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%

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% = 0.4020 ÷ 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:

You Try

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:

You Try

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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Solution:

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Solution:

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

You Try

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.

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: 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.