Percent Calculations

Why This Matters

Percent calculations show up everywhere on your Pre Algebra assessments—and in real life. You're being tested on your ability to move fluidly between fractions, decimals, and percents, apply formulas to find missing values, and interpret what percent changes actually mean. These skills form the foundation for algebra, statistics, and every math course that follows.

Here's the key insight: percent problems aren't random. They all come down to understanding the relationship between a part, a whole, and a percent. Once you see that every problem is just asking you to find one of these three pieces, the calculations become predictable. Don't just memorize formulas—know which formula to use based on what's missing and what's given.

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Converting Between Forms

Before you can solve any percent problem, you need to move comfortably between fractions, decimals, and percentages. These are just three different ways to express the same value—the skill is knowing which form makes your calculation easiest.

Fraction-Decimal-Percent Conversions

• Fraction to percent—multiply by 100 (or equivalently, convert to decimal first, then shift the decimal point two places right)
• Decimal to percent—multiply by 100 and add the % symbol; $0.75 = 75\%$
• Percent to fraction—divide by 100 and simplify; $25\% = \frac{25}{100} = \frac{1}{4}$

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Finding a Part, Whole, or Percent

The core of percent problems is the relationship: Part = Whole × Percent (as a decimal). Every problem gives you two of these values and asks for the third. Recognizing which piece is missing determines your approach.

Calculating Percentage of a Number

• Formula: Part = Whole × Percent (as decimal)—this finds how much a percentage represents
• Example—to find 20% of 50, calculate $50 \times 0.20 = 10$
• Common applications—discounts, tips, tax amounts, and test scores all use this calculation

Finding the Whole from a Part

• Formula: Whole = Part ÷ Percent (as decimal)—use when you know the part and the percent
• Example—if 30 is 60% of a number, then $30 \div 0.60 = 50$
• Watch for this setup—"___ is __% of what number?" signals you're solving for the whole

Using Proportions for Percent Problems

• Set up the ratio—$\frac{\text{Part}}{\text{Whole}} = \frac{\text{Percent}}{100}$
• Cross-multiply to solve—this method works for any missing piece and helps visualize the relationship
• Best for complex problems—when the formula approach feels confusing, proportions provide a reliable backup

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Calculating Percent Change

Percent change measures how much a value has increased or decreased relative to where it started. This concept appears constantly in data interpretation, finance questions, and real-world applications.

Percentage Increase and Decrease

• Increase formula—$\text{Percent Increase} = \frac{\text{Increase}}{\text{Original}} \times 100$
• Decrease formula—$\text{Percent Decrease} = \frac{\text{Decrease}}{\text{Original}} \times 100$
• Critical detail—always divide by the original value, not the new value; this is a common error

Finding Percent Change (General Formula)

• Universal formula—$\text{Percent Change} = \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100$
• Positive result = increase; negative result = decrease—the sign tells you the direction of change
• Real-world use—price changes, population shifts, and grade improvements all use this calculation

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Money Applications

Percent calculations power most financial math. These problems apply everything above to real scenarios involving money—and they're heavily tested because they demonstrate practical understanding.

Calculating Simple Interest

• Formula: $I = P \times r \times t$—where $I$ = interest, $P$ = principal (starting amount), $r$ = rate (as decimal), $t$ = time
• Rate conversion is essential—5% becomes 0.05; forgetting this step is the #1 error
• Example—$\$500$ at 4% for 3 years: $500 \times 0.04 \times 3 = \$60$ interest

Finding Discounts and Sales Tax

• Discount calculation—multiply original price by discount percent, then subtract from original
• Sales tax calculation—multiply price by tax rate, then add to original
• Shortcut for final price—for 20% off, multiply by 0.80 directly; for 8% tax, multiply by 1.08

Calculating Tip and Commission

• Tip formula—$\text{Tip} = \text{Bill} \times \text{Tip Percent (as decimal)}$
• Commission works identically—salespeople earn a percent of their sales; $\$2000$ in sales at 6% commission = $\$120$
• Mental math trick—find 10% by moving the decimal, then adjust (15% = 10% + half of 10%)

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Problem-Solving Strategies

Word problems test whether you can identify which percent calculation to use, not just whether you can execute it. The skill is translating English into math.

Solving Word Problems with Percentages

• Identify the three pieces—every problem involves a part, whole, and percent; determine which two you have
• Translate keywords—"of" means multiply; "is" means equals; "what" is your variable
• Check reasonableness—if finding 25% of something, your answer should be about ¼ of the original

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Quick Reference Table

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Self-Check Questions

1. What's the key difference between finding "20% of 80" and finding "what percent is 20 of 80"—and which formula do you use for each?

2. If a shirt originally costs $45 and is on sale for 30% off, what two different methods could you use to find the final price?

3. Compare percent increase and percent decrease: why must you always divide by the original value rather than the new value?

4. A salesperson earns $180 in commission on $3,000 in sales. What is their commission rate, and which formula structure (part/whole/percent) did you use to find it?

5. If you know that $24 represents 15% of your total budget, explain step-by-step how you would find the total budget—and what common error might give you the wrong answer.