Decimals, Fractions, and Percents. Session Outcomes: To identify equivalences between fractions,...

Decimals, Fractions, andPercents

Session Outcomes:

• To identify equivalences between fractions,

decimals and percent.

• To identify the relationship between

fractions, decimals and percent.

2

Definitions • Decimal - Any number shown with a decimal point; a

number based upon tenths or hundredths. ( 0.2, 0.375, 86.4 )

• Fraction - One or more of the equal parts of a whole; a number usually expressed in the form a/b. ( 1/3, 2 5/8, 7/4 )

• Percent- The word percent comes from Latin and means “ for each hundred” or “ per hundred.” ( 70%, 6.5%, 4.09% )

What are fractions, decimals,and percents?

• Fractions, decimals and percents are different ways of representing part of a

whole.

• We could say 50% of something, or ½, or 0.5. These are all the same quantity.

4

What are fractions?• A fraction describes part of

a whole when the whole is cut into equal parts.

• This circle has been cut into four equal parts. These equal parts are called fourths. A fourth is written as: ¼

• The three red sections equal three fourths, which would

be written as: ¾5

¼

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Fractions

Fractions can be converted into other forms…

Decimals

Percents

Same Value Different FormsA same value can be written in different forms. For example:

½ fraction

.5 decimal

50% percent

8

Convert a fraction into a decimal

Divide the top of the fraction(numerator) by the

bottom (denominator) of the fraction:

½ = 1÷2 = 0.5

9

Time to try converting a fraction into a decimal.

1. Fraction to convert = 3/4

2. Divide 3(numerator) ÷ 4(denominator)

3. Answer = .75

3/4 is equivlent to .75

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Convert a decimalinto a percentage

Multiply the decimal by 100:

0.25 x 100 = 25%

11

Time to try converting a decimal into a percent.

1. Decimal to convert = .75

2. Multiply the decimal by 100

.75 x 100

3. Answer = 75%

.75 is equivalent to 75%

12

75 out of 100 squares are shaded. .75 or 75% or 75/100 (which reduces to ¾)

13

HINTSFraction to a decimal = Divide the numerator (top) by the denominator (bottom).

Decimal to a percent = Multiple the decimal by 100.

FRACTIONS DECIMALS PERCENTS

3/4 .75 75%

2/8 .25

.50

.40

.30

25%

50%

30%

40%

3/6

3/10

2/5

More Problems…

Write as a Decimal:

90%

Answer:

0.9

Write as a Decimal:

115.9 %

Answer:

1.159

Write as a Decimal:

9 %

Answer:

0.09

Write as a Decimal:

0.3 %

Answer:

0.003

Write as a Decimal:

⅗

Answer:

0.6

Write as a Decimal:

2⅗

Answer:

2.6

Write as a Decimal:

8/33

Answer: ___

0.24

Write as a Decimal:

4 ²⁷/₁₂₅

Answer:

4.216

Write as a Percent:

0.452

Answer:

45.2%

Write as a Percent:

0.006

Answer:

0.6%

Write as a Percent:

3.63

Answer:

363%

Write as a Fraction:

25%

Answer:

1/4

Write as a Fraction:

70%

Answer:

7/10

Write as a Fraction:

58%

Answer:

29/50

Write as a Fraction:

2.2

Answer:

2 ⅕

Write as a Fraction:

0.08

Answer:

2/25

Write as a Fraction:

0.005

Answer:

1/200

Write as a Percent:

1/2

Answer:

50%

Write as a Percent:

1/8

Answer:

12.5%

Write as a Percent:

2/3

Answer:

66.6%

Write as a Percent:

2 ⅟₁₀

Answer:

210%

Competition Problems

65% of the Student Council members voted yes on a survey. What fraction voted no?

Answer:

7/20

Express 5/8 as a percent.

Answer:

62.5%

On Brennan’s test, he got 22 out of 25 questions correct. What percentage did he answer incorrectly?

Answer:

12%

Find 80% of 115. Find 50% of that number.

Answer:

46

What is 75.5% of 60?

Answer:

45.3

What is 35% of 60% of 180?

Answer:

37.8

Three-fifths of the students made at least an 83 on the exam. What percent of the students did not make at

least an 83?

Answer:

40%

ChangingRepeating Decimal

to aFraction

Let’s recall how we write repeating decimals.

0.1111111… = 0.10.2222222… = 0.20.3333333… = 0.30.4444444… = 0.40.5555555… = 0.50.6666666… = 0.60.7777777… = 0.70.8888888… = 0.80.9999999… = 0.9

Use your calculator and check a few fractions:

1/9 = 0.1111111…2/9 = 0.2222222…3/9 = 1/3 = 0.333333…4/9 = 0.4444444…5/9 = 0.5555555…6/9 = 2/3 = 0.66666666…7/9 = 0.7777777…8/9 = 0.8888888…

So… Can you see the pattern?

0.1111111… = 0.10.2222222… = 0.20.3333333… = 0.30.4444444… = 0.40.5555555… = 0.50.6666666… = 0.60.7777777… = 0.70.8888888… = 0.80.9999999… = 0.9

So… Can you see the pattern?

0.1111111… = 0.1 = 1/90.2222222… = 0.2 = 2/90.3333333… = 0.3 = 3/9 = 1/30.4444444… = 0.4 = 4/90.5555555… = 0.5 = 5/90.6666666… = 0.6 = 6/9 = 2/30.7777777… = 0.7 = 7/90.8888888… = 0.8 = 8/90.9999999… = 0.9 = 9/9 = 1

Use your calculator and check a few more fractions:

12/99 = 0.12121212…23/99 = 0.23232323…34/99 = 0.34343434…45/99 = 0.45454545…56/99 = 0.56565656…67/99 = 0.67676767…78/99 = 0.78787878…89/99 = 0.89898989…

Use your calculator and check a few more fractions:

24/99 = 0.24242424…52/99 = 0.52525252…71/99 = 0.71717171…38/99 = 0.38383838…63/99 = 0.63636363…41/99 = 0.41414141…19/99 = 0.19191919…85/99 = 0.85858585…

Use your calculator and check a few more fractions:

248/999 = 0.248248248248…527/999 = 0.527527527527…2471/9999 = 0.247124712471…3278/9999 = 0.327832783278…62653/99999 = 0.626536265362653…45341/99999 = 0.453414534145341…157899/999999 = 0.157899157899157899…8521967/9999999 = 0.852196785219678521967…

So… Can you see the pattern?

If all the decimals repeat, then just place a “9” under each number to complete the fraction!

1/9 = 0.1111111…34/99 = 0.3434343434…248/999 = 0.248248248248…527/999 = 0.527527527527…2471/9999 = 0.247124712471…3278/9999 = 0.327832783278…62653/99999 = 0.626536265362653…45341/99999 = 0.453414534145341…157899/999999 = 0.157899157899157899…8521967/9999999 = 0.852196785219678521967…

Let’s look at a little more complicated repeating decimal.

0.455555555…0.277777777…0.166666666…

Notice that the 1st decimal does not repeat, but the remaining do.

0.455555555… = 0.4 + 0.055555…

Notice that the 1st decimal does not repeat, but the remaining do.

0.4 + 0.055555… = 4/10 + 0.0(5/9)= 2/5 + (5/9)·(1/10)= 2/5 + (5·1)/(9·10)

= 2/5 + 5/90= 36/90 + 5/90

= 41/90

(use a calculator to check your answer)

We can do this with any repeating decimal.

0.127455555555…

= 0.1274 + 0.000055555…= 1274/10000 + 0.0000(5/9)= 1274/10000 + (5/9)·(1/10000)

= 1274/10000 + 5/90000

= 11466/90000 + 5/90000

= 11471/90000

We can do this with any repeating decimal.

0.127453535353…

= 0.1274 + 0.0000535353…= 1274/10000 + 0.0000(53/99)= 1274/10000 + (53/99)·(1/10000)

= 1274/10000 + 53/990000

= 126126/990000 + 53/990000

= 126179/990000

Write as a Fraction: __

0.3

Answer:

1/3

Write as a Fraction: ____

0.15

Answer:

5/33

Write as a Fraction: __

66.6%

Answer:

2/3

Write as a Fraction: __

0.35

Answer:

16/45

Write as a Fraction: ____

0.357

Answer:

59/165