Section 4.1: Percents

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Learning Objectives

• Convert decimals to percents
• Convert percents to decimals
• Solve application problems involving percents

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Definition: Percents

Percent means "per hundred" or "out of 100." It's a way to express a fraction or ratio as parts of 100. So,

The symbol % is derived from the Latin words "per centum".

 

To Convert a Decimal to a Percent

Method 1: Multiply the decimal by 100, then and add the percent sign (%). For example:

• 0.75 → 0.75 × 100 = 75 → 75%
• 0.4 → 0.4 × 100 = 40 → 40%
• 0.09 → 0.09 × = 9 → 9%

Method 2: Move the decimal point two places to the right, adding zeros if needed, then add the percent sign (%). For example:

• 0.75 → move decimal point two places right → 75 → 75%
• 0.4 → move decimal point two places right → 40 → 40%
• 0.09 → move decimal point two places right → 9 → 9%

 

Example #4.1.1 🤔

Express each decimal as a percent.

1. 0.56
2. 0.056
3. 0.27
4. 0.8
5. 0.08
6. 8
7. 1.2 
8. 1

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✅ Solution:

• Method 1: Multiply the decimal by 100, then add the percent sign (%).
• Method 2: Move the decimal point two places to the right, adding zeros if needed, then add the percent sign (%).

1. 0.56 → 0.56 × 100 = 56 → 56%    or    0.56 → move decimal two places right → 56 → 56%
2. 0.056 → 0.056 × 100 = 5.6 → 5.6%    or    0.056 → move decimal two places right → 5.6 → 5.6%
3. 0.27 → 0.27 × 100 = 27 → 27%    or    0.27 → move decimal two places right → 27 → 27%
4. 0.8 → 0.8 × 100 = 80 → 80%    or    0.8 → move decimal two places right, add 1 zero → 80 → 80%
5. 0.08 → 0.08 × 100 = 8 → 8%    or    0.08 → move decimal two places right → 8 → 8%
6. 8 → 8 × 100 = 800 → 800%    or    8 → move decimal two places right, add 2 zeros → 800 → 800%
7. 1.2 → 1.2 × 100 = 120 → 120%    or    1.2 → move decimal two places right, add 1 zero → 120 → 120%
8. 1 → 1 × 100 = 100 → 100%    or    1 → move decimal two places right, add 2 zeros → 100 → 100%

 

To Convert a Percent to a Decimal

Method 1: Drop the percent sign, then divide by 100. For example:

• 25% → 25 ÷ 100 = 0.25
• 70% → 70 ÷ 100 = 0.70 (or 0.7)
• 3% → 3 ÷ 100 = 0.03

Method 2: Drop the percent sign, then move the decimal point two places to the left, adding zeros if needed. For example:

• 25% → move decimal point two places left → 0.25
• 70% → move decimal point two places left → 0.70
• 3% → move decimal point two places left → 0.03

 

Example #4.1.2 🤔

Express each decimal as a percent.

1. 47%
2. 90%
3. 9%
4. 7.75%
5. 0.01%
6. 150%
7. 6\(\frac{1}{4}\)%
8. 2\(\frac{3}{8}\)%

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✅ Solution:

• Method 1: Drop the percent sign, then divide by 100.
• Method 2: Drop the percent sign, then move the decimal point two places to the left, adding zeros if needed.

1. 47% → 47 ÷ 100 = 0.47    or    47% → 47 → move decimal two places left → 0.47
2. 90% → 90 ÷ 100 = 0.90 (or 0.9)    or    90% → 90 → move decimal two places left → 0.90 (or 0.9)
3. 9% → 9 ÷ 100 = 0.09    or    9% → 9 → move decimal two places left, add 1 zero → 0.09%
4. 7.75% → 7.75 ÷ 100 = 0.0775    or    7.75% → 7.75 → move decimal two places left, add 1 zero → 0.0775
5. 0.01% → 0.01 ÷ 100 = 0.0001    or    0.01% → 0.01 → move decimal two places left, add 2 zeros → 0.0001
6. 150% → 150 ÷ 100 = 1.5    or    150% → 150 → move decimal two places left → 1.50 (or 1.5)
7. 6\(\frac{1}{4}\)% → 6.25 ÷ 100 = 0.0625    or    6\(\frac{1}{4}\)% = 6.25% → 6.25 → move decimal two places left, add 1 zero → 0.0625
8. 2\(\frac{3}{8}\)% → 2.375 ÷ 100 = 0.02375    or    2\(\frac{3}{8}\)% = 2.375% → 2.375 → move decimal two places left, add 1 zero → 0.02375

 

Tax calculations represent one of the most frequent and practical uses of percent in daily life. One type of tax is a state sales tax. Sales tax is a consumption tax that affects nearly every purchase we make in our daily lives. When you buy clothing, electronics, restaurant meals, or most other goods and services, you pay not only the listed price but also an additional percentage called sales tax. This tax is collected by the business at the time of purchase and then forwarded to government authorities, making retailers essentially tax collectors for the state and local governments.

The way sales tax works is straightforward: it's calculated as a percentage of your purchase price and added to your total bill. For example, if you buy a $50 item in an area with 8% sales tax, you'll pay an additional $4 in tax ($50 × 0.08 = $4), making your total cost $54. The tax rate you pay depends on where you shop, as it combines state sales tax set by your state government with any additional local taxes imposed by counties, cities, or special districts. This is why you might notice different tax rates when shopping in different cities or counties. For example, as of 2025, in Orange County, CA., the city of Costa Mesa has a 7.75% sales tax rate, whereas the city of Fountain Valley has an 8.75% sales tax rate and the city of Santa Ana has a 9.25% sales tax rate. (California's base state sales tax rate is 7.25%, which includes a 6% base rate plus a mandatory 1.25% local rate. So, Costa Mesa added an additional 0.5% tax rate, while Fountain Valley added and additional 1.5% tax rate, and Santa Ana added an additional 2.0% tax rate).

Sales tax serves as a crucial source of revenue for government services that benefit the community. The money collected helps fund public schools, maintain roads and infrastructure, support police and fire departments, and provide various local services. While most goods and services are subject to sales tax, many states exempt essential items like groceries and prescription medications to reduce the tax burden on basic necessities. Understanding sales tax helps you budget more accurately and become a more informed consumer and citizen.

 

How to Calculate Sales Tax & Total Cost:

Tax Amount = Purchase Price × Tax Rate
Total Cost = Purchase Price + Tax Amount

• Step 1: Convert the tax rate from percent to decimal
• Step 2: Multiply the purchase price by the decimal tax rate
• Step 3: Add the tax amount to the original price

Alternative Method: The Multiplier Approach:

Instead of calculating tax separately, you can find the total in one step:

• Total Cost = Purchase Price × (1 + Tax Rate)

 

Sales tax calculation uses a straightforward percent formula that you'll encounter countless times as an informed consumer and citizen.

How to Calculate Sales Tax

Step 1: Convert the tax rate from percent to decimal

• Move decimal point two places to the left (or divide by 100)
• Example: 8.25% becomes 0.0825

Step 2: Multiply the purchase price by the decimal tax rate

• This gives you the tax amount in dollars

Step 3: Add the tax amount to the original price

• This gives you the total amount you'll pay

 

Example #4.1.3 🤔

Find the sales tax and the total cost on a 65” television that costs $1,699.99. The tax rate is 8%.

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✅ Solution:

• Step 1: Convert percent to decimal:  8% → 0.08
• Step 2: Calculate tax amount:  $1,699.99 × 0.08 = $135.9992
  • Rounding to the nearest cent, we have $136.00
• Step 3: Find total cost:  $1,699.99 + $136.00 = $1,835.99

So, the tax is $1,699.99 and the total cost is $1,835.99.

Example #4.1.4 🤔

Find the sales tax and total cost on a new 2024 Toyota Tundra Limited Edition that costs $61,711. Assume a tax rate of 7.75%.

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✅ Solution:

• Step 1: Convert percent to decimal:  7.75% → 0.0775
• Step 2: Calculate tax amount:  $61,711 × 0.0775 = $4,782.6025
  • Rounding to the nearest cent, we have $4,782.61
• Step 3: Find total cost:  $61,711 + $4,782.61 = $66,493.61

So, the tax is $4,782.61 and the total cost is $66,493.61.

Example #4.1.5 🤔

Find the total cost including tax of a Samsung Bespoke Ultra Capacity Front Load Washer and Electric Dryer Set that costs $2,098.00. Assume the tax rate is 9.25%.

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✅ Solution:

So, since we are not asked to calculate tax, we can use the the alternate method to calculate the total. 

We know 9.25% written as a decimal is 0.0925.

Now, use the alternate method.

\[\begin{align} \text{Total Cost} &= \text{Purchase Price × (1 + Tax Rate)}\nonumber\\ \text{Total Cost} &= \text\$2,098.00 × (1 + 0.0925)\nonumber\\ &= \text\$2.098.00 × (1.0925)\nonumber\\ &= \text\$2,292.065\nonumber\\ \end{align}\]

So, the total cost rounded to the nearest cent is $2,292.07.

 

Another practical application of percent is a down payment. A down payment is an upfront payment made when purchasing expensive items, typically expressed as a percentage of the total purchase price. Down payments are commonly required when making major purchases that involve financing. Whether you're buying a home, purchasing a car, investing in major appliances or furniture for your living space, or acquiring business equipment for a company, lenders typically expect you to pay a percentage of the total cost upfront. This initial payment demonstrates your financial commitment to the purchase and reduces the amount you need to borrow, making it a standard practice across many industries and types of significant investments.

 

How to Calculate a Down Payment

Down Payment Amount = Purchase Price × Down Payment Rate

Loan Amount/Balance = Purchase Price – Down Payment

 

Example #4.1.6 🤔

If someone wanted to put a 10% down payment on a used boat that cost $7,000, what is the down payment and the balance after the down payment?

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✅ Solution:

We know 10% written as a decimal is 0.10.

So,

\[\begin{align} \text{Down Payment Amount} &= \text{Purchase Price × Down Payment Rate} \nonumber \\  &= \text{\(\$7,000\)} × (0.10) \nonumber \\ &= \text\$700 \nonumber \\ \end{align}\]

and

\[\begin{align} \text{Loan Amount/Balance} &= \text{Purchase Price – Down Payment} \nonumber \\  &= \text{\(\$7,000\) – \(\$700\)} \nonumber \\ &= \text\$6,300 \nonumber \\ \end{align}\]

So, the down payment is $700 and the balance is $6,300.

Example #4.1.7 🤔

A couple wants to put a 5% down payment on a new home that has a purchase price of $1,200,000. What is the down payment and the balance after the down payment?

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✅ Solution:

We know 5% written as a decimal is 0.05.

So,

\[\begin{align} \text{Down Payment Amount} &= \text{Purchase Price × Down Payment Rate} \nonumber \\  &= \text{\(\$1,200,000\)} × (0.05) \nonumber \\ &= \text\$60,000 \nonumber \\ \end{align}\]

and

\[\begin{align} \text{Loan Amount/Balance} &= \text{Purchase Price – Down Payment} \nonumber \\  &= \text{\(\$1,200,000\) – \(\$60,000\)} \nonumber \\ &= \text\$1,140,000 \nonumber \\ \end{align}\]

So, the down payment is $60,000 and the balance is $1,140,000.

Section 4.1: Percents [In-Class Exercises]

1. Express each decimal as a percent.

1. 0.61
2. 0.7
3. 0.07
4. 4
5. 1.72
6. 0.12345
7. 1 
8. 0.0001

2. Express each percent as a decimal.

1. 60%
2. 2%
3. 8.25%
4. 0.001%
5. 125%
6. 17.9%
7. 3\(\frac{1}{8}\)%
8. 12\(\frac{3}{4}\)%

3. A Black Friday sale has the Nintendo Switch 2 on sale for $429.99. If your city sales tax rate is 8.75%, what is the sales tax on the purchase of a brand new Nintendo Switch 2?
4. Find the total cost including tax of a Jackery HomePower 3000 Solar Generator + 2 SolarSaga 200W Portable Solar Panels that costs $1,599.00. Assume the tax rate is 10.25%.
5. Private Mortgage Insurance (PMI) is a required, non-owner-protective fee for conventional loans with less than a 20% down payment, typically costing 0.46%–1.5% of the loan amount annually. It protects the lender if the buyer defaults. A 1,400 square foot, 3 bedroom, 2 bath condo in Huntington Beach, CA is selling for $775,000. What would one have to pay as a down payment to avoid paying PMI? What would be the balance after the down payment?

Answers

1. 61%, 70%, 7%, 400%, 172%, 12.345%, 100%, 0.01%
2. 0.60, 0.02, 0.0825, 0.00001, 1.25, 0.179, 0.03125, 0.1275 
3. $37.62
4. $1,762.90
5. $155,000;  $620,000

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