Percent Problems

Finding percent of a number, percent increase/decrease

Percent Problems

Finding a Percent of a Number

Method: Convert percent to decimal, then multiply.

Example: What is 25% of 60? $0.25 \times 60 = 15$

The Percent Equation

$\text{Part} = \text{Percent} \times \text{Whole}$

Can be rearranged:

$\text{Percent} = \frac{\text{Part}}{\text{Whole}}$
$\text{Whole} = \frac{\text{Part}}{\text{Percent}}$

Percent Increase

$\text{Percent Increase} = \frac{\text{New} - \text{Original}}{\text{Original}} \times 100\%$

Example: Price increases from $\$ 20 $to$ $25$: $\frac{25 - 20}{20} \times 100\% = \frac{5}{20} \times 100\% = 25\%$

Percent Decrease

$\text{Percent Decrease} = \frac{\text{Original} - \text{New}}{\text{Original}} \times 100\%$

Example: Price drops from $\$ 40 $to$ $30$: $\frac{40 - 30}{40} \times 100\% = \frac{10}{40} \times 100\% = 25\%$

Tips and Discounts

Discount: amount saved Sale price = Original price - Discount

Tip/Tax: added amount Total = Original + Tip (or Tax)

Common Percent Problems

Finding a sale price: Original $\times$ (1 - percent off)
Finding total with tax: Original $\times$ (1 + tax rate)

📚 Practice Problems

1Problem 1easy

❓ Question:

What is 30% of 80?

💡 Show Solution

Convert percent to decimal and multiply:

$30\% = 0.30$

$0.30 \times 80 = 24$

Answer: $24$

2Problem 2medium

❓ Question:

A shirt costs $\$ 40$. It is on sale for 25% off. What is the sale price?

💡 Show Solution

Method 1: Find discount, then subtract $\text{Discount} = 0.25 \times 40 = 10$ $\text{Sale price} = 40 - 10 = 30$

Method 2: Multiply by (1 - 0.25) $\text{Sale price} = 40 \times 0.75 = 30$

Answer: $\$ 30$

3Problem 3hard

❓ Question:

A population increased from 250 to 300. What is the percent increase?

💡 Show Solution

Use percent increase formula:

$\text{Percent Increase} = \frac{\text{New} - \text{Original}}{\text{Original}} \times 100\%$

$= \frac{300 - 250}{250} \times 100\%$

$= \frac{50}{250} \times 100\%$

$= 0.2 \times 100\% = 20\%$

Answer: 20% increase

4Problem 4medium

❓ Question:

A jacket originally costs $80. It goes on sale for 30% off. What is the sale price?

💡 Show Solution

Method 1: Find discount, then subtract Step 1: Find 30% of $80. 0.30 × 80 =$ 24 discount

Step 2: Subtract from original. $80 -$ 24 = $56

Method 2: Find what you pay directly Step 1: If 30% off, you pay 70%. 100% - 30% = 70%

Step 2: Find 70% of $80. 0.70 × 80 =$ 56

Answer: $56

5Problem 5hard

❓ Question:

You got 17 out of 20 questions correct on a test. What is your percentage score? Then, if the teacher adds 5 bonus points to your raw score, what is your new percentage?

💡 Show Solution

Part 1: Original percentage Step 1: Set up the fraction. 17/20

Step 2: Convert to decimal. 17 ÷ 20 = 0.85

Step 3: Convert to percent. 0.85 = 85%

Part 2: With bonus points Step 1: New score. 17 + 5 = 22 points earned Total still out of 20 (bonus goes above 100%)

Step 2: Calculate percent. 22/20 = 1.10 = 110%

Answer: Original: 85%, With bonus: 110%

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