Basic Operations & Applications Unit

Solving Arithmetic Problems Involving Percent Types of percent problems -What (number) is m% of n? -m is what percent of n? -m is n% of what (number)? -Percent off -Percent change (increase/decrease) -Tax added

What (number) is m% of n? Example 1: What is 20% of 50? Solution: 50(0.20) = 10 Note: Change percent to decimal by moving decimal point two places to the left. Then multiply number by decimal Example 2: What is 35% of 70? Solution: 70(0.35) = 24.5

You Try What is 18% of 40? Solution: 40(0.18) = 7.2

m is what percent of n? Example 1: 25 is what percent of 90? Solution: 25 ÷ 90 = = 27.8% Note: Divide first number by second number. Then change decimal to percent by moving decimal point two places to right. Example 2: 45 is what percent of 110? Solution: 45 ÷ 110 = = 40.9%

You Try 9 is what percent of 60? Solution: 9 ÷ 60 = 0.15 = 15%

m is n% of what (number)? Example 1: 20 is 40% of what number? Solution: 40% = ÷ 0.40 = 50 Note: Change percent to decimal. Then divide number by decimal. Example 2: 100 is 72% of what number? Solution: 72% = ÷ 0.72 = 138.9

You Try 70 is 60% of what number? Solution: 60% = ÷ 0.60 = 116.7

Solving Basic Percent Problems Algebraically Transferring words to symbols – “What”  x “is”  = “of”  multiply or times “out of”  divide

Examples What is 20% of 50? Solution – x = 0.20 * 50 x = 10 So, 10 is 20% of 50.

More Examples 25 is what percent of 90? Solution – 25 = x * 90 25/90 = x * 90/ = x 27.8% = x So, 25 is 27.8% of 90.

More Examples 20 is 40% of what number? Solution – 20 = 0.40 * x 20/0.40 = 0.40/0.40 * x 50 = x

Percent Word Problems Solution: Rephrase question – What is 70% of 40? 40(0.70) = 28 So, they need to win 28 games.

More Examples Solution: Rephrase question – 56 is what percent of 60? 56 ÷ 60 = = 93.3%

More Examples Solution: Rephrase question – 65 is 40.6% of what number? 65 ÷ ≈ $ So, Alexis received about $ on her birthday.

You Try

Homework Problems Do problems 1 – 18 of BOA practice problems.

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.

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.

You Try

Homework Problems Do problems 19 – 26 of BOA practice problems

Percent Change Example 1: Find the percent change from 54 feet to 87.7 feet. Solution: 87.7 – 54 = ÷ 54 = = 62.4% increase Note: Find the difference between the two values. This number represents the increase or decrease. Divide this number by the first value. Convert decimal to percent.

More Examples Example 2: Find the percent change from 61 miles to 47 miles. Solution: = ÷ 61 = = 23.0% decrease

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

You Try Find the percent change from 80m to 28m. Solution:

Percent Change Algorithm 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

When Given Percent Change Example 1: From 83 tons to x tons with a 71.1% decrease. Find x. Solution: 100% -71.1% = 28.9% = (0.289) = tons = x Note: Subtract the percent decrease from 100%. Convert the difference in percent to decimal and then multiply by first value. If given the second value, then divide.

More Examples Example 2: From 3 minutes to x minutes with a 70% increase. Find x. Solution: 100% + 70% = 170% = (1.70) = 5.1 minutes = x Note: Add the percent increase to 100%. Convert the sum of percent to decimal and then multiply by first value. If given the second value, then divide.

Solving When Given Percent Change Algebraically Example 1 – Find x. From 83 tons to x tons with 71.1% decrease. Solution – = (83 – x) / * 83 = 83 – x = 83 – x – 83 = - x = - x = x

More Examples Example 2: From 3 minutes to x minutes with a 70% increase. Find x. Solution – 0.70 = (x – 3) / * 3 = x – = x – = x 5.1 = x So, 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:

Percent Change Word Problems Solution: 25 – 7 = ÷ 7 = 2.57 = 257% increase

More Examples Solution: 32 – 9 = ÷ 32 = 0.72 = 72% decrease

More Examples Solution: 100% + 14% = 114% = ÷ 1.14 = 21.9 Note: Since the second value is given, then we divide instead of multiplying.

More Examples Solution: 100% % = 27.5% = (0.275) = 11

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Homework Problems Do problems 27 – 46 of BOA practice problems.

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.

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

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

Homework Problems Do problems 47 – 52 of BOA practice problems.

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

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

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

Homework Problems Do problems of BOA practice problems.

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 attended 3.What do I know? -What rink charges is an expense -What we charged is income -Profit = income – expense 4.Solve the problem I = 8(300) – 600 – 3(300) = 2400 – 600 – 900 = 900 So, we made a $900 profit from our skating party

You Try

More Examples Solution: 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 Illinois 3.What do I know? -Multiply each fraction by 60 to find the actual amount of players from each state 4.Solve the problem 1/5(60) = 12 California 1/6(60) = 10 NY 1/12(60) = 5 IL = – 27 = 33 So, 33 are from other states.

You Try Solution: 1.I am trying to find out how much must Kuumba Lynx pay for gas to avoid the penalty fee. 2.What’s given from problem? -¼ tank of gas is in van -½ tank was in the van when it was rented -A tank of gas holds 24 gallons -Gas costs $4.19 per gallon 3.What do I know? -Find how many gallons of gas Kuumba Lynx needs to buy -Multiply the # of gallons by $ Solve the problem -½ - ¼ = ¼ -¼ (24) = 6 gallons -6 ($4.19) = $25.14 So, Kuumba Lynx must buy $25.14 worth of gas before returning the vehicle to avoid the penalty fee.

Homework Problems Do problems 60 – 65 of BOA practice problems.

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 Example of proportion: 20 miles/1 hr = 40 miles/2 hrs

Solving Proportions 2 ways to solve: Arithmetic solution: 1.Divide 6 by 4 6 ÷ 4 = Multiply 2 by 1.5 2(1.5) = 3 3.So, x = 3

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

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

You Try (Choose your method)

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.

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.

You Try

Homework Problems Do problems of BOA practice problems.

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

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.

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

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

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 ft60 sec60 min 1 mile 1 sec1 min 1 hr5280 feet

Note: 1.Set up original ratio 2.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 Examples 80 yards3 feet1 min 1 yard60 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.

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)

Homework Problems Do problems 86 – 98 of BOA practice problems.