 0.

0.35%

Hence 0.35% = 0.0035.


= 0.00.35

=3 0 0 0.

1 0 0.3,000%

= 30.00.

Hence 3,000% = 30.00 or 30.

Example F. The cost of a dozen eggs changed from $2.40 to $3.00. What is the increase in the price? What is the % of increase in the price?


= 0.351 0 0.

0.35%

Hence 0.35% = 0.0035.


= 0.00.35

=3 0 0 0.

1 0 0.3,000%

= 30.00.

Hence 3,000% = 30.00 or 30.


The increase in the price is 3.00 – 2.40 = $0.60.


= 0.351 0 0.

0.35%

Hence 0.35% = 0.0035.


= 0.00.35

=3 0 0 0.

1 0 0.3,000%

= 30.00.

Hence 3,000% = 30.00 or 30.



The % of increase in the price is:0.60the price hike

original price= 2.40


= 0.351 0 0.

0.35%

Hence 0.35% = 0.0035.


= 0.00.35

=3 0 0 0.

1 0 0.3,000%

= 30.00.

Hence 3,000% = 30.00 or 30.





= =14

= 0.25 = 25%6.24.


= 0.351 0 0.

0.35%

Hence 0.35% = 0.0035.


= 0.00.35

=3 0 0 0.

1 0 0.3,000%

= 30.00.

Hence 3,000% = 30.00 or 30.





= =14

= 0.25 = 25%

So this is a 25 % increase in the price.

6.24.


= 0.351 0 0.

0.35%

Hence 0.35% = 0.0035.


= 0.00.35

=3 0 0 0.

1 0 0.3,000%

= 30.00.

Hence 3,000% = 30.00 or 30.





= =14

= 0.25 = 25%

So this is a 25 % increase in the price.

6.24.

Question: If the price falls from $3.00 to $2.40, what is the % of price drop? (Ans: 20%)

6 percentages, conversion between fractions, decimals and percentages

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