1. Price Lists

Definition: A price list is a table or chart showing the prices of different items or services.

Key Skills:

• Reading and interpreting price information from lists or tables
• Calculating total cost when purchasing multiple items
• Comparing prices of different items

Formula for Total Cost:

\( \text{Total Cost} = \text{Price}_1 + \text{Price}_2 + \text{Price}_3 + \ldots \)

or \( \text{Total Cost} = \sum (\text{Quantity} \times \text{Unit Price}) \)

Example: Apple = $2, Banana = $1.50, Orange = $1.75

Total Cost = \( 2 + 1.50 + 1.75 = \$5.25 \)

2. Unit Prices

Definition: Unit price is the cost per single unit of an item (per ounce, per liter, per item, etc.).

Main Formula:

\( \text{Unit Price} = \frac{\text{Total Price}}{\text{Number of Units}} \)

Example 1: A 12-pack of soda costs $6

\( \text{Unit Price} = \frac{\$6}{12} = \$0.50 \text{ per can} \)

Example 2: 16 oz of cereal costs $4.80

\( \text{Unit Price} = \frac{\$4.80}{16} = \$0.30 \text{ per ounce} \)

Why It Matters: Unit prices help compare products of different sizes to find the best value.

3. Unit Prices with Unit Conversions

Strategy: Convert units first, then calculate unit price for accurate comparison.

Common Conversions:

• Weight: 1 pound (lb) = 16 ounces (oz)
• Volume: 1 gallon = 4 quarts = 8 pints = 16 cups
• Metric: 1 kilogram = 1000 grams; 1 liter = 1000 milliliters

Example: Compare prices:

• Option A: 2 lbs for $8
• Option B: 24 oz for $5

Solution: Convert Option A to ounces: \( 2 \text{ lbs} = 32 \text{ oz} \)

Option A: \( \frac{\$8}{32} = \$0.25 \text{ per oz} \)

Option B: \( \frac{\$5}{24} = \$0.21 \text{ per oz} \) (better value)

4. Unit Prices: Find the Total Price

Formula:

\( \text{Total Price} = \text{Unit Price} \times \text{Number of Units} \)

Example 1: If apples cost $1.50 per pound, how much do 5 pounds cost?

\( \text{Total Price} = \$1.50 \times 5 = \$7.50 \)

Example 2: Juice costs $0.25 per ounce. Find the cost of 48 ounces.

\( \text{Total Price} = \$0.25 \times 48 = \$12.00 \)

Real-World Application: Used when buying produce, bulk items, or gas by volume.

5. Percent of a Number: Tax, Discount, and More

Sales Tax:

\( \text{Tax Amount} = \text{Original Price} \times \frac{\text{Tax Rate}}{100} \)

\( \text{Total Cost} = \text{Original Price} + \text{Tax Amount} \)

or \( \text{Total Cost} = \text{Original Price} \times (1 + \frac{\text{Tax Rate}}{100}) \)

Example: A $50 item with 8% tax

\( \text{Tax} = \$50 \times 0.08 = \$4 \)

\( \text{Total} = \$50 + \$4 = \$54 \)

Discount:

\( \text{Discount Amount} = \text{Original Price} \times \frac{\text{Discount Rate}}{100} \)

\( \text{Sale Price} = \text{Original Price} - \text{Discount Amount} \)

or \( \text{Sale Price} = \text{Original Price} \times (1 - \frac{\text{Discount Rate}}{100}) \)

Example: A $80 jacket with 25% discount

\( \text{Discount} = \$80 \times 0.25 = \$20 \)

\( \text{Sale Price} = \$80 - \$20 = \$60 \)

Tip:

\( \text{Tip Amount} = \text{Bill} \times \frac{\text{Tip Rate}}{100} \)

\( \text{Total Bill} = \text{Bill} + \text{Tip} \)

Example: $45 meal with 20% tip

\( \text{Tip} = \$45 \times 0.20 = \$9 \)

\( \text{Total} = \$45 + \$9 = \$54 \)

6. Find the Percent: Tax, Discount, and More

General Formula:

\( \text{Percent Rate} = \frac{\text{Part}}{\text{Whole}} \times 100 \)

Finding Tax Rate:

\( \text{Tax Rate} = \frac{\text{Tax Amount}}{\text{Original Price}} \times 100 \)

Example: Tax is $6 on a $75 purchase

\( \text{Tax Rate} = \frac{6}{75} \times 100 = 8\% \)

Finding Discount Rate:

\( \text{Discount Rate} = \frac{\text{Discount Amount}}{\text{Original Price}} \times 100 \)

Example: A $40 discount on a $200 item

\( \text{Discount Rate} = \frac{40}{200} \times 100 = 20\% \)

Finding Tip Rate:

\( \text{Tip Rate} = \frac{\text{Tip Amount}}{\text{Bill}} \times 100 \)

Example: $12 tip on a $60 bill

\( \text{Tip Rate} = \frac{12}{60} \times 100 = 20\% \)

7. Sale Prices: Find the Original Price

Main Formula:

\( \text{Original Price} = \frac{\text{Sale Price}}{1 - \frac{\text{Discount Rate}}{100}} \)

Alternative Formula (working backwards):

\( \text{Original Price} = \frac{\text{Sale Price}}{\text{Decimal Multiplier}} \)

where Decimal Multiplier = \( 1 - \text{Discount as decimal} \)

Example 1: A shirt is on sale for $36 after a 20% discount. Find the original price.

\( \text{Original Price} = \frac{\$36}{1 - 0.20} = \frac{\$36}{0.80} = \$45 \)

Example 2: After a 30% discount, an item costs $49. What was the original price?

\( \text{Original Price} = \frac{\$49}{0.70} = \$70 \)

Check Your Work: Multiply original price by (1 - discount rate) to verify.

8. Multi-Step Problems with Percents

Strategy: Break the problem into smaller steps and solve in order.

Common Scenario: Discount THEN Tax

1. Step 1: Calculate the discount amount
2. Step 2: Subtract discount from original price
3. Step 3: Calculate tax on the discounted price
4. Step 4: Add tax to get final price

Formula:

\( \text{Final Price} = [\text{Original} \times (1 - \frac{d}{100})] \times (1 + \frac{t}{100}) \)

where \( d \) = discount rate, \( t \) = tax rate

Example: A $100 item has a 20% discount, then 8% tax is added.

Step 1: Discount = \( \$100 \times 0.20 = \$20 \)

Step 2: Price after discount = \( \$100 - \$20 = \$80 \)

Step 3: Tax = \( \$80 \times 0.08 = \$6.40 \)

Step 4: Final price = \( \$80 + \$6.40 = \$86.40 \)

Another Scenario: Meal + Tip + Tax

Note: Tip is typically calculated on the pre-tax amount, then tax is added to the meal only.

9. Estimate Tips

Quick Mental Math Strategies:

Method 1: Using 10%

1. Find 10% by moving decimal point one place left
2. Multiply or adjust to get desired tip percentage

Example: Estimate 15% tip on $42

10% of $42 = $4.20

5% = half of 10% = $2.10

15% tip ≈ $4.20 + $2.10 = $6.30

Common Tip Percentages:

• 10%: Move decimal one place left
• 15%: Find 10%, then add half of that
• 20%: Find 10%, then double it
• 25%: Divide bill by 4

Rounding Strategy:

Round the bill to a friendly number first, then calculate

Example: 20% tip on $38.75

Round to $40 → 10% = $4 → 20% ≈ $8

10. Simple Interest

Definition: Interest calculated only on the principal (original amount).

Main Formula:

\( I = P \times r \times t \)

or \( I = \frac{P \times r \times t}{100} \) (when rate is in percent)

Where:

• \( I \) = Interest earned or paid
• \( P \) = Principal (starting amount)
• \( r \) = Interest rate (as decimal or percent)
• \( t \) = Time (in years)

Total Amount Formula:

\( A = P + I = P + (P \times r \times t) \)

\( A = P(1 + rt) \)

Example: Find the simple interest on $500 at 4% annual rate for 3 years.

\( P = \$500 \), \( r = 0.04 \), \( t = 3 \)

\( I = 500 \times 0.04 \times 3 = \$60 \)

\( \text{Total Amount} = \$500 + \$60 = \$560 \)

Other Useful Forms:

Find Principal: \( P = \frac{I}{rt} \)

Find Rate: \( r = \frac{I}{Pt} \)

Find Time: \( t = \frac{I}{Pr} \)

11. Compound Interest

Definition: Interest calculated on the principal AND previously earned interest.

Main Formula:

\( A = P \left(1 + \frac{r}{n}\right)^{nt} \)

Where:

• \( A \) = Final amount (principal + interest)
• \( P \) = Principal (starting amount)
• \( r \) = Annual interest rate (as decimal)
• \( n \) = Number of times interest is compounded per year
• \( t \) = Time (in years)

Compound Interest Earned:

\( I = A - P \)

Common Compounding Periods:

• Annually: \( n = 1 \)
• Semi-annually: \( n = 2 \)
• Quarterly: \( n = 4 \)
• Monthly: \( n = 12 \)
• Daily: \( n = 365 \)

Special Case - Compounded Annually:

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

Example: $1,000 invested at 5% annual rate, compounded annually for 3 years.

\( P = \$1000 \), \( r = 0.05 \), \( n = 1 \), \( t = 3 \)

\( A = 1000(1 + 0.05)^3 = 1000(1.05)^3 = 1000(1.157625) = \$1,157.63 \)

\( I = \$1,157.63 - \$1,000 = \$157.63 \)

Example 2: $500 at 6% compounded quarterly for 2 years.

\( P = \$500 \), \( r = 0.06 \), \( n = 4 \), \( t = 2 \)

\( A = 500\left(1 + \frac{0.06}{4}\right)^{4 \times 2} = 500(1.015)^8 = \$563.39 \)

\( I = \$563.39 - \$500 = \$63.39 \)

Simple vs. Compound Interest

Quick Reference Formulas

💡 Key Tips for Consumer Math Success

• ✓ Always convert percents to decimals before calculating (divide by 100)
• ✓ For unit prices, compare the same units (convert first if needed)
• ✓ Tax is added; discount is subtracted from the original price
• ✓ In multi-step problems, apply discount first, then tax
• ✓ Compound interest grows faster than simple interest over time
• ✓ Check if time units match the interest rate period (annual rate needs time in years)
• ✓ Round money amounts to 2 decimal places (cents)
• ✓ Use estimation to check if your answer is reasonable