decimals, fractions, and percents. session outcomes: to identify equivalences between fractions,...
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Decimals, Fractions, andPercents
Session Outcomes:
• To identify equivalences between fractions,
decimals and percent.
• To identify the relationship between
fractions, decimals and percent.
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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.
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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
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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
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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%
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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%
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75 out of 100 squares are shaded. .75 or 75% or 75/100 (which reduces to ¾)
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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