1. Basic Operations & Applications Unit

2. 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.

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

4. You Try #1 – Reduce 24/66 to lowest terms. Solution –

5. 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 = 7/25 So, Askia spent seven twenty-fifths of the money that his grandmother gave him.

6. 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 = 5/6 So, five out of six students in the class thought that they did not have a good math teacher when they were in elementary school.

7. 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 –

8. 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.

9. 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.

10. 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.

11. 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 –

12. 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.

13. 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 = ?/35 4 x 5/7 x 5 = 20/35 2 x 7/5 x 7 = 14/35 Now, add the numerators. 20/ /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 = ?/24 1/6 = ?/24 3x3/8x3 = 9/24 1x4/6x4 = 4/24 Now, subtract the numerators. 9/24 – 4/24 = 5/24

14. 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 1/3 The LCM between 5 and 3 is 15. 3/5 = ?/15 and 1/3 = ?/15 3x3/5x3 = 9/15 and 1x5/3x5 = 5/15 9/15 + 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.

15. 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 –

16. 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

17. You Try
#1) 2/7 x 5/8 = ? Solution –
#2) 3/4 x 2/5 = ? Solution –

18. 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

19. You Try
#3) 7/12 ÷ 2/5 = ? Solution -
#4) 2/3 ÷ 8/9 = ? Solution -

20. 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.

21. 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 · x ¼ ÷ 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.

22. 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 –

23. 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

24. 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?
Answer – The underlined place value is the hundredths place.

25. 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 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 ≈ Example 4 – Round to the nearest tenths place. 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 ≈ 3.6

26. 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 – the place value that is farthest to the right is the hundredths place 48/100 = 12/25

27. You Try
#1 – Convert 10/17 to a decimal. Round to the nearest hundredths place. Solution –
#2 – Convert to a fraction. Solution –

28. 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 80/100 When 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 = = 37.5/100 => 37.5 out of 100 => 37.5% Example 2 – Convert 5/12 to a percent. 5/12 = 0.416̅ = 41.6̅/100 => 41.6̅ out of 100 => 41.6̅% Note: If a fraction is more than 1, then it is more than 100% since 1 = 100%.

29. 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

30. You Try #1 – Convert 8/5 to a percent. Solution –
#2 – Convert 110% to a fraction. Solution –

31. Solving Arithmetic Problems Involving Percent
Types of percent problems
Basic percent
Double percent
Percent off
Percent change (increase/decrease)
Tax added

32. 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

33. 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

34. You Try
What is 18% of 40? Solution –

35. Basic Percent: m is what percent of n?
Example 1: 25 is what percent of 90? Solution: 25 = (x/100) · = 90x/ · 100 = (90x/100) · = 90x 2500/90 = 90x/ = x Example 2: 45 is what percent of 110? Solution: 45 = (x/100) · = 110x/ · 100 = (110x/100) · = 110x 4500/110 = 110x/ = x Note: In the above problems, x is a percent.

36. You Try
9 is what percent of 60? Solution -

37. Basic Percent: m is n% of what (number)?
Example 1: 20 is 40% of what number? Solution: 20 = (40/100) · x 20 = 40x/ · 100 = (40x/100) · = 40x 2000/40 = 40x/40 50 = x Example 2: 100 is 72% of what number? 100 = (72/100) · x 100 = 72x/ · 100 = (72x/100) · = 72x 10000/72 = 72x/ = x

38. You Try
70 is 60% of what number? Solution -

39. Percents In Context
Solution: Rephrase question – What is 70% of 40? x = (70/100) · 40 x = 2800/100 x = 28 So, they need to win 28 games.

40. More Examples
Solution: Rephrase question – 56 is what percent of 60? 56 = (x/100) · = 60x/ · 100 = (60x/100) · = 60x 5600/60 = 60x/ = x So, Joshua answered 93.3% of the questions correctly.

41. More Examples
Solution: Rephrase question – 65 is 40.6% of what number? 65 = (40.6/100) · x 65 = 40.6x/ · 100 = (40.6x/100) · = 40.6x 6500/40.6 = 40.6x/ = x So, Alexis received about $ on her birthday.

42. You Try
Solution –

43. You Try
Solution -

44. You Try
Solution -

45. Percent Off
Solution - 100% - 15% = 85% = 0.85 $120 (0.85) = $102 Note: Since the discount is 15%, subtract 15% from 100%. Convert that percent to a decimal and then multiply that decimal by the regular price.

46. More Examples
Solution - $ $13.99 = $ % - 20% = 80% = 0.80 $30.98 (0.80) = $24.78 Note: Add the regular prices. Subtract 20% from 100%. Convert percent to decimal and then multiply by sum of the regular prices.

47. You Try
Solution –

48. 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 $ $0.19 = $4.04 Note: Convert percent to decimal and then multiply by regular price. Add product to regular price.

49. More Examples
Example 2: Find the total cost of a sled if the regular price is $ and tax is 6%. Solution: $ (0.06) = $9.00 $ $9.00 = $158.95

50. You Try
Find the total cost of a purse if the regular price is $39.50 and tax is 2%. Solution -

51. 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) = $ % = 0.02 $17.15(0.02) = $0.34 $ $0.34 = $17.49

52. More Examples
Example 2: Find the total cost of a cell phone on sale for 30% off if the regular price is $ and tax is 3%. Solution: 100% - 30% = 70% = 0.70 $ (0.70) = $94.15 $94.15 (0.03) = $2.82 $ $2.82 = $96.97

53. You Try
Find the total cost of concert tickets on sale for 42% off if the regular price is $ and tax is 1%. Solution –

54. Percent Change
Percent Change = (big number – small number) / first number Example 1 - Find the percent change from 54 feet to 87.7 feet. Solution – Percent Change = (87.7 – 54) / 54 = = 62.4% increase

55. More Examples
Example 2: Find the percent change from 61 miles to 47 miles. Solution: (61-47)/61 = = 22.95% ≈ 23.0% decrease

56. You Try
Find the percent change from 57 inches to 83 inches. Solution –

57. 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?

58. 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.

59. More Examples
Solution: (32 – 9)/32 = = 71.9% decrease So, there is a 71.9% decrease in rainfall from July to August.

60. You Try
Solution –

61. You Try
Solution –

62. 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) = 100(83 – x) = 8300 – 100x – = -100x /-100 = -100x/ = x So, from 83 tons to tons is a 71.1% decrease.

63. 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 – = 100x 510/100 = 100x/ = x So, from 3 minutes to 5.1 minutes is a 70% increase.

64. You Try
#1) From 93.4 hours to x hours with 47.5% decrease. Find x. Solution -

65. You Try
#2) From 13 meters to x meters with a 376.9% increase. Find x. Solution -

66. When Given Percent Change in Context
Solution: 14/100 = (25 – x)/x 14x = 100(25 – x) 14x = 2500 – 100x + 100x +100x 114x = x/114 = 2500/114 x = 21.9 So, in the previous year Derrick Rose averaged 21.9 points per game.

67. More Examples
Solution: 72.5/100 = (40 – x)/ (40) = 100(40 – x) 2900 = 4000 – 100x = -100x -1100/-100 = -100x/ = x So, Adam Dunn has hit 11 home runs this year.

68. You Try
Solution -

69. You Try
Solution -

70. Rate and Proportion What is rate?
- Comparison of one quantity to another (ratio)
Usually stated as one quantity per another
What 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.

71. Solving Proportions 2 ways to solve: Arithmetic solution:
Divide 6 by 4
6 ÷ 4 = 1.5
Multiply 2 by 1.5
2(1.5) = 3
3. So, x = 3

72. Example Continued
Algebraic Solution: - Cross multiply to set up equation and then solve for x 4x = 6(2) 4x = 12 x = 3

73. More Examples
Arithmetic Solution: 2 ÷ 4 = 0.5 5(0.5) = 2.5 So, n = 2.5 Algebraic Solution: 5(2) = 4n 10 = 4n 2.5 = n

74. You Try (Choose your method)
Solution -

75. Setting Up and Solving Proportions
Solution: 1 pkg => $3 x pkgs => $9 1/x = 3/9 Algebraic solution: 3x = 9 x = 3 So, she can buy 3 packages for $9.

76. More examples
Solution: 1 bag => $2 x bags => $20 1/x = 2/20 2x = 20 x = 10 So, you can buy 10 bags for $20.

77. You Try

78. You Try

79. 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

80. Examples of Solving Basic Percent Problems by Setting Up Proportions
Example 1 – What is 25% of 30? Solution – 25 / 100 = x / x = 25(30) 100x = x/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 = x/70 = 5500/70 x = 78.8 So, 55 is 70% of 78.8.

81. You Try #1 – Twenty-eight is 45% of what number? Solution –
#2 – What number is 12% of 90? Solution –

82. 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.

83. 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 = x/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.

84. 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 –

85. Unit Conversion
Convert 44 inches to feet. (Hint: 12 inches = 1 foot) Solution: Set up proportion – inches/feet = inches/feet 44/x = 12/1 12x = 44 x = 3.67 feet So, 44 inches is 3.67 feet

86. More examples
Convert 2.5 hours to minutes. (Hint: 1 hour = 60 minutes) 2.5/x = 1/60 x = 2.5(60) x = 150 So, 2.5 hours is 150 minutes.

87. You Try
Convert 94 ounces to pounds. (Hint: 1 pound = 16 ounces) Solution -

88. You Try
Convert 0.2 hours to minutes. (Hint: 1 hour = 60 min) Solution –

89. 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: miles/5280 hr = miles per hour So, 90 feet per second is miles per hour
90 ft
60 sec
60 min
1 mile
1 sec
1 min
1 hr
5280 feet

90. Note: Set up original ratio
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.)
Continue the process until the desired units are the only units left
Multiply all numbers in numerator and multiply all numbers in denominator
Divide numerator by denominator

91. More Examples
80 yards
3 feet
1 min
1 yard
60 sec
Convert 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.

92. 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 -

93. Multi-step Arithmetic Problems
Solution:
Restate question – How much money did we make?
What is given from problem?
Rink charges $600 up front
Rink charges $3 per person
We charged $8 per person
300 people attended
What do I know?
What rink charges is an expense
What we charged is income
Profit = income – expense
Solve the problem
P = 8(300) – 600 – 3(300)
= 2400 – 600 – 900 = 900
So, we made a $900 profit from our skating party

94. You Try
Solution -

95. More Examples Solution -
I am trying to find out how many players are from other states.
What’s given?
There are 60 players
1/5 are from California
1/6 are from New York
1/12 are from Illinois
What do I know?
Multiply each fraction by 60 to find the actual amount of players from each state
Solve the problem
1/5(60) = 12 California
1/6(60) = 10 NY
1/12(60) = 5 IL
= 27
60 – 27 = 33
So, 33 are from other states.

96. You Try
Solution –