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) = 10Note: 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 = 0.278 = 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 = 0.409 = 40.9%
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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% = 0.4020 ÷ 0.40 = 50Note: Change percent to decimal. Then divide number by
decimal.Example 2: 100 is 72% of what number?Solution:72% = 0.72100 ÷ 0.72 = 138.9
You Try
70 is 60% of what number?Solution: 60% = 0.6070 ÷ 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 * 50x = 10
So, 10 is 20% of 50.
More Examples
25 is what percent of 90?
Solution – 25 = x * 9025/90 = x * 90/90 0.278 = x27.8% = 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) = 28So, they need to win 28
games.
More Examples
Solution:Rephrase question – 56 is
what percent of 60?56 ÷ 60 = 0.933 = 93.3%
More Examples
Solution:Rephrase question – 65 is
40.6% of what number?65 ÷ 0.406 ≈ $160.00So, Alexis received about
$160.00 on her birthday.
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Homework Problems
Do problems 1 – 18 of BOA practice problems.
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
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 = 33.733.7 ÷ 54 = 0.624 = 62.4% increaseNote: 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 -47 = 1414 ÷ 61 = 0.230 = 23.0% decrease
You Try
Find the percent change from 57 inches to 83 inches.
Solution:
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Find the percent change from 80m to 28m.Solution:
Percent Change Algorithm
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
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% = 0.28983 (0.289) = 23.99 tons = xNote: 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% = 1.703(1.70) = 5.1 minutes = xNote: 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 – 0.711 = (83 – x) / 83
0.711 * 83 = 83 – x 59.01 = 83 – x 59.01 – 83 = - x
-23.99 = - x 23.99 = x
More Examples
Example 2: From 3 minutes to x minutes with a 70% increase. Find x.
Solution – 0.70 = (x – 3) / 30.70 * 3 = x – 3 2.1 = x – 3
2.1 + 3 = 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:
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2. From 13 meters to x meters with a 376.9% increase. Find x.
Solution:
Percent Change Word Problems
Solution:25 – 7 = 1818 ÷ 7 = 2.57 = 257%
increase
More Examples
Solution:32 – 9 = 2323 ÷ 32 = 0.72 = 72%
decrease
More Examples
Solution:100% + 14% = 114% = 1.1425 ÷ 1.14 = 21.9Note: Since the second
value is given, then we divide instead of multiplying.
More Examples
Solution:100% - 72.5% = 27.5% =
0.27540 (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$3.85 + $0.19 = $4.04Note: 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 $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:
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) = $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:
Homework Problems
Do problems 53 -57 of BOA practice problems.
Multi-step Arithmetic ProblemsSolution: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 problemI = 8(300) – 600 – 3(300) = 2400 – 600 – 900 = 900So, we made a $900 profit from our skating
party
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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 state4. 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 TrySolution: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 gallon3. What do I know?- Find how many gallons of gas Kuumba Lynx
needs to buy- Multiply the # of gallons by $4.194. 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 anotherWhat is a proportion?- 2 or more rate/ratios set equal to each otherExample 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 = 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 x
4x = 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)
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.
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Homework Problems
Do problems 66 -85 of BOA practice problems.
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.
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Convert 94 ounces to pounds.(Hint: 1 pound = 16 ounces)
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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: 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)
Homework Problems
Do problems 86 – 98 of BOA practice problems.