Basic Operations & Applications

Unit

What Are Fractions?

• Fractions are representations used to compare one quantity to another quantity. Fractions are also known as ratios.

• The fraction bar, “―” or “/”, can be translated to mean “out of,” “per,” or “divided by.”

• The quantity that is above or to the left of the fraction bar is the numerator.

• The quantity that is below or to the right of the fraction bar is the denominator .

• 5/7 can be translated to five-sevenths, five out of seven, five per seven, or five divided by seven.

Reducing Fractions

• When we reduce fractions, we look for the greatest common factor, GCF, between the numerator and denominator.

• We then divide both the numerator and denominator by the GCF.

• If the GCF is 1, then the fraction is already reduced.

Example 1 – Reduce 10/16 to lowest terms.

Solution – The GCF between 10 and 16 is 2.So, 10÷2/16÷2 = 5/8

Example 2 – Reduce 21/45 to lowest terms.

Solution – The GCF between 21 and 45 is 3.So, 21÷3/45÷3 = 7/15

You Try

#1 – Reduce 24/66 to lowest terms.

Solution –

#2 Reduce 56/80 to lowest terms.

Solution –

Reducing Fractions in Context

Example 1 – Askia spent $14 out of the $50 that his grandmother gave him for his trip. What fraction, in lowest terms, of the money that his grandmother gave him did he spend?

Solution – Since 2 is the GCF between 14 and 50, we divide both 14 and 50 by 2 to reduce the fraction.14/50 = 14÷2/50÷2 = 7/25So, Askia spent seven twenty-fifths of the money that his grandmother gave him.

Reducing Fractions in Context

Example 2 – When each student in the class was asked if they thought that they had a good math teacher when they were in elementary school, 4 students said “yes” and 20 students said “no.” What fraction, in lowest terms, of the students in the class thought that they did not have a good math teacher when they were in elementary school?

Solution – Since 4 students said “yes” and 20 students said “no,” there is a total of 24 students in the class. So, 20 out of 24 students said “no.” Since 4 is the GCF between 20 and 24, we divide both and 20 and 24 by 4.20÷4/24÷4 = 5/6So, five out of six students in the class thought that they did not have a good math teacher when they were in elementary school.

You Try

#3 - Eighteen out of the 60 teachers at Uplift have Twitter accounts. In a recent report, it was stated that three out of every five teachers at Uplift do not have a Twitter account. Prove why this is or is not an accurate statement.

Solution –

Equivalent Fractions

• Equivalent fractions are various ways of representing the same ratio.

• To produce a fraction that is equivalent to another fraction, multiply both the numerator and denominator by the SAME number. That number can be any number other than 0.

Checking Fraction Equivalency

Example 1 – Determine whether or not 2/5 and 14/35 are equivalent.

Solution – Since we multiply 2 by 7 to produce 14 and 5 by 7 to produce 35, then 2/5 and 14/35 are equivalent.

Example 2 – Determine whether or not 3/8 and 12/40 are equivalent.

Solution – Since we multiply 3 by 4 to produce 12 but we multiply 8 by 5 to produce 40, then 3/8 and 12/40 are not equivalent.

Fraction Equivalency in Context

Example 3 - Trevon runs 2 miles in 15 minutes. Augustine runs 6 miles in 45 minutes. Do Trevon and Augustine run at the same pace?

Solution – 2/15 = 6/45 ?Since we multiply 2 by 3 to produce 6 and we multiply 15 by 3 to produce 45, 2/15 is equivalent to 6/45. Therefore, Trevon and Augustine do run at the same pace.

You Try

#1 – Determine whether or not 5/6 and 25/36 are equivalent.

Solution -

#2 – A high school varsity basketball game consists of four 8-minute quarters. If Devin averages 3 points every 4 minutes and Jerrard averages 21 points per game, determine whether or not Jerrard and Devin average the same amount of points per game.

Solution –

Adding and Subtracting Fractions

• In order to add or subtract fractions, the denominators of the fractions must be the same.

• In order to make the denominators the same, we must first find a common multiple or preferably the least common multiple (LCM) between the denominators.

• The LCM is the smallest number that two or more numbers have in common when multiples of those numbers are produced.

• Once we’ve found the LCM, then we produce equivalent fractions with the LCM being the new denominator.

• When the equivalent fractions are produced, we add the numerators while the denominators remain the same.

Example of Adding and Subtracting Fractions

Example 1: 4/7 + 2/5

Solution –The LCM between 7 and 5 is 35.

So, 35 will be the new denominator.4/7 = ?/35 2/5 = ?/354 x 5/7 x 5 = 20/352 x 7/5 x 7 = 14/35

Now, add the numerators.20/35 + 14/35 = 34/35

Example 2: 3/8 – 1/6

Solution – The LCM between 8 and 6 is 24.

So, 24 will be the new denominator.3/8 = ?/241/6 = ?/243x3/8x3 = 9/241x4/6x4 = 4/24

Now, subtract the numerators.9/24 – 4/24 = 5/24

Adding and Subtracting Fractions in Context

Example 3 – At Uplift 3/5 of the girls play sports, 1/3 of the girls are in other after-school programs, and the rest of the girls are not involved in any activities. What fraction of the girls at Uplift are not involved in any activities?

Solution – Add 3/5 and 1/3The LCM between 5 and 3 is 15.

3/5 = ?/15 and 1/3 = ?/153x3/5x3 = 9/15 and 1x5/3x5 = 5/159/15 + 5/15 = 14/15

Since 15/15 is equivalent to 1 which represents all of the girls at Uplift, we subtract 14/15 from 15/15.

15/15 – 14/15 = 1/15

So, 1/15 of the girls at Uplift are not involved in any activities.

You Try

#1 4/9 + 5/6 = ?

Solution -

#2 – Malcolm X spent 1/8 of his day reading and writing, 1/12 of his day lecturing, and 1/6 of his day doing community service. What fraction of his day did he have left to do other things?

Solution –

Multiplying Fractions

• When multiplying fractions, we multiply numerators by numerators and denominators by denominators. Then reduce the product fraction to lowest terms if possible.

Example 1: 2/9 x 3/5 = ?2/9 x 3/5 = 2x3/9x5 = 6/45

Since 3 is the GCF between 6 and 45, we divide both 6 and 45 by 3.

6÷3/45÷3 = 2/15

You Try

#1) 2/7 x 5/8 = ?

Solution –

#2) 3/4 x 2/5 = ?

Solution –

Dividing Fractions• When we divide fractions, we must multiply the first fraction

by the reciprocal of the second fraction. Then reduce the product to lowest terms.

Example 2: 3/8 ÷ 5/6 = ?

Solution – Multiply 3/8 by the reciprocal of 5/6 which is 6/5.

3/8 x 6/5 = 18/40

The GCF between 18 and 40 is 2, so divide both 18 and 40 by 2.18÷2/40÷2 = 9/20

You Try

#3) 7/12 ÷ 2/5 = ?

Solution -

#4) 2/3 ÷ 8/9 = ?

Solution -

Multiplying Fractions in Context

Example 3 - Artezia spends 3/8 of her day in school. Two-thirds of the time that she is in school she spends thinking about what she is going to eat for dinner. What fraction of the day does she spend thinking about what she is going to eat for dinner?

Solution – We are trying to find what is 2/3 of 3/8, so we multiply.

3/8 x 2/3 = 6/24 = ¼

So, Artezia spends ¼ of her day thinking about what she is going to eat for dinner.

Example 4 – Three-fifths of CPS high school graduates graduate from college within 5 years. Of those students ¼ of them are African-American, but only two-sevenths of those students are African-American males. What fraction of CPS high graduates who graduate from college within 5 years are African-American males?

Solution – We are trying to find what is 2/7 of ¼ of 3/5, so we multiply.

3/5 x ¼ x 2/7 = 6/140 = 3/70

So, 3/70 of or 3 out of every 70 CPS high graduates who graduate from college within 5 years are African-American males.

Dividing Fractions in Context

Example 5 – How many eighths are in 2/3?

Solution - We are trying to find out how many times does 1/8 go into 2/3, so we divide 2/3 by 1/8.

2/3 ÷ 1/8 = 2/3 x 8/1 = 16/3

So, 16/3 or 5 1/3 eighths are in 2/3. In other words, 1/8 goes into 2/3 sixteen-thirds or five and one-third times.

Example 6 – Eight-ninths of the Uplift students who have been to New York City have visited Time Square. This is ¼ of the students at Uplift. What fraction of the students at Uplift have been to New York City?

Solution – We are trying to out ¼ is 8/9 of what number, so we can set up an algebraic equation where x represents the fraction of students at Uplift who have been to NYC.

¼ = 8/9 · x¼ ÷ 7/9 = 8/9 ÷ 8/9 · x x = ¼ ÷ 8/9 x = ¼ · 8/7 x = 8/28 = 2/7

So, two-sevenths of the Uplift students have been to NYC.

You Try

#5) Five-sixths of teenage drivers turn their cell phone off before they drive. Three-tenths of the teenage drivers who do not turn their cell phone off before they drive also text while they drive. What fraction of teenage drivers text while they drive?

Solution –

#6) Four-fifths of all Lincoln car owners who have had engine trouble with their car had the engine trouble after their warranty expired. This is ¾ of all Lincoln car owners. What fraction of Lincoln car owners have ever had engine trouble with their car?

Solution –

Converting Fractions to Decimals

When converting fractions to decimals, we divide the numerator by the denominator.

Example 1 – Convert 3/8 to a decimal.

Solution – 3 ÷ 8 = 0.375

Recognizing Decimal Places

• Decimal places are the place values that numbers occupy with respect to the decimal point.

• thousands hundreds tens ones . tenths hundredths thousandths

Example 2 – What place value is underlined in the following number?

1257.683

Answer – The underlined place value is the hundredths place.

Rounding to the Nearest Decimal Place

When we round to the nearest decimal place, we recognize the place value to which we are rounding and then identify the number immediately to its right.

If that number is more than 4, then the number occupying the place value to which we must round goes up by 1.If that number is 4 or less, then the number occupying the place value to which we must round stay the same.

Example 3 – Round 0.7859 to nearest thousandths place.

Answer –

0.785│9: 5 occupies the thousandths place. 9 is immediately to the right of 5. Since 9 is more than 4, we must round 5 up to 6

0.7859 ≈ 0.786

Example 4 – Round 3.6219 to the nearest tenths place.

Answer –

3.6│219: 6 occupies the tenths place. 2 is immediately to the right of 6. Since 2 is not more than 4, 6 must not change.

3.6219 ≈ 3.6

Converting Decimals to Fractions

When we convert decimals to fractions, we must first recognize the place value that is farthest to the right that the decimal occupies.

Then we put the numbers from the decimal in the numerator of the fraction, and we put the number that the place value that is farthest to the right represents in the denominator of the fraction.

Lastly, we reduce the fraction to lowest terms.

Example 5 – Covert 0.48 to a fraction.

Solution -

0.48 – the place value that is farthest to the right is the hundredths place

48/100 = 12/25

You Try

#1 – Convert 10/17 to a decimal. Round to the nearest hundredths place.

Solution –

#2 – Convert 0.225 to a fraction.

Solution –

Converting Fractions to Percents

Percents, symbolized by “%,” are another representation of fractions since percent means “some number out of 100.”

For example, 80% means 80 out of 100 or 80/100When we convert fractions to percents, we must find the equivalent fraction whose

denominator is 100. First, we convert the fraction to a decimal. Then we write the decimal as a fraction with 100

as the denominator using the hundredths place as the point of reference.

Example 1 – Convert 3/8 to a percent.

Solution – 3/8 = 0.375 = 37.5/100 => 37.5 out of 100 => 37.5%

Example 2 – Convert 5/12 to a percent.

Solution – 5/12 = 0.416U = 41.6U/100 => 41.6U out of 100 => 41.6@%

Note: If a fraction is more than 1, then it is more than 100% since 1 = 100%.

Converting Percents to Fractions

When we convert percents to fractions, we write the percent as some number out of hundred and then reduce the fraction to lowest terms if possible.

Example 3 – Convert 32% to a fraction.

Solution – 32% = 32/100 = 8/25

You Try

#1 – Convert 8/5 to a percent.

Solution –

#2 – Convert 110% to a fraction.

Solution –

Solving Arithmetic Problems Involving Percent

Types of percent problems- Basic percent - Double percent- Percent off- Percent change (increase/decrease)- Tax added

Solving Basic Percent Problems Algebraically

Transferring words to symbols – “What” x“is” =“of” multiply or times“out of” divide

“percent” per 100 out of 100

Basic Percent: What (number) is m% of n?

Example 1: What is 20% of 50?Solution: x = (20/100) · 50

x = 10

Example 2: What is 35% of 70?Solution: x = (35/100) · 70

x = 24.5

You Try

What is 18% of 40?

Solution –

Basic Percent: m is what percent of n?

Example 1: 25 is what percent of 90?Solution: 25 = (x/100) · 90

25 = 90x/100 25 · 100 = (90x/100) · 100 2500 = 90x

2500/90 = 90x/90 27.8 = x

Example 2: 45 is what percent of 110?Solution: 45 = (x/100) · 110 45 = 110x/100 45 · 100 = (110x/100) · 100 4500 = 110x 4500/110 = 110x/110 40.9 = x

Note: In the above problems, x is a percent.

You Try

9 is what percent of 60?

Solution -

Basic Percent: m is n% of what (number)?

Example 1: 20 is 40% of what number?Solution: 20 = (40/100) · x 20 = 40x/100 20 · 100 = (40x/100) · 100 2000 = 40x2000/40 = 40x/40 50 = x

Example 2: 100 is 72% of what number?Solution:

100 = (72/100) · x100 = 72x/100 100 · 100 = (72x/100) · 100 10000 = 72x 10000/72 = 72x/72 138.9 = x

You Try

70 is 60% of what number?

Solution -

Percents In Context

Solution:Rephrase question –

What is 70% of 40?

x = (70/100) · 40 x = 2800/100 x = 28So, they need to

win 28 games.

More Examples

Solution:Rephrase question – 56 is

what percent of 60? 56 = (x/100) · 60

56 = 60x/100 56 · 100 = (60x/100) · 100 5600 = 60x5600/60 = 60x/60 93.3 = xSo, Joshua answered

93.3% of the questions correctly.

More Examples

Solution:Rephrase question – 65 is

40.6% of what number? 65 = (40.6/100) · x 65 = 40.6x/100

65 · 100 = (40.6x/100) · 100 6500 = 40.6x

6500/40.6 = 40.6x/40.6 160.1 = x

So, Alexis received about $160.00 on her birthday.

You Try

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Solution -

Percent Off

Solution - 100% - 15% = 85% =

0.85$120 (0.85) = $102Note: Since the discount

is 15%, subtract 15% from 100%. Convert that percent to a decimal and then multiply that decimal by the regular price.

More Examples

Solution - $16.99 + $13.99 =

$30.98100% - 20% = 80% =

0.80$30.98 (0.80) = $24.78Note: Add the regular

prices. Subtract 20% from 100%. Convert percent to decimal and then multiply by sum of the regular prices.

You Try

Solution –

Tax Added

Example 1 - Find the total cost of a goldfish if the regular price is $3.85 and tax is 5%.

Solution:5% = 0.05($3.85)(0.05) = $0.19$3.85 + $0.19 = $4.04Note: Convert percent to decimal and then

multiply by regular price. Add product to regular price.

More Examples

Example 2: Find the total cost of a sled if the regular price is $149.95 and tax is 6%.

Solution: $149.95 (0.06) = $9.00$149.95 + $9.00 = $158.95

You Try

Find the total cost of a purse if the regular price is $39.50 and tax is 2%.

Solution -

Tax Added and Percent Off

Example 1: Find the total cost of a shirt on sale for 30% off if the regular price is $24.50 and tax is 2%.

Solution:100% - 30% = 70% = 0.70$24.50(0.70) = $17.152% = 0.02$17.15(0.02) = $0.34$17.15 + $0.34 = $17.49

More Examples

Example 2: Find the total cost of a cell phone on sale for 30% off if the regular price is $134.50 and tax is 3%.

Solution:100% - 30% = 70% = 0.70$134.50 (0.70) = $94.15$94.15 (0.03) = $2.82$94.15 + $2.82 = $96.97

You Try

Find the total cost of concert tickets on sale for 42% off if the regular price is $159.95 and tax is 1%.

Solution –

Percent Change

Percent Change = (big number – small number) / first

numberExample 1 - Find the percent change from

54 feet to 87.7 feet.Solution –

Percent Change = (87.7 – 54) / 54 = 0.624 = 62.4% increase

More Examples

Example 2: Find the percent change from 61 miles to 47 miles.

Solution: (61-47)/61 = 0.2295 = 22.95% ≈ 23.0% decrease

You Try

Find the percent change from 57 inches to 83 inches.

Solution –

You Try

#1 - Find the percent change from 80m to 28m.

Solution -

#2 – There is a big push in the US to raise minimum wage to at least $11/hour. In Illinois, minimum wage is $8.25/hour. What would be the percent change in minimum wage if it was raised to $11/hour in Illinois?

Solution -

Percent Change In Context

Solution - (25 – 7)/7 = 2.57 =

257% increase So, there was a

257% increase in students who scored 20+ on the ACT from two years ago to last year.

More Examples

Solution:(32 – 9)/32 = 0.719

= 71.9% decreaseSo, there is a

71.9% decrease in rainfall from July to August.

You Try

Solution –

You Try

Solution –

When Given Percent Change

Example 1: From 83 tons to x tons with a 71.1% decrease. Find x.

Solution: 71.1/100 = (83 – x) / 83 71.1(83) = 100(83 – x) 5901.3 = 8300 – 100x

5901.3 = 8300 – 100x – 8300 - 8300 -2398.7 = -100x -2398.7/-100 = -100x/-100 23.99 = x So, from 83 tons to 23.99 tons is a 71.1% decrease.

More Examples

Example 2: From 3 minutes to x minutes with a 70% increase. Find x.

Solution: 70/100 = (x – 3) / 3 70(3) = 100(x – 3) 210 = 100x – 300

210 = 100x – 300 + 300 + 300

510 = 100x 510/100 = 100x/100 5.1 = xSo, from 3 minutes to 5.1 minutes is a 70%

increase.

You Try

#1) From 93.4 hours to x hours with 47.5% decrease. Find x.

Solution -

You Try

#2) From 13 meters to x meters with a 376.9% increase. Find x.

Solution -

When Given Percent Change in Context

Solution: 14/100 = (25 – x)/x 14x = 100(25 – x) 14x = 2500 – 100x + 100x +100x 114x = 2500114x/114 = 2500/114 x = 21.9So, in the previous year

Derrick Rose averaged 21.9 points per game.

More Examples

Solution: 72.5/100 = (40 – x)/40 72.5(40) = 100(40 – x) 2900 = 4000 –

100x -4000 -4000 -1100 = -100x-1100/-100 = -100x/-100 11 = xSo, Adam Dunn has hit

11 home runs this year.

You Try

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You Try

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Rate and ProportionWhat is rate?- Comparison of one quantity to another (ratio)- Usually stated as one quantity per anotherWhat is a proportion?- 2 or more rate/ratios set equal to each other

Examples of proportions:

1/2 = 2/4 1 out of 2 is proportional to 2 out of 4 20 miles/1 hr = 40 miles/2 hrs 20 miles in 1 hours is

proportional to 40 miles in 2 hours

Note: When two things are proportional, they are similar.

Solving Proportions

2 ways to solve:Arithmetic solution:1. Divide 6 by 4

6 ÷ 4 = 1.52. Multiply 2 by 1.5

2(1.5) = 33. So, x = 3

Example Continued

Algebraic Solution:- Cross multiply to

set up equation and then solve for x4x = 6(2)4x = 12x = 3

More Examples

Arithmetic Solution:2 ÷ 4 = 0.55(0.5) = 2.5So, n = 2.5Algebraic Solution:5(2) = 4n10 = 4n2.5 = n

You Try (Choose your method)

Solution -

Setting Up and Solving Proportions

Solution:1 pkg => $3x pkgs => $91/x = 3/9Algebraic solution:3x = 9 x = 3So, she can buy 3

packages for $9.

More examples

Solution:1 bag => $2 x bags => $201/x = 2/202x = 20 x = 10So, you can buy 10

bags for $20.

You Try

You Try

Solving Percent Problems by Setting Up Proportions

• Another way to solve basic percent problems is by setting up a proportion.

• For example, a percent question may be asked, “what (number) is a% of b?”

- on the left side of the proportion we’d write the percent as a fraction (a / 100)

- the value that immediately follows “of” always goes in the denominator on the right side of the proportion

- the value that immediately comes before or after “is” always goes in the numerator of the right side of the proportion.

- so, a / 100 = x / b • Note: the unknown value will always have an “x” in its

location

Examples of Solving Basic Percent Problems by Setting Up Proportions

Example 1 – What is 25% of 30?

Solution – 25 / 100 = x / 30 100x = 25(30) 100x = 750100x/100 = 750/100 x = 7.5

So, 7.5 is 25% of 30.

Example 2 – Fifty-five is 70% of what number?

Solution – 70 / 100 = 55 / x 70x = 100(55) 70x = 5500 70x/70 = 5500/70 x = 78.8

So, 55 is 70% of 78.8.

You Try

#1 – Twenty-eight is 45% of what number?

Solution –

#2 – What number is 12% of 90?

Solution –

Unit Rate

• Unit rate is a “per 1” or “out of 1” ratio.

• For example, if a 128-ounce container of juice costs $3.99, then the unit rate would tell us what is the cost per ounce of that container of juice.

• FYI, grocery stores are required by law to provide the unit rate of every food item on the shelf.

How To Find the Unit Rate

To find the unit rate, we must set up a proportion in which the other ratio has a “1” in the denominator and an “x” in the numerator. Then solve for x.

Example 1 – A 128-ounce container of juice costs $3.99. How much does the juice cost per ounce?

Solution –

3.99 / 128 = x / 1 128x = 3.99(1) 128x = 3.99128x/128 = 3.99/128 x = 0.03

So, the juice costs $0.03 per ounce.

Note: Make sure that the unit that come after “per” is always in the denominator on both sides of the proportion.

You Try

#1 – If Kevin Durant scored 36 points in a game and a game is 48 minutes long, then how many points per minute did he score?

Solution –

#2 – If we need 5/3 of a cup of water for every 3 servings of rice, then how many cups of water would we need for 1 serving of rice?

Solution –

Unit Conversion

Convert 44 inches to feet.(Hint: 12 inches = 1 foot)Solution: Set up proportion – inches/feet =

inches/feet44/x = 12/112x = 44 x = 3.67 feetSo, 44 inches is 3.67 feet

More examples

Convert 2.5 hours to minutes.(Hint: 1 hour = 60 minutes)2.5/x = 1/60 x = 2.5(60)

x = 150So, 2.5 hours is 150 minutes.

You Try

Convert 94 ounces to pounds.(Hint: 1 pound = 16 ounces)

Solution -

You Try

Convert 0.2 hours to minutes.(Hint: 1 hour = 60 min)

Solution –

Multi-step Unit Conversions

Convert 90 feet per second to miles per hour.

(Hint: 1 mile = 5280 feet, 1 hour = 60 min, 1 min = 60 sec)

Solution: 324000 miles/5280 hr =61.36 miles per hourSo, 90 feet per second is

61.36 miles per hour

90 ft 60 sec

60 min

1 mile

1 sec 1 min

1 hr 5280 feet

Note:

1. Set up original ratio2. Convert one unit at a time by setting up

another ratio with units to be converted diagonal from each other. (For example, if inches are in numerator of one ratio, then inches should be in denominator of other ratio.)

3. Continue the process until the desired units are the only units left

4. Multiply all numbers in numerator and multiply all numbers in denominator

5. Divide numerator by denominator

More Examples80 yards 3 feet 1 min

1 min 1 yard 60 secConvert 80 yards per

minute to feet per second.

(Hint: 1 yard = 3 feet, 1 min = 60 sec)

Solution: 240 feet/60 sec = 4 feet per second

So, 80 yards per minute is 4 feet per second.

You Try

Convert 40 yards per 4 seconds to miles per hour. (Hint: 1 mile = 1760 yards, 60 seconds = 1 minute, 60 minutes = 1 hour)

Solution -

Multi-step Arithmetic Problems

Solution:1. Restate question – How much

money did we make?2. What is given from problem?- Rink charges $600 up front- Rink charges $3 per person- We charged $8 per person- 300 people attended3. What do I know?- What rink charges is an

expense- What we charged is income- Profit = income – expense4. Solve the problemP = 8(300) – 600 – 3(300) = 2400 – 600 – 900 = 900So, we made a $900 profit from our

skating party

You Try

Solution -

More ExamplesSolution - 1. I am trying to find out how many

players are from other states.2. What’s given?- There are 60 players- 1/5 are from California- 1/6 are from New York- 1/12 are from Illinois3. What do I know?- Multiply each fraction by 60 to find

the actual amount of players from each state

4. Solve the problem1/5(60) = 12 California1/6(60) = 10 NY1/12(60) = 5 IL12 + 10 + 5 = 2760 – 27 = 33 So, 33 are from other states.

You Try

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