34 conversion between decimals, fractions and percentages

$: 34 conversion between decimals, fractions and percentages$
Conversion Between Decimals, Fractions and Percentages

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.


Example A. Convert the fractions into a decimal.81

Conversion Between Decimals, Fractions and PercentagesTo convert fractions to decimals, we attach 0’s after the decimal point and perform long division.

Example A. Convert the fractions into a decimal.81

)8 1.Perform long division,

Conversion Between Decimals, Fractions and PercentagesTo convert fractions to decimals, we attach 0’s after the decimal point and perform long division.

attach 0’s 0 0 0

Example A. 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.

=21Here 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.

=21Here 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 B. Calculate the 0.375 x 1600 using fractions.

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



830.375 =

Some problems are easier to do using fractions.


83so 0.375 x 1600 =


830.375 = x 1600 =



83so 0.375 x 1600 =


830.375 = 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 C. Convert the following decimals to fractions. a. 0.023



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



0 . 0 2 31 .

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



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.




0 . 0 2 31 .0 0 0

0 . 0 2 31 .

= . .




0 . 0 2 31 .0 0 0 1000

23=

0 . 0 2 31 .

= . .




0 . 0 2 31 .0 0 0 1000

23=

0 . 0 2 31 .

= . .

b. 37. 25




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




0 . 0 2 31 .0 0 0 1000

23=

0 . 0 2 31 .

= . .

b. 37. 25

0 . 2 51 .


0. 2 5 =




0 . 0 2 31 .0 0 0 1000

23=

0 . 0 2 31 .

= . .

b. 37. 25

0 . 2 51 .


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

= ..




0 . 0 2 31 .0 0 0 1000

23=

0 . 0 2 31 .

= . .

b. 37. 25

0 . 2 51 .


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

= .. 100

25=




0 . 0 2 31 .0 0 0 1000

23=

0 . 0 2 31 .

= . .

b. 37. 25

0 . 2 51 .


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

= .. 100

25= =


41



0 . 0 2 31 .0 0 0 1000

23=

0 . 0 2 31 .

= . .

b. 37. 25

0 . 2 51 .


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

= .. 100

25= = . Therefore 37.25 = 37


41

41



0 . 0 2 31 .0 0 0 1000

23=

0 . 0 2 31 .

= . .

b. 37. 25

0 . 2 51 .


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

= .. 100

25= = . Therefore 37.25 = 37 41


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 .



1. insert “1.” as the denominator, 2. move the decimal points two places to the right to make the denominator “100.”, write the result using the % notation.

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





# . # # # 1 .

. . 0 0=

move right, expand denom. to “100”




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


# . # # # 1 .

. . 0 0=


4.5 =




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


# . # # # 1 .

. . 0 0=


4.5 =




Example D. a. Convert 4.5 to %. b. Convert 0.045 to %.4.51. = 4.50.

1.00.


# . # # # 1 .

. . 0 0=


move right, expand to “100”

4.5 =





1.00. = 450%

Hence 4.50 = 450.%.


# . # # # 1 .

. . 0 0=



4.5 =





1.00. = 450% 0.045 = 1.0.045

Hence 4.50 = 450.%.


# . # # # 1 .

. . 0 0=



4.5 =





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 =





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 =





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 =





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 =





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 E. 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.


= 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.



The % of increase in the price is: 0.60the price hike original price = 2.40 = =

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%)

34 conversion between decimals, fractions and percentages

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