Chapter 6

Percentages and Their Applications

Parts, wholes, changes, and everyday price decisions

A percent describes a part relative to a whole, whether you are checking a survey or planning a purchase. Learn to choose the correct base before calculating discounts, taxes, tips, and changes. All scenarios and rates are hypothetical; round money to the nearest cent, with a half cent rounded up, and retain unrounded intermediate rates unless told otherwise.

6.1

Percent, Rate, and Base

Learning objectives

After this section you will be able to:

Convert among fractions, decimals, and percents.
Identify the part, rate, and base in a situation.
Solve for a missing part, rate, or base.

Real-life situation

A club surveys its members

Use this hypothetical Philadelphia-area scenario to connect the calculation to a decision. All people, businesses, amounts, and rates in this chapter are invented; tax and payroll rules are teaching assumptions, not current law.

A club surveys its members

Members surveyed: 80
Members choosing a weekend event: 28
Weekend share: 35%

Figure 6.1 A club surveys its members (hypothetical)

The concept

A percent is a ratio with a base of 100. The statement 35% means 35 out of every 100, not necessarily that exactly 100 people were counted. The same share is 0.35 or 35/100 = 7/20.

To change a percent to a decimal, divide its numerical value by 100. To change a decimal to a percent, multiply by 100 and attach the percent sign. For a fraction, divide numerator by denominator first. Values below 1% and above 100% are valid: context determines whether they make sense.

Every basic percent problem has a part, a base, and a rate. The base is the entire quantity described by the phrase "percent of." In the club survey, 80 is the base and 28 is the part. Labeling these quantities prevents accidentally dividing by the part.

Use part = base × rate with the rate in decimal form. If the rate or base is missing, undo multiplication by division. Check the scale: a rate below 100% should give a part smaller than a positive base. A zero base cannot be used to calculate a meaningful part-to-base rate.

Fraction → decimal → percent

7/20 = 0.35 = 35%

Count 7 shaded parts out of 20 equal parts: 7/20.
Divide numerator by denominator: 7 ÷ 20 = 0.35.
Multiply the decimal by 100: 35%. This matches 28 of 80 surveyed members.

Original WealthWiseGuys diagram · Hypothetical teaching quantities.

Money Words

Plain-language words you will use in this lesson — learn them before we reach the formulas.

discount
An amount taken off a regular price.
sales tax
A percentage added by the government to the price of many purchases.
markup
The amount added to a cost to set a selling price; the markup rate uses cost as its base.

Key terms

Percent: A ratio expressed per hundred.
Base: The whole quantity used as the reference.
Part: The amount being compared with the whole.
Rate: The part-to-base ratio, often written as a percent.

Formulas & rules

Find a part
Part = base × decimal rate
Find a rate
Decimal rate = part ÷ base
Find a base
Base = part ÷ decimal rate

Worked examples

Example 1Read the survey

What percent of the 80 club members chose the weekend event if 28 selected it?

Solution

1Identify the part as 28 members and the base as 80 members.
2Divide: 28 ÷ 80 = 0.35.
3Convert: 0.35 × 100 = 35%.

Answer: 35% of surveyed members chose the weekend event.

Example 2Recover a fundraising goal

A club has raised $180, which is 30% of its goal. What is the full goal?

Solution

1Convert 30% to 0.30.
2Divide the part by the rate: 180 ÷ 0.30 = 600.
3Check: 600 × 0.30 = 180.

Answer: The goal is $600.00.

Practice exercises 6.1

Write 3/8 as a decimal and percent.
Find 12% of $250.
A team wins 18 of 24 games. Find its winning percentage.
$45 is 15% of what amount?
A student writes 0.6% as 0.6. Correct the error and find 0.6% of $500.

Section 6.1 recap

Name the base before choosing an operation.
Use decimal rates in calculations; multiply by 100 only to report a percent.

Answer key with explanations: Section 6.1

Practice exercises

3 ÷ 8 = 0.375; 0.375 × 100 = 37.5%.
12% = 0.12; 250 × 0.12 = $30.00.
18 ÷ 24 = 0.75; multiply by 100 to get 75%.
45 ÷ 0.15 = $300.00; 300 × 0.15 = 45 checks the result.
Divide by 100: 0.6% = 0.006, not 0.6. Then 500 × 0.006 = $3.00.

Recap quiz

1.25: 125 ÷ 100 = 1.25; a percent can exceed 100%.
12: 60 × 0.20 = 12.
30%: 9 ÷ 30 = 0.30 = 30%.

6.2

Percent Change and Percentage Points

Learning objectives

After this section you will be able to:

Compute increases and decreases using the original amount.
Distinguish percent change from percentage-point change.
Find an original amount from a changed amount.

Real-life situation

A phone plan changes price

Old monthly price: $40.00
New monthly price: $46.00
Increase: $6.00, or 15%

Figure 6.2 A phone plan changes price (hypothetical)

The concept

Percent change compares a difference with the original amount. Calculate new minus old first, then divide by old. A positive result describes an increase and a negative result describes a decrease. If the original amount is zero, ordinary percent change is undefined.

The denominator matters. A price rising from $40 to $46 increased by 15%, but a drop from $46 to $40 would use $46 as its base. Reversing the direction changes the percentage even when the dollar difference stays the same.

When comparing two rates, subtraction gives percentage points. A survey share moving from 20% to 25% rises by 5 percentage points. Its relative increase is 5 ÷ 20 = 25%. These statements answer different questions and should not be interchanged.

To work backward, use a multiplier. After a 15% increase, the new amount equals 1.15 times the old amount. After a 15% decrease it equals 0.85 times the old amount. Successive changes multiply; equal percentage increases and decreases do not generally cancel.

Compare with the original price

Original plan$ 40

New plan$ 46

Bars share a zero baseline.

Figure 6.2.V Compare with the original price. Hypothetical teaching data.

Money Words

Plain-language words you will use in this lesson — learn them before we reach the formulas.

profit margin
Profit shown as a percent of revenue; a 10% margin means $10 of profit per $100 of sales.
return
The money or gain an investment produces, often shown as a percent.
ROI
Return on investment — the gain compared to what you put in, as a percent: (gain ÷ cost) × 100.

Key terms

Percent change: Difference divided by the original amount, expressed as a percent.
Original amount: The starting value that supplies the comparison base.
Multiplier: The factor applied to the original value.
Percentage point: One unit of difference between values already expressed as percents.

Formulas & rules

Percent change
Percent change = (new − old) ÷ old × 100%
Increase multiplier
New = old × (1 + decimal increase rate)
Decrease multiplier
New = old × (1 − decimal decrease rate)
Rate difference
Percentage-point change = new percent − old percent

Worked examples

Example 1Compare two prices

A phone plan rises from $40 to $46 monthly. Find its dollar and percent increase.

Solution

1Subtract: 46 − 40 = $6.00.
2Use the original base: 6 ÷ 40 = 0.15.
3Convert: 0.15 × 100 = 15%.

Answer: The monthly price increases $6.00, or 15%.

Example 2Undo a reduction

A console accessory costs $72 after a 20% reduction. What was its original price?

Solution

1The remaining share is 100% − 20% = 80% = 0.80.
2Solve 72 = original × 0.80: original = 72 ÷ 0.80 = $90.00.
3Check the discount: 90 × 0.20 = $18.00; 90 − 18 = $72.00.

Answer: The original price was $90.00.

Practice exercises 6.2

A ticket rises from $25 to $30. Find percent change.
A price drops from $80 to $60. Find percent change.
A response rate rises from 30% to 36%. Give both comparisons.
After a 10% increase a fee is $55. Find the old fee.
A $100 price rises 20%, then falls 20%. Does it return to $100?

Section 6.2 recap

Use the original amount as the denominator of percent change.
Distinguish a difference in rates from a relative change in rates.

Answer key with explanations: Section 6.2

Practice exercises

Difference = 30 − 25 = 5; 5 ÷ 25 × 100% = 20% increase.
(60 − 80) ÷ 80 × 100% = −25%, a 25% decrease.
36 − 30 = 6 percentage points; 6 ÷ 30 × 100% = 20% relative increase.
55 ÷ 1.10 = $50.00; 50 × 1.10 = 55.
100 × 1.20 = 120; 120 × 0.80 = $96.00. It is $4.00 lower because the second base is larger.

Recap quiz

−20%: (40 − 50) ÷ 50 × 100% = −20%.
2: Subtract the rates as stated: 10 − 8 = 2 percentage points.
0.75: The sale price is 75% of the original, so divide by 0.75.

6.3

Discounts, Sales Tax, and Tips

Learning objectives

After this section you will be able to:

Calculate a sale price and a tax-inclusive total.
Apply a stated tipping base correctly.
Compare successive discounts with a single discount.

Real-life situation

A weekend meal budget

Meal before tax: $40.00
Hypothetical tax: 6% = $2.40
Tip on pretax meal: 18% = $7.20
Total: $49.60

Figure 6.3 A weekend meal budget (hypothetical)

The concept

A discount reduces a stated price by a percent of that price. Find the discount amount first, round it to cents, and subtract it. The amount remaining is the sale price; a 25% discount leaves 75% of the original price.

In this chapter, taxable purchases are taxed after discounts. Find the tax using the discounted taxable subtotal, round that tax to cents, and add it to the subtotal. Actual taxability varies, so the fictional rate in a problem is an assumption rather than a statement of local law.

A tip also needs a clearly identified base. Here, a meal tip is calculated on the pretax meal amount, not on the tax. Add meal, tax, and tip as separate components. If people share the bill equally, divide the final total and account for any leftover cent.

Successive discounts apply to changing bases. A 20% discount followed by 10% off the reduced price leaves 0.80 × 0.90 = 0.72 of the original, an equivalent 28% discount, not 30%. For actual currency transactions, round each discount before moving to the next step; this can affect a final cent.

From meal price to amount paid

1
Meal
$40.00
2
Add tax and pretax tip
$2.40 + $7.20
3
Total
$49.60

Figure 6.3.V From meal price to amount paid. Hypothetical teaching data.

Money Words

Plain-language words you will use in this lesson — learn them before we reach the formulas.

yield
The income an investment pays, shown as a percent of its price.
interest rate
The percent used to calculate how much interest is earned or charged.
APR
Annual percentage rate — the yearly cost of borrowing, including interest and some fees, shown as a percent.

Key terms

Discount: A reduction from a stated price.
Sale price: Price remaining after a discount.
Taxable subtotal: Amount to which the problem says tax applies.
Tip base: Amount used to calculate a gratuity.
Successive discounts: Discounts applied one after another to changing amounts.

Formulas & rules

Discount amount
Discount = price × decimal discount rate
Sale price
Sale price = original price − discount
Tax and total
Total = taxable subtotal + rounded tax
Tip
Tip = stated tip base × decimal tip rate

Worked examples

Example 1Discount before tax

A $60 backpack is 25% off. A hypothetical 6% tax applies to its sale price. Find the final total.

Solution

1Discount: 60 × 0.25 = $15.00.
2Sale price: 60 − 15 = $45.00.
3Tax: 45 × 0.06 = $2.70.
4Total: 45 + 2.70 = $47.70.

Answer: The final total is $47.70.

Example 2A meal with a stated tip base

A $40 meal has hypothetical 6% tax and an 18% tip on its pretax price. What is paid?

Solution

1Tax: 40 × 0.06 = $2.40.
2Tip: 40 × 0.18 = $7.20.
3Add the components: 40 + 2.40 + 7.20 = $49.60.

Answer: The payment is $49.60.

Practice exercises 6.3

Find the price of a $90 jacket after 30% off, before tax.
Add hypothetical 5% tax to a $24 taxable subtotal.
A $50 meal has hypothetical 8% tax and a 20% pretax tip. Find the total.
Apply 20% off, then another 10% off, to $100. Find the equivalent discount.
Split a $53.00 final bill equally among three friends to the nearest cent, keeping the sum correct.

Section 6.3 recap

Calculate each percent on its stated base and round each monetary component.
Successive percentage discounts multiply; they do not simply add.

Answer key with explanations: Section 6.3

Practice exercises

Discount = 90 × 0.30 = $27.00; sale price = 90 − 27 = $63.00.
Tax = 24 × 0.05 = $1.20; total = 24 + 1.20 = $25.20.
Tax = $4.00 and tip = $10.00; 50 + 4 + 10 = $64.00.
First price = $80.00; second discount = $8.00; final = $72.00. Savings = $28.00, or 28% of $100.
53 ÷ 3 = 17.666...; three payments of $17.67 would total $53.01. Use $17.67, $17.67, and $17.66, totaling $53.00.

Recap quiz

$47.70: Discount = $5; sale price = $45; tax = $2.70; total = $47.70.
$3.00: 20 × 0.15 = $3.00.
$81.00: 100 × 0.90 = 90; 90 × 0.90 = $81.00.

Chapter 6 Organizer and Review

Chapter Organizer

Every section at a glance: key formulas, key terms, and the main idea.

6.1 Percent, Rate, and Base
Key formulas
- Find a part: Part = base × decimal rate
- Find a rate: Decimal rate = part ÷ base
- Find a base: Base = part ÷ decimal rate
Key terms
Percent, Base, Part, Rate
Remember
Name the base before choosing an operation.
6.2 Percent Change and Percentage Points
Key formulas
- Percent change: Percent change = (new − old) ÷ old × 100%
- Increase multiplier: New = old × (1 + decimal increase rate)
- Decrease multiplier: New = old × (1 − decimal decrease rate)
- Rate difference: Percentage-point change = new percent − old percent
Key terms
Percent change, Original amount, Multiplier, Percentage point
Remember
Use the original amount as the denominator of percent change.
6.3 Discounts, Sales Tax, and Tips
Key formulas
- Discount amount: Discount = price × decimal discount rate
- Sale price: Sale price = original price − discount
- Tax and total: Total = taxable subtotal + rounded tax
- Tip: Tip = stated tip base × decimal tip rate
Key terms
Discount, Sale price, Taxable subtotal, Tip base, Successive discounts
Remember
Calculate each percent on its stated base and round each monetary component.

Cumulative Review

Mixed practice from this chapter and the two chapters before it.

Section 6.1
A student writes 0.6% as 0.6. Correct the error and find 0.6% of $500.
Show answer
Divide by 100: 0.6% = 0.006, not 0.6. Then 500 × 0.006 = $3.00.
Section 6.2
A $100 price rises 20%, then falls 20%. Does it return to $100?
Show answer
100 × 1.20 = 120; 120 × 0.80 = $96.00. It is $4.00 lower because the second base is larger.
Section 6.3
Split a $53.00 final bill equally among three friends to the nearest cent, keeping the sum correct.
Show answer
53 ÷ 3 = 17.666...; three payments of $17.67 would total $53.01. Use $17.67, $17.67, and $17.66, totaling $53.00.
Review of Section 4.3
Current balance $1,050.00, pending debits $220.45. Available balance?
Show answer
$829.55
Review of Section 5.3
A number plus 18 is 47. Find the number.
Show answer
n + 18 = 47 → 29

All prices, stores, people, and accounts in this chapter are hypothetical and for teaching only. Product references are generic categories; no brand endorses or is endorsed by this textbook.

Chapter 5 Chapter 7