Financial Literacy and Applied Modeling

Income, Taxation, and Net Pay

When people work, they earn income. But the amount they earn is not always the amount they actually get to keep. Some money is taken out for taxes and other required deductions. The money left after those deductions is called net pay.

In this lesson, you will learn how to find gross income, calculate deductions using percentages, and determine net pay. These are important real-life math skills because they help people understand paychecks and plan their budgets.

1. Important vocabulary

• Income: money a person earns.
• Gross income: the total amount earned before any money is taken out.
• Tax: money taken out to help pay for government services.
• Deduction: any amount subtracted from gross income.
• Net pay: the amount left after deductions are taken out.

You can think of it like this:

$$\text{Net Pay} = \text{Gross Income} - \text{Total Deductions}$$

2. Finding gross income

Gross income depends on how a person is paid. A person might earn money by the hour, by the week, or by salary.

If someone is paid by the hour, use:

$$\text{Gross Income} = \text{Hourly Rate} \times \text{Hours Worked}$$

If someone earns the same amount each week or month, that amount may already be their gross income for that pay period.

For example, if a worker earns \(\$15\) per hour and works \(20\) hours, then their gross income is:

$$15 \times 20 = 300$$

So the gross income is \(\$300\).

3. Understanding deductions as percentages

Many deductions are given as percentages. For example, income tax might be \(12\%\), social security might be \(6\%\), and local tax might be \(2\%\).

To find a percentage of an amount, change the percent to a decimal and multiply.

Examples:

• \(10\% = 0.10\)
• \(6\% = 0.06\)
• \(2\% = 0.02\)

Then use:

$$\text{Deduction Amount} = \text{Percent as Decimal} \times \text{Gross Income}$$

If the gross income is \(\$300\) and the tax rate is \(10\%\), then:

$$0.10 \times 300 = 30$$

So the tax deduction is \(\$30\).

4. Finding total deductions

Usually, more than one deduction is taken out of a paycheck. To find the total deductions, calculate each deduction separately and then add them together.

For example, suppose a paycheck has:

• Income tax: \(10\%\)
• Social security: \(5\%\)
• Local tax: \(2\%\)

If the gross income is \(\$400\):

• Income tax: \(0.10 \times 400 = 40\)
• Social security: \(0.05 \times 400 = 20\)
• Local tax: \(0.02 \times 400 = 8\)

Total deductions:

$$40 + 20 + 8 = 68$$

Net pay:

$$400 - 68 = 332$$

So the worker takes home \(\$332\).

5. A shortcut when all deductions are percentages

If all deductions are percentages of the same gross income, you can add the percentages first.

Using the example above:

$$10\% + 5\% + 2\% = 17\%$$

Then find \(17\%\) of \(\$400\):

$$0.17 \times 400 = 68$$

This gives the same total deductions. Then subtract from gross income:

$$400 - 68 = 332$$

This shortcut works well when all deductions are based on the gross income.

6. Worked examples

Example 1: One deduction

Mia earns \(\$12\) per hour and works \(15\) hours this week. Her income tax is \(10\%\). Find her gross income and net pay.

Step 1: Find gross income.

$$12 \times 15 = 180$$

Gross income is \(\$180\).

Step 2: Find the tax deduction.

$$0.10 \times 180 = 18$$

The tax deduction is \(\$18\).

Step 3: Find net pay.

$$180 - 18 = 162$$

Answer: Mia’s net pay is \(\$162\).

Example 2: Two deductions

Jordan has a gross income of \(\$250\). His paycheck has \(8\%\) income tax and \(4\%\) social security. Find his net pay.

Step 1: Find each deduction.

$$0.08 \times 250 = 20$$

Income tax is \(\$20\).

$$0.04 \times 250 = 10$$

Social security is \(\$10\).

Step 2: Add deductions.

$$20 + 10 = 30$$

Total deductions are \(\$30\).

Step 3: Subtract from gross income.

$$250 - 30 = 220$$

Answer: Jordan’s net pay is \(\$220\).

Example 3: Hourly pay with three deductions

Ava earns \(\$18\) per hour and works \(25\) hours. Her deductions are:

• Income tax: \(12\%\)
• Social security: \(6\%\)
• Local tax: \(2\%\)

Find her net pay.

Step 1: Find gross income.

$$18 \times 25 = 450$$

Gross income is \(\$450\).

Step 2: Find each deduction.

$$0.12 \times 450 = 54$$

Income tax is \(\$54\).

$$0.06 \times 450 = 27$$

Social security is \(\$27\).

$$0.02 \times 450 = 9$$

Local tax is \(\$9\).

Step 3: Add deductions.

$$54 + 27 + 9 = 90$$

Total deductions are \(\$90\).

Step 4: Find net pay.

$$450 - 90 = 360$$

Answer: Ava’s net pay is \(\$360\).

Example 4: Using the percentage shortcut

Leo earns \(\$500\) gross income. His deductions are \(9\%\) income tax, \(5\%\) social security, and \(1\%\) local tax. Find his net pay.

Step 1: Add the deduction rates.

$$9\% + 5\% + 1\% = 15\%$$

Step 2: Find total deductions.

$$0.15 \times 500 = 75$$

Total deductions are \(\$75\).

Step 3: Subtract from gross income.

$$500 - 75 = 425$$

Answer: Leo’s net pay is \(\$425\).

7. Common mistakes to avoid

• Forgetting to change a percent to a decimal. For example, \(7\%\) should be written as \(0.07\), not \(7\).
• Subtracting the percent instead of the deduction amount. You subtract dollars, not percentages.
• Using net pay instead of gross income to find deductions. Most deductions are based on the gross income.
• Not adding all deductions. If there are several taxes, find them all before subtracting.

8. Helpful step-by-step method

1. Find the gross income.
2. Write each deduction percent as a decimal.
3. Multiply the gross income by each decimal to find each deduction.
4. Add all deductions together.
5. Subtract total deductions from gross income to get net pay.

9. Why this matters

Understanding income, taxation, and net pay helps you read a paycheck and know how much money is actually available to spend or save. A job may sound like it pays a certain amount, but taxes and deductions mean the take-home amount is smaller.

This math is part of financial literacy. It helps people make smart choices about work, saving, and spending.

Summary

Gross income is the full amount earned before deductions. Deductions such as income tax, social security, and local taxes are often found by multiplying gross income by a percent written as a decimal. After adding all deductions, subtract them from gross income to find net pay.

Remember the main formula:

$$\text{Net Pay} = \text{Gross Income} - \text{Total Deductions}$$

If you follow the steps carefully, you can solve paycheck problems with confidence.

Budgets and Fixed vs. Variable Expenses

Have you ever wondered where money goes each month? A budget helps answer that question. A budget is a plan for how money will be earned, spent, and saved.

Learning to make a budget is an important life skill. It helps people make smart choices, avoid overspending, and work toward goals like saving for a game, a bike, or a trip.

One big idea in budgeting is understanding the difference between fixed expenses and variable expenses. When you know which costs stay the same and which can change, it becomes much easier to control your money.

1. What is a budget?

A budget compares the money coming in with the money going out.

• Income: money you receive, such as allowance, gift money, or money from a job.
• Expenses: money you spend.
• Savings: money you choose to keep for future use.

A simple budget idea is:

$$\text{Income} = \text{Expenses} + \text{Savings}$$

If your expenses are too high, your savings will be smaller. If you reduce some spending, you can save more.

2. Fixed expenses

Fixed expenses are costs that stay the same each month, or stay close to the same amount. These are easier to predict when making a budget.

Examples of fixed expenses can include:

• Rent or mortgage
• A monthly phone plan
• Internet bill
• Bus pass
• Streaming subscription

If a family pays \(\$60\) every month for internet, that is a fixed expense because it does not usually change from month to month.

3. Variable expenses

Variable expenses are costs that can change from one month to the next. These expenses are less predictable.

Examples of variable expenses can include:

• Food or snacks
• Clothing
• Entertainment
• School supplies
• Electricity or water bills that may change

If you spend \(\$15\) on snacks one week and \(\$25\) the next week, snack spending is a variable expense.

4. Why does the difference matter?

It is helpful to separate fixed and variable expenses because they affect your choices in different ways.

• Fixed expenses are usually harder to change quickly.
• Variable expenses are often easier to adjust.

For example, if you want to save more money this month, you may not be able to change a phone bill right away. But you may be able to spend less on snacks, games, or movies.

5. How to build a simple budget

To build a budget, follow these steps:

1. Find your total income.
2. List your fixed expenses.
3. List your variable expenses.
4. Add all expenses.
5. Subtract expenses from income.
6. Decide how much to save.

You can write this as:

$$\text{Money Left Over} = \text{Income} - \text{Total Expenses}$$

If the money left over is positive, you can save it or spend part of it carefully. If it is negative, your budget is not balanced and you need to reduce spending.

6. What is a balanced budget?

A balanced budget means you are not planning to spend more money than you earn.

In other words:

$$\text{Income} \geq \text{Expenses} + \text{Savings}$$

If your income is less than your expenses, you have a problem because you are spending too much.

7. Savings ratio

A savings ratio tells what part of your income is being saved. It can be written as a fraction, decimal, or percent.

The formula is:

$$\text{Savings Ratio} = \frac{\text{Savings}}{\text{Income}}$$

To turn it into a percent, multiply by \(100\).

For example, if you save \(\$20\) out of \(\$100\):

$$\frac{20}{100} = 0.20 = 20\%$$

That means you saved 20% of your income.

Worked Example 1: Identifying fixed and variable expenses

Mia has these monthly expenses:

• Phone plan: \(\$25\)
• Bus pass: \(\$30\)
• Snacks: \(\$18\)
• Movie tickets: \(\$12\)

Which expenses are fixed, and which are variable?

Step 1: Look for costs that stay the same.

• Phone plan: fixed
• Bus pass: fixed

Step 2: Look for costs that may change.

• Snacks: variable
• Movie tickets: variable

Answer: Fixed expenses are the phone plan and bus pass. Variable expenses are snacks and movie tickets.

Worked Example 2: Finding money left over

Jordan earns \(\$120\) in a month from allowance and chores.

His expenses are:

• Music subscription: \(\$10\)
• Club fee: \(\$15\)
• Snacks: \(\$20\)
• Games: \(\$25\)

How much money does Jordan have left over?

Step 1: Add the expenses.

$$10 + 15 + 20 + 25 = 70$$

Total expenses are \(\$70\).

Step 2: Subtract expenses from income.

$$120 - 70 = 50$$

Answer: Jordan has \(\$50\) left over. He could choose to save some or all of it.

Worked Example 3: Creating a balanced budget

Sofia has monthly income of \(\$90\).

Her fixed expenses are:

• Phone plan: \(\$20\)
• Art class fee: \(\$25\)

Her variable expenses are:

• Snacks: \(\$18\)
• Entertainment: \(\$22\)

She wants to save \(\$15\). Is her budget balanced?

Step 1: Add fixed expenses.

$$20 + 25 = 45$$

Step 2: Add variable expenses.

$$18 + 22 = 40$$

Step 3: Add total expenses and savings.

$$45 + 40 + 15 = 100$$

Step 4: Compare with income.

Income is \(\$90\), but planned spending and saving total \(\$100\).

$$90 < 100$$

The budget is not balanced.

Step 5: Fix the budget.

Sofia needs to reduce spending by:

$$100 - 90 = 10$$

Since variable expenses are easier to change, she could lower entertainment from \(\$22\) to \(\$12\).

Answer: No, the budget is not balanced. She must cut \(\$10\) from spending or save less.

Worked Example 4: Finding a savings ratio

Leo earns \(\$200\) in a month. After paying all expenses, he saves \(\$35\).

What is his savings ratio?

Step 1: Write the ratio.

$$\frac{\text{Savings}}{\text{Income}} = \frac{35}{200}$$

Step 2: Change to a decimal.

$$\frac{35}{200} = 0.175$$

Step 3: Change to a percent.

$$0.175 \times 100 = 17.5\%$$

Answer: Leo's savings ratio is \(0.175\), or 17.5%.

8. Tips for improving a budget

• Write down all income and expenses.
• Separate fixed costs from variable costs.
• Check which variable expenses can be reduced.
• Set a savings goal each month.
• Review your budget often, because variable expenses can change.

9. Common mistakes to avoid

• Forgetting small purchases like snacks or drinks
• Mixing up fixed and variable expenses
• Planning to spend more than your income
• Not including savings in the budget

10. Final idea

A budget is not just a list of numbers. It is a tool for making choices. When you understand fixed and variable expenses, you can see where your money goes and make a better plan for saving.

If you want to improve your budget, start by looking at the variable expenses. Even small changes can help you reach your goals faster.

Retail Math: Markups, Discounts, and Sales Tax

Have you ever seen a store sign that says "20% off" or looked at a receipt and noticed sales tax added at the end? That is retail math. Retail math helps us understand how prices change from the original price to the final amount paid at the register.

In this lesson, you will learn how to work with markups, discounts, and sales tax. You will also learn how to do more than one percent change in order, because in real shopping situations, these steps often happen one after another.

Important idea: Percent changes are usually based on the current price, not always the original price. That means the order matters.

1. What is a markup?

A markup is an amount added to a price. Stores use markups to raise the price of an item above what it cost them. For example, if a store buys a water bottle for \(\$10\) and adds a 30% markup, the selling price becomes higher than \(\$10\).

To find a markup amount, multiply the original price by the markup percent written as a decimal.

$$ \text{Markup Amount} = \text{Original Price} \times \text{Markup Rate} $$

Then add the markup to the original price.

$$ \text{New Price} = \text{Original Price} + \text{Markup Amount} $$

You can also do both steps at once by multiplying by \(1 + \text{markup rate}\).

$$ \text{New Price} = \text{Original Price} \times (1 + \text{markup rate}) $$

For example, a 25% markup means multiply by \(1.25\).

• 10% markup \(\rightarrow 1.10\)
• 25% markup \(\rightarrow 1.25\)
• 40% markup \(\rightarrow 1.40\)

2. What is a discount?

A discount is an amount taken away from a price. When a store says an item is 15% off, that means you subtract 15% of the current price.

To find the discount amount:

$$ \text{Discount Amount} = \text{Price} \times \text{Discount Rate} $$

Then subtract the discount.

$$ \text{Sale Price} = \text{Price} - \text{Discount Amount} $$

You can also do this in one step by multiplying by \(1 - \text{discount rate}\).

$$ \text{Sale Price} = \text{Price} \times (1 - \text{discount rate}) $$

For example, a 20% discount means multiply by \(0.80\).

• 10% discount \(\rightarrow 0.90\)
• 20% discount \(\rightarrow 0.80\)
• 35% discount \(\rightarrow 0.65\)

3. What is sales tax?

Sales tax is extra money added to the purchase price. It is usually added after discounts at the register.

To find the tax amount:

$$ \text{Tax Amount} = \text{Price} \times \text{Tax Rate} $$

Then add the tax to the price.

$$ \text{Final Price} = \text{Price} + \text{Tax Amount} $$

Or use one step:

$$ \text{Final Price} = \text{Price} \times (1 + \text{tax rate}) $$

For example:

• 5% tax \(\rightarrow 1.05\)
• 7% tax \(\rightarrow 1.07\)
• 8.25% tax \(\rightarrow 1.0825\)

4. Changing a percent to a decimal

To use percents in calculations, change the percent to a decimal by dividing by 100.

• \(15\% = 0.15\)
• \(8\% = 0.08\)
• \(6.5\% = 0.065\)

5. The order of steps matters

In many shopping problems, the steps happen in a certain order:

1. Start with the original price.
2. Apply a markup if the price is being increased before selling.
3. Apply any discount to the current price.
4. Apply sales tax at the end.

This matters because each percent is taken from the price that comes right before it.

For example, taking 20% off and then adding 8% tax is not the same as just combining them into 12%. Each step changes the price before the next step happens.

6. Worked Example 1: Finding a markup

A store buys a notebook for \(\$12\) and uses a 25% markup. What is the selling price?

Step 1: Change 25% to a decimal.

$$ 25\% = 0.25 $$

Step 2: Find the markup amount.

$$ 12 \times 0.25 = 3 $$

The markup is \(\$3\).

Step 3: Add the markup to the original price.

$$ 12 + 3 = 15 $$

Answer: The selling price is \(\$15\).

You could also do it in one step:

$$ 12 \times 1.25 = 15 $$

7. Worked Example 2: Finding a discount

A hoodie costs \(\$40\). It is on sale for 30% off. What is the sale price?

Step 1: Change 30% to a decimal.

$$ 30\% = 0.30 $$

Step 2: Find the discount amount.

$$ 40 \times 0.30 = 12 $$

The discount is \(\$12\).

Step 3: Subtract the discount from the original price.

$$ 40 - 12 = 28 $$

Answer: The sale price is \(\$28\).

One-step method:

$$ 40 \times 0.70 = 28 $$

8. Worked Example 3: Discount, then sales tax

A pair of shoes costs \(\$60\). The store offers 20% off, and then 5% sales tax is added. What is the final price?

Step 1: Find the discounted price.

$$ 20\% = 0.20 $$

$$ 60 \times 0.20 = 12 $$

The discount is \(\$12\).

$$ 60 - 12 = 48 $$

After the discount, the price is \(\$48\).

Step 2: Find the sales tax on the discounted price.

$$ 5\% = 0.05 $$

$$ 48 \times 0.05 = 2.40 $$

The tax is \(\$2.40\).

Step 3: Add the tax.

$$ 48 + 2.40 = 50.40 $$

Answer: The final register price is \(\$50.40\).

Using multipliers, you could also write:

$$ 60 \times 0.80 \times 1.05 = 50.40 $$

9. Worked Example 4: Markup, discount, and sales tax

A store gets a game for \(\$50\). The store adds a 40% markup. Then the game is put on sale for 10% off. Finally, 8% sales tax is added at checkout. What is the final price?

Step 1: Apply the markup.

$$ 40\% = 0.40 $$

$$ 50 \times 1.40 = 70 $$

After the markup, the price is \(\$70\).

Step 2: Apply the discount.

$$ 10\% = 0.10 $$

$$ 70 \times 0.90 = 63 $$

After the discount, the price is \(\$63\).

Step 3: Apply the sales tax.

$$ 8\% = 0.08 $$

$$ 63 \times 1.08 = 68.04 $$

Answer: The final register price is \(\$68.04\).

10. A helpful shortcut: multipliers

When several percent changes happen in a row, multipliers can save time.

• For a markup, use \(1 + \text{rate}\)
• For a discount, use \(1 - \text{rate}\)
• For sales tax, use \(1 + \text{rate}\)

Example of a chain of changes:

Original price \(= 80\), 25% off, then 6% tax:

$$ 80 \times 0.75 \times 1.06 $$

First multiply:

$$ 80 \times 0.75 = 60 $$

Then multiply again:

$$ 60 \times 1.06 = 63.60 $$

The final price is \(\$63.60\).

11. Common mistakes to avoid

• Forgetting to change percent to decimal: 15% means \(0.15\), not 15.
• Adding or subtracting the percent instead of using it on the price: For 20% off \(\$50\), do not do \(50 - 20\). Find 20% of 50 first.
• Taxing the wrong amount: Sales tax is usually added after the discount, not before.
• Combining percent changes incorrectly: A 20% discount and 5% tax do not mean just 15% off.
• Using the original price every time: After one change, the next percent uses the new current price.

12. How to solve retail math problems step by step

1. Read carefully and find the starting price.
2. Underline each percent change and note whether it is a markup, discount, or tax.
3. Change each percent to a decimal.
4. Work in the correct order, updating the price after each step.
5. Round money to the nearest cent if needed.

13. Quick check

Try these on your own:

• A \(\$25\) item has a 20% markup. What is the new price?
• A \(\$90\) jacket is 30% off. What is the sale price?
• A \(\$50\) item is 10% off, then 6% tax is added. What is the final price?

Answers:

• \($30\)
• \($63\)
• \($47.70\)

14. Summary

Retail math helps you find how prices change in stores. A markup increases a price, a discount decreases a price, and sales tax adds extra cost at the end.

To solve these problems, change each percent to a decimal and apply each step in order. Remember that each new percent is based on the current price. If you follow the order carefully, you can find the final register price with confidence.

Simple vs. Compound Interest

Money can grow when you save or invest it, and money can also increase when you borrow it. The extra money added is called interest.

There are two common types of interest: simple interest and compound interest. They may sound similar, but they work in different ways.

In this lesson, you will learn what each type means, how to calculate it, and why compound interest usually grows faster over time.

1. Important words to know

• Principal: the starting amount of money
• Interest: the extra money earned or owed
• Rate: the percent used to calculate interest
• Time: how long the money is saved or borrowed
• Total amount: principal plus interest

We often use these variables:

• \(P\) = principal
• \(r\) = interest rate written as a decimal
• \(t\) = time in years
• \(I\) = interest
• \(A\) = total amount

To change a percent to a decimal, divide by 100.

• \(5\% = 0.05\)
• \(8\% = 0.08\)
• \(12\% = 0.12\)

2. What is simple interest?

Simple interest is interest calculated only on the original principal. That means the same amount of interest is added each year.

This creates linear growth. Linear growth means the total increases by equal amounts over equal times.

The formula for simple interest is:

$$I = Prt$$

Once you find the interest, you can find the total amount:

$$A = P + I$$

Because simple interest adds the same amount each year, it does not “snowball.”

3. What is compound interest?

Compound interest means interest is calculated on the principal and also on past interest.

This means your money can grow faster over time. Each time interest is added, the balance gets bigger. Then the next interest calculation uses that bigger balance.

This creates a snowball effect. The amount grows more and more quickly as time goes on.

If interest is compounded once each year, the total amount after \(t\) years is:

$$A = P(1+r)^t$$

The interest earned is then:

$$I = A - P$$

For 7th Grade, we will mostly compare simple interest with compound interest that is added yearly.

4. The big difference

• Simple interest: interest is based only on the starting principal.
• Compound interest: interest is based on the starting principal and the interest already added.

So:

• Simple interest grows by the same amount each year.
• Compound interest grows by larger and larger amounts over time.

5. Worked Example 1: Finding simple interest

A student puts \(\$500\) in a savings account earning \(6\%\) simple interest each year for \(3\) years. How much interest is earned?

Step 1: Write the values.

• \(P = 500\)
• \(r = 0.06\)
• \(t = 3\)

Step 2: Use the formula.

$$I = Prt$$ $$I = 500(0.06)(3)$$ $$I = 90$$

Answer: The interest earned is \(\$90\).

Step 3: Find the total amount if needed.

$$A = P + I = 500 + 90 = 590$$

So after 3 years, the account has \(\$590\).

Notice: The account earns the same amount each year:

• Year 1: \(\$30\)
• Year 2: \(\$30\)
• Year 3: \(\$30\)

6. Worked Example 2: Finding compound interest

Now use the same starting amount: \(\$500\) at \(6\%\) interest for \(3\) years, but this time it is compounded yearly.

Step 1: Write the formula.

$$A = P(1+r)^t$$

Step 2: Substitute the values.

$$A = 500(1+0.06)^3$$ $$A = 500(1.06)^3$$

Step 3: Calculate.

$$1.06^3 \approx 1.191016$$ $$A \approx 500(1.191016)$$ $$A \approx 595.51$$

Step 4: Find the interest.

$$I = A - P = 595.51 - 500 = 95.51$$

Answer: The total amount is about \(\$595.51\), so the interest earned is about \(\$95.51\).

Compare:

• Simple interest after 3 years: \(\$90\)
• Compound interest after 3 years: about \(\$95.51\)

Compound interest earns more because each year the new interest is calculated on a larger balance.

7. Seeing the snowball effect

Let’s look at the yearly balances for the compound interest example.

• Start: \(\$500.00\)
• After Year 1: \(500 \times 1.06 = 530.00\)
• After Year 2: \(530.00 \times 1.06 = 561.80\)
• After Year 3: \(561.80 \times 1.06 \approx 595.51\)

The interest added each year is not the same:

• Year 1 interest: \(\$30.00\)
• Year 2 interest: \(\$31.80\)
• Year 3 interest: about \(\$33.71\)

That is why compound interest is called a snowball effect. The growth gets bigger over time.

8. Worked Example 3: Comparing simple and compound interest

Suppose \(\$1{,}000\) is invested at \(5\%\) for \(4\) years. Compare simple interest and compound interest.

Simple Interest

$$I = Prt$$ $$I = 1000(0.05)(4) = 200$$ $$A = 1000 + 200 = 1200$$

With simple interest, the total amount is \(\$1{,}200\).

Compound Interest

$$A = P(1+r)^t$$ $$A = 1000(1.05)^4$$ $$A = 1000(1.21550625)$$ $$A \approx 1215.51$$ $$I = 1215.51 - 1000 = 215.51$$

With compound interest, the total amount is about \(\$1{,}215.51\).

Comparison:

• Simple interest earned: \(\$200\)
• Compound interest earned: about \(\$215.51\)

At first, the difference may seem small. But over many years, compound interest can become much larger.

9. Worked Example 4: Interest on money you owe

Interest does not only help savings. It can also make debt grow.

A person borrows \(\$800\) at \(10\%\) compound interest yearly for \(2\) years. How much will be owed after 2 years?

Step 1: Use the formula.

$$A = P(1+r)^t$$ $$A = 800(1+0.10)^2$$ $$A = 800(1.10)^2$$ $$A = 800(1.21)$$ $$A = 968$$

Answer: After 2 years, the amount owed is \(\$968\).

The interest is:

$$I = 968 - 800 = 168$$

This example shows why borrowing money can become expensive, especially when interest compounds.

10. Quick tips for solving problems

1. Read carefully to see whether the problem says simple or compound interest.
2. Find the principal, rate, and time.
3. Turn the percent into a decimal.
4. Use the correct formula.
5. Check whether the question asks for interest only or the total amount.

11. Common mistakes to avoid

• Forgetting to change percent to decimal
  Example: \(7\%\) should be \(0.07\), not \(7\).
• Mixing up interest and total amount
  Remember: \(I\) is just the extra money, while \(A\) is the whole amount.
• Using the simple interest formula for compound interest
  Compound interest must include repeated growth.
• Thinking compound interest adds the same amount each year
  It usually adds more each year because the balance keeps growing.

12. A simple way to remember the difference

• Simple interest = same base every time
  The interest always uses the original principal.
• Compound interest = growing base every time
  The interest uses a balance that keeps getting bigger.

You can also think of it this way:

• Simple interest grows like adding equal steps.
• Compound interest grows like a snowball rolling downhill.

13. Brief summary

Simple interest is calculated only on the original principal, so it grows by the same amount each year. Its formula is \(I = Prt\).

Compound interest is calculated on the principal and the interest already added, so it grows faster over time. For yearly compounding, its formula is \(A = P(1+r)^t\).

When comparing the two, compound interest usually gives more growth for savings and can also make debt increase faster. That is why it is important to understand both types of interest in real life.

Lesson: Credit, Loans, and Amortization Intuition

Money can help us buy things now, even if we do not have enough cash today. But when we borrow money, we usually have to pay back more than we borrowed. That extra amount is called interest.

In this lesson, you will learn how credit, loans, and payments over time work. You will also build intuition for amortization, which means paying off a loan little by little through regular payments.

Understanding this helps you answer important questions like:

• How much will I really pay if I borrow money?
• Why can small payments make a loan last a very long time?
• How does interest change the total cost?

1. What is credit?

Credit means being allowed to borrow money now and pay it back later. A common example is a credit card. When you use a credit card, the bank pays first, and then you pay the bank back.

If you do not pay the full amount right away, the bank usually charges interest. That means your balance can grow.

2. What is a loan?

A loan is money you borrow and agree to repay over time. Examples include borrowing money for a bike, a car, or school expenses.

Important parts of a loan are:

• Principal: the amount you first borrow
• Interest: the extra money charged for borrowing
• Payment: the amount you pay each time
• Balance: the amount you still owe

For example, if you borrow \(\$200\), then the principal is \(\$200\). If you repay \(\$220\) in total, then \(\$20\) of that total is interest.

3. What is interest?

Interest is like the price of borrowing money. The lender charges it because they are letting you use their money for a while.

Interest is often given as a percent. For example, a loan might have a yearly interest rate of \(12\%\).

To estimate one year of simple interest, you can use:

$$\text{Interest} = \text{Principal} \times \text{Rate}$$

If you borrow \(\$100\) at \(10\%\) for one year:

$$\text{Interest} = 100 \times 0.10 = 10$$

So you would owe:

$$100 + 10 = 110$$

4. Why loans can cost more than expected

Many people look only at the monthly payment. But a smaller monthly payment can mean you pay for a longer time. That often means paying more total interest.

So when comparing loans, it is smart to ask:

• What is the monthly payment?
• How many payments will I make?
• How much will I pay altogether?

The total amount paid can be found by:

$$\text{Total Paid} = \text{Payment Amount} \times \text{Number of Payments}$$

Then:

$$\text{Total Interest} = \text{Total Paid} - \text{Principal}$$

5. What does amortization mean?

Amortization means paying off a loan through regular payments over time. Each payment usually does two jobs:

• pays some interest
• pays down some of the principal

At the beginning of a loan, a bigger part of each payment often goes toward interest. Later, more of each payment goes toward the principal.

This is why loans can feel slow to shrink at first.

6. A simple way to picture one payment

Suppose you owe \(\$500\), and interest for the month is \(\$10\). If you make a payment of \(\$40\):

• \(\$10\) pays the interest
• the remaining \(\$30\) reduces the balance

Your new balance would be:

$$500 - 30 = 470$$

Even though you paid \(\$40\), the balance only dropped by \(\$30\) because part of the payment covered interest.

7. Minimum payments and credit card debt

Credit cards often allow a minimum payment. This is the smallest amount you are allowed to pay each month.

Paying only the minimum may sound helpful, but it can be very expensive. Why?

• The balance stays around for a long time.
• Interest keeps being added.
• You may end up paying much more than the original amount.

If your payment is only a little larger than the interest charged, the debt shrinks very slowly.

Worked Example 1: Finding total cost of a loan

Maria borrows \(\$300\) to buy a tablet. She pays \(\$27\) each month for 12 months.

Step 1: Find the total paid.

$$27 \times 12 = 324$$

Maria pays \(\$324\) in total.

Step 2: Find the interest paid.

$$324 - 300 = 24$$

Answer: Maria paid \(\$24\) in interest.

This means the tablet really cost her \(\$324\), not just \(\$300\).

Worked Example 2: One payment on a loan

Jalen owes \(\$200\). This month, the interest added is \(\$8\). He makes a payment of \(\$25\).

Step 1: Subtract the interest part from the payment.

$$25 - 8 = 17$$

So \(\$17\) goes toward the principal.

Step 2: Find the new balance.

$$200 - 17 = 183$$

Answer: After the payment, Jalen still owes \(\$183\).

Notice that he paid \(\$25\), but the balance fell by only \(\$17\).

Worked Example 3: Why a bigger payment helps

A credit card balance is \(\$400\). This month, \(\$12\) of interest is added.

Case A: Minimum payment of \(\$20\)

• Interest paid: \(\$12\)
• Amount reducing balance: \(20 - 12 = 8\)

New balance:

$$400 - 8 = 392$$

Case B: Larger payment of \(\$50\)

• Interest paid: \(\$12\)
• Amount reducing balance: \(50 - 12 = 38\)

New balance:

$$400 - 38 = 362$$

Answer: The larger payment reduces the debt much faster.

In Case A, the balance drops by only \(\$8\). In Case B, it drops by \(\$38\). This is why paying more than the minimum can save a lot of money.

Worked Example 4: Comparing two loan choices

A store offers two ways to pay for a \(\$600\) bike.

• Plan 1: \(\$55\) per month for 12 months
• Plan 2: \(\$35\) per month for 20 months

Plan 1 total:

$$55 \times 12 = 660$$

Interest on Plan 1:

$$660 - 600 = 60$$

Plan 2 total:

$$35 \times 20 = 700$$

Interest on Plan 2:

$$700 - 600 = 100$$

Answer: Plan 1 costs less overall, even though the monthly payment is higher.

This shows an important idea: a lower monthly payment does not always mean a better deal.

8. Warning signs to watch for

When borrowing money, be careful if you notice:

• very high interest rates
• very long repayment times
• payments that are so small the balance barely changes
• only looking at the monthly payment instead of the total cost

9. Smart habits when thinking about credit

• Borrow only what you can afford to repay.
• Try to pay more than the minimum.
• Look at the total amount paid, not just each monthly payment.
• Remember that interest makes things cost more over time.

10. Key ideas to remember

• Credit lets you buy now and pay later.
• Loans must be repaid with interest.
• Interest is the extra cost of borrowing.
• Amortization means paying off debt through regular payments over time.
• Each payment often covers interest first, then lowers the principal.
• Small payments can make debt last much longer.
• Paying more each month can reduce the total interest paid.

Brief Summary

Borrowing money can be useful, but it has a cost. The cost comes from interest, which makes you repay more than you borrowed. When you understand how payments, balance, and interest work together, you can make smarter choices and avoid debt that lasts too long.

Lesson: Fermi Problems and Magnitude Estimation

Sometimes in math, you are asked a question that seems impossible to answer exactly.

For example:

• How many ping pong balls could fit in a classroom?
• How many steps do students at your school take in one day?
• How many coins would fill a backpack?

These are called Fermi problems. A Fermi problem is a question where you probably do not know the exact answer, but you can make a smart estimate by breaking the problem into smaller, easier parts.

In this lesson, you will learn how to use reasonable assumptions, simple geometry, and powers of ten to estimate very large quantities.

What is magnitude estimation?

Magnitude means the size of something. In magnitude estimation, you are trying to figure out about how big or small a quantity is.

You are not trying to get the exact answer. You are trying to get an answer that is close enough to make sense.

For example, if someone asks how many minutes are in a year, you might not know right away. But you can estimate:

$$365 \times 24 \times 60$$

This gives

$$365 \times 1440 = 525{,}600$$

So there are about 500,000 minutes in a year. That is a magnitude estimate.

Why Fermi problems matter

Fermi problems help you practice skills that are useful in real life:

• making reasonable guesses
• using what you already know
• breaking big problems into smaller steps
• checking whether an answer makes sense
• working with very large numbers

These skills are also helpful in financial literacy and applied modeling. In money situations, people often estimate costs, savings, time, and amounts before making decisions.

The basic steps for solving a Fermi problem

1. Understand the question. Decide what is being estimated.
2. Break the problem into parts. Turn one huge question into several smaller ones.
3. Make reasonable assumptions. Use values that are sensible, even if they are not exact.
4. Do the math. Multiply, divide, or add your estimates.
5. Check whether the answer is reasonable. Ask: Is it too big? Too small? Does it fit the situation?

Important idea: assumptions

An assumption is a value you choose because it seems reasonable.

For example, if you are estimating how many students can fit in an auditorium, you might assume:

• each chair is about 2 feet wide
• there are about 20 rows
• each row has about 25 chairs

Then your estimate would be

$$20 \times 25 = 500$$

That does not mean the exact number is 500. It means 500 is a reasonable estimate based on your assumptions.

Using powers of ten

Powers of ten help us describe very large or very small numbers more easily.

Here are some examples:

• \(10^1 = 10\)
• \(10^2 = 100\)
• \(10^3 = 1000\)
• \(10^4 = 10{,}000\)
• \(10^5 = 100{,}000\)
• \(10^6 = 1{,}000{,}000\)

When estimating, it is often helpful to round numbers to nearby powers of ten.

For example:

• 48 is about \(50\), which is about \(5 \times 10\)
• 320 is about \(300\), which is \(3 \times 10^2\)
• 9,800 is about \(10{,}000\), which is \(10^4\)

This makes mental math easier.

Estimating with area and volume

Many Fermi problems involve space, so area and volume are useful.

Area tells how much flat surface something covers.

$$\text{Area of rectangle} = \text{length} \times \text{width}$$

Volume tells how much space a 3D object takes up.

$$\text{Volume of rectangular prism} = \text{length} \times \text{width} \times \text{height}$$

If you are estimating how many objects fit in a space, you can often:

• find the volume of the space
• estimate the volume of one object
• divide

But in real life, objects do not pack perfectly. There are gaps between them. So your answer may need to be adjusted.

Worked Example 1: How many pencils are in a box?

Suppose a teacher has a large box of pencils, and you want to estimate how many pencils are inside.

Let us assume:

• the box is about 30 cm long, 20 cm wide, and 10 cm tall
• one pencil is about 20 cm long
• a pencil takes up about 1 cm by 1 cm of space when packed

Since the pencils are about 20 cm long, they would likely lie along the 30 cm length of the box.

Now estimate how many pencils fit in one layer:

$$20 \times 10 = 200$$

That is because the box width is about 20 cm and the height is about 10 cm, so each layer could hold about \(20 \times 10\) pencils.

Since the pencils are 20 cm long and the box is 30 cm long, only one main pencil length fits along that direction.

So the estimate is about 200 pencils.

This may not be exact, but it gives the right size of answer.

Worked Example 2: How many ping pong balls fit in a classroom?

This is a classic Fermi problem.

Assume the classroom is a rectangular prism with dimensions:

• length = 9 m
• width = 8 m
• height = 3 m

First find the volume of the classroom:

$$9 \times 8 \times 3 = 216 \text{ cubic meters}$$

Now estimate the size of a ping pong ball. A ping pong ball is about 4 cm across, which is \(0.04\) m.

To make the estimate simpler, pretend each ball fits inside a small cube that is \(0.04\) m on each side.

The volume of one small cube is

$$0.04 \times 0.04 \times 0.04 = 0.000064 \text{ cubic meters}$$

Now divide:

$$\frac{216}{0.000064} = 3{,}375{,}000$$

So about 3,375,000 ping pong balls could fit if there were no furniture and everything packed perfectly.

But real classrooms have desks, shelves, and empty gaps between balls. So a more realistic estimate might be closer to 2,000,000 to 3,000,000 ping pong balls.

Notice that the exact answer is not the goal. The goal is finding the correct magnitude, which is in the millions.

Worked Example 3: How many text messages could a group of students send in a year?

Suppose you want to estimate how many text messages 500 students might send in one year.

Let us assume:

• each student sends about 20 text messages per day
• there are 365 days in a year
• there are 500 students

First find how many messages one student sends in a year:

$$20 \times 365 = 7300$$

Now multiply by 500 students:

$$7300 \times 500 = 3{,}650{,}000$$

So the students might send about 3,650,000 text messages in a year.

Using magnitude, we can say this is about \(4 \times 10^6\), or about 4 million.

Worked Example 4: Estimating a snack budget for a school event

Fermi problems are also useful in financial literacy.

Suppose a school is planning an event for about 300 students. You want to estimate how much money is needed for snack bags.

Assume each snack bag contains:

• 1 granola bar for about \(\$1\)
• 1 juice box for about \(\$2\)
• 1 small fruit pack for about \(\$1\)

So one snack bag costs about

$$1 + 2 + 1 = 4$$

Now multiply by 300 students:

$$300 \times 4 = 1200$$

The school should plan for about \(\$1200\).

If prices change a little, the exact total may differ, but the estimate helps with planning.

Rounding to make estimates easier

When working with Fermi problems, rounding is very helpful.

For example, suppose you need to estimate:

$$48 \times 197$$

You could round:

• \(48 \approx 50\)
• \(197 \approx 200\)

Then compute:

$$50 \times 200 = 10{,}000$$

The exact answer is different, but 10,000 is a good estimate.

How to know if an estimate is reasonable

After you solve a Fermi problem, always ask yourself whether your answer makes sense.

Here are some questions to check:

• Is my answer too big or too small for the situation?
• Did I use units correctly?
• Did I forget empty space, gaps, or real-world obstacles?
• Would a different reasonable assumption change the answer a little, or a lot?

For example, if you estimated that only 50 ping pong balls fit in a classroom, that would be far too small. If you estimated 50 billion, that would be far too large. A number in the millions makes much more sense.

Common mistakes to avoid

• Trying to be exact. Fermi problems are about reasonable estimates, not perfect answers.
• Using unrealistic assumptions. Your guesses should fit the real world.
• Forgetting units. Keep track of meters, centimeters, dollars, days, and so on.
• Ignoring empty space. Objects usually do not fit together perfectly.
• Not checking the final answer. Always ask if the result makes sense.

Helpful strategy: write the problem as a multiplication chain

Many Fermi problems can be written as several smaller factors multiplied together.

For example:

$$\text{messages per student per day} \times \text{days per year} \times \text{number of students}$$

Or:

$$\text{volume of room} \div \text{volume of one object}$$

This helps organize your thinking and makes large problems easier to solve.

Why different people may get different answers

In Fermi problems, two students may get different answers and both could still be correct if their assumptions are reasonable.

For example, one student may assume a classroom is 8 m by 7 m by 3 m, while another assumes 10 m by 8 m by 3 m. Both are possible classroom sizes, so their final estimates may differ.

What matters most is:

• the assumptions are sensible
• the math is correct
• the final answer has the right general size

Summary

Fermi problems ask you to estimate large or hard-to-find quantities by using logic and simple math.

To solve them, break the problem into smaller parts, make reasonable assumptions, use multiplication or division, and check whether your answer makes sense.

Magnitude estimation helps you understand the general size of an answer, often using powers of ten like thousands, millions, or more.

These skills are useful in school, science, and real-life planning, including budgeting, shopping, and solving everyday problems.

Critiquing Misleading Data in Media

Every day, we see data in news articles, advertisements, social media posts, and videos. Charts and graphs can help us understand information quickly. But sometimes, the way data is shown can be misleading. That means it can make us believe something that is not fully true or not shown fairly.

Learning how to critique misleading data helps you become a smarter reader and decision-maker. In financial literacy, this is especially important because graphs and claims can affect how people think about prices, savings, sales, or business success.

In this lesson, you will learn how to spot common tricks used in media, including truncated axes, disproportionate scaling, and manipulative data framing.

1. What does it mean for data to be misleading?

Data is misleading when it is presented in a way that gives the wrong impression. The numbers themselves might be correct, but the graph, scale, labels, or wording may make the change look bigger, smaller, better, or worse than it really is.

For example, if one store says, “Our sales doubled!” that sounds huge. But if sales went from 2 items to 4 items, that is very different from going from 200 items to 400 items. The statement is true, but it leaves out important context.

2. Watch out for truncated axes

An axis is one of the number lines on a graph. On a bar graph, the vertical axis often shows amounts, and the horizontal axis often shows categories or time.

A truncated axis happens when the graph does not start at 0. Instead, it starts at a higher number, such as 50 or 90. This can make small differences look very large.

Suppose two companies have profits of \(95\) dollars and \(100\) dollars. The actual difference is only:

$$100 - 95 = 5$$

If a graph starts at 90 instead of 0, one bar will look much taller than the other, even though the difference is just 5 dollars.

When you see a graph, ask:

• Does the axis start at 0?
• If not, is there a good reason?
• Does the graph make a small difference look much bigger?

Line graphs sometimes use truncated axes too. This can be useful for showing small changes clearly, but it should be labeled carefully so viewers are not tricked.

3. Watch out for disproportionate scaling

Scaling means how the numbers are spaced on the graph. A graph has disproportionate scaling when the spaces between numbers are not equal or when pictures are stretched in a way that exaggerates changes.

For example, a vertical axis might go from 0 to 10, then 10 to 20, then jump to 100. If the spacing looks equal, the graph can confuse the reader.

Another example is using pictures or icons. If one picture is twice as tall and twice as wide as another, it may look like it represents 4 times as much, not 2 times as much. That is because area changes, not just height.

If one square has side length \(2\) and another has side length \(4\), their areas are:

$$2^2 = 4 \quad \text{and} \quad 4^2 = 16$$

So the larger square has:

$$\frac{16}{4} = 4$$

times the area, even though the side length only doubled. This is one way images can mislead people.

Ask these questions:

• Are the intervals on the axis equal?
• Are symbols or pictures making the differences look too large?
• Does the graph use 3D shapes or oversized images that distract from the actual numbers?

4. Watch out for manipulative data framing

Data framing means how information is described or organized. The same data can sound very different depending on the words used.

For example, a headline might say, “School lunch prices increased by 50%!” That sounds dramatic. But if the price changed from \(\$2.00\) to \(\$3.00\), then the increase is:

$$3.00 - 2.00 = 1.00$$

Yes, a \(\$1.00\) increase is 50% of \(\$2.00\), but the headline may be trying to create a stronger reaction by using percent only.

Another framing trick is leaving out part of the story. A company might say, “Most customers saved money,” but not say how many customers were studied, how much they saved, or whether some customers lost money.

Good questions to ask are:

• Is the claim using percent, but not showing the actual numbers?
• Is important information missing?
• Does the wording try to make you feel excited, worried, or rushed?
• Who made the graph or claim, and what do they want you to believe?

5. Compare absolute change and percent change

To understand data fairly, it helps to look at both absolute change and percent change.

Absolute change is the amount something increases or decreases:

$$\text{absolute change} = \text{new value} - \text{original value}$$

Percent change compares the change to the original amount:

$$\text{percent change} = \frac{\text{new value} - \text{original value}}{\text{original value}} \times 100\%$$

Both are useful, but seeing only one can be misleading.

Worked Example 1: Spotting a truncated axis

A news post shows a bar graph comparing two stores.

• Store A monthly profit: \(\$480\)
• Store B monthly profit: \(\$500\)

The vertical axis starts at \(450\), not \(0\).

Step 1: Find the real difference.

$$500 - 480 = 20$$

The difference is only \(\$20\).

Step 2: Think about the graph.

Since the axis starts at \(450\), Store A is shown as height \(30\), and Store B is shown as height \(50\).

Step 3: Critique it.

The graph makes \(\$20\) look much bigger than it really is. The data may be correct, but the display is misleading because of the truncated axis.

Worked Example 2: Understanding percent and actual numbers

An advertisement says, “Our app users increased by 100% in one month!”

Suppose the app had \(50\) users at first and \(100\) users later.

Step 1: Find the absolute change.

$$100 - 50 = 50$$

Step 2: Find the percent change.

$$\frac{100 - 50}{50} \times 100\% = \frac{50}{50} \times 100\% = 100\%$$

Step 3: Critique it.

The claim is true, but the ad gives only the percent change. Going from \(50\) to \(100\) users is a much smaller business than going from \(5{,}000\) to \(10{,}000\) users, even though both are 100% increases. Without the actual numbers, the claim may sound more impressive than it is.

Worked Example 3: Disproportionate picture graph

A magazine uses coin pictures to compare savings.

• Person A saved \(\$100\)
• Person B saved \(\$200\)

The picture for Person B is twice as tall and twice as wide as the picture for Person A.

Step 1: Compare the actual amounts.

Person B saved:

$$200 - 100 = 100$$

more dollars than Person A.

Also, Person B saved:

$$\frac{200}{100} = 2$$

times as much as Person A.

Step 2: Compare the picture sizes.

If height and width both double, the area becomes 4 times as great.

Step 3: Critique it.

The picture makes \(\$200\) look like 4 times \(\$100\), even though it is only 2 times as much. This is misleading scaling.

Worked Example 4: Manipulative wording in a headline

A headline says, “Snack prices are skyrocketing!” The article gives these prices:

• Last year: \(\$1.20\)
• This year: \(\$1.50\)

Step 1: Find the absolute change.

$$1.50 - 1.20 = 0.30$$

Step 2: Find the percent change.

$$\frac{1.50 - 1.20}{1.20} \times 100\% = \frac{0.30}{1.20} \times 100\% = 25\%$$

Step 3: Critique the framing.

The price did increase, and \(25\%\) is a real increase. But the word “skyrocketing” is strong emotional language. A change of \(\$0.30\) may or may not feel huge, depending on the situation. The wording is trying to create a dramatic reaction.

6. A checklist for critiquing media data

When you see a chart, graph, or claim, use this checklist:

1. Read the title carefully. What is the graph trying to prove?
2. Check the axes. Do they start at 0? Are intervals equal?
3. Look at the labels. Are units clear, like dollars, people, or percent?
4. Compare the actual numbers. Do the visuals match the data?
5. Watch for loaded words. Does the wording sound dramatic or one-sided?
6. Ask what is missing. Is there enough information to understand the full story?
7. Think about the source. Is it news, an ad, a business, or social media?

7. Why this matters in real life

Misleading data can affect important choices. People may decide where to shop, what to buy, how to save money, or which company to trust based on graphs and claims they see.

If you learn to question how data is presented, you are less likely to be fooled. You do not have to reject every graph. Instead, you should look carefully, ask questions, and decide whether the data is being shown fairly.

Summary

Media can present data in misleading ways, even when the numbers are technically correct. A truncated axis can make small differences look large, disproportionate scaling can exaggerate changes, and manipulative framing can use wording or missing details to influence the reader.

To critique data well, always check the graph scale, compare the real numbers, and think about how the information is being described. A careful reader looks beyond the picture and asks, “What is this data really showing?”