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Percent Problems
in Context

Master percent calculations for real-world scenarios: discounts, tax, tips, and more. Build the skills to solve any percent word problem with confidence.

ISEE Middle Level

What Is a Percent?

A percent means "per hundred." When we write \( 25\% \), we mean \( 25 \) out of every \( 100 \). It is simply a special ratio with a denominator of \( 100 \).

\( 4 \text{ out of } 10 = 40\% \)

Three Equivalent Forms

Percent
\( 75\% \)
The "per hundred" form used in everyday language.
Fraction
\( \frac{75}{100} = \frac{3}{4} \)
Divide by \( 100 \) and simplify.
Decimal
\( 0.75 \)
Move the decimal point two places left.

Why Percents Matter

Percents appear everywhere in daily life: sales tax, discounts, tips at restaurants, test scores, and interest rates. On the ISEE, you will see percent problems wrapped inside real-world stories.

Shopping
"A jacket is \( 30\% \) off its \$80 price."
Dining
"Leave a \( 20\% \) tip on a \$45 bill."
School
"You scored \( 18 \) out of \( 20 \). What percent?"

The Core Percent Formula

Part = Percent × Whole
\( \text{Part} = \text{Percent (as decimal)} \times \text{Whole} \)

Three Question Types

1
Find the Part: "What is \( 20\% \) of \( 150 \)?" Multiply: \( 0.20 \times 150 = 30 \).
2
Find the Percent: "\( 30 \) is what percent of \( 150 \)?" Divide: \( \frac{30}{150} = 0.20 = 20\% \).
3
Find the Whole: "\( 30 \) is \( 20\% \) of what number?" Divide: \( \frac{30}{0.20} = 150 \).

"Is" vs "Of" in Percent Problems

Key Word Math Meaning Example
"is" = (equals) "12 is ..." → \( 12 = \ldots \)
"of" × (multiply) "20% of 60" → \( 0.20 \times 60 \)
"what" unknown variable "What percent?" → \( x \)

How to Solve Any Percent Problem

1
Read carefully. Identify the Part, Percent, and Whole. Which one is missing?
2
Convert the percent to a decimal by dividing by \( 100 \). Example: \( 35\% \to 0.35 \).
3
Set up the equation: \( \text{Part} = \text{Percent} \times \text{Whole} \).
4
Solve for the missing value using multiplication or division.
5
Check your answer. Does it make sense in context? A \( 10\% \) tip should be small, not larger than the bill!

Example 1: Finding the Part

A pair of sneakers costs \$80. The store offers a \( 25\% \) discount. How much money do you save?

$80 (Whole) $20 25% 75% remaining

Example 2: Finding the Percent

On a test with \( 40 \) questions, Maria answered \( 34 \) correctly. What percent did she get right?

34 correct 6
\( 34 \div 40 = 0.85 = 85\% \)

Example 3: Finding the Whole

A tip of \$9 represents \( 15\% \) of the restaurant bill. What was the total bill?

? (Whole) $9 15%

Multi-Step: Discount + Tax

A \$60 backpack is on sale for \( 20\% \) off. After the discount, \( 8\% \) sales tax is added. What is the final price?

1
Find the discount: \( 0.20 \times 60 = \$12 \).
2
Subtract: \( 60 - 12 = \$48 \) (sale price).
3
Find the tax: \( 0.08 \times 48 = \$3.84 \).
4
Add: \( 48 + 3.84 = \) \( \$51.84 \).

Percent Increase and Decrease

Percent Increase
$$ \frac{\text{New} - \text{Original}}{\text{Original}} \times 100 $$ The value went up.
Percent Decrease
$$ \frac{\text{Original} - \text{New}}{\text{Original}} \times 100 $$ The value went down.

Example: Percent Change

A town's population grew from \( 4{,}000 \) to \( 4{,}500 \). What is the percent increase?

4,000 (Original) 4,500 (New) +500
\( 500 \div 4{,}000 = 0.125 = 12.5\% \)

Common Mistakes to Avoid

✕ Forgetting to Convert
Using \( 25 \) instead of \( 0.25 \) when multiplying. Always convert the percent to a decimal first.
✕ Wrong Denominator
Dividing by the new value instead of the original in percent change problems.
✕ Tax on Full Price
Applying sales tax to the original price instead of the discounted price when both apply.

Percent Problem Summary

Find... Formula Example
Part \( \text{P} \times \text{W} \) \( 0.30 \times 50 = 15 \)
Percent \( \frac{\text{Part}}{\text{Whole}} \times 100 \) \( \frac{15}{50} \times 100 = 30\% \)
Whole \( \frac{\text{Part}}{\text{Percent}} \) \( \frac{15}{0.30} = 50 \)
% Change \( \frac{\text{Change}}{\text{Original}} \times 100 \) \( \frac{10}{40} \times 100 = 25\% \)

Question 1 of 5

A shirt originally costs \$40. It is on sale for \( 15\% \) off. What is the sale price?

Question 2 of 5

Marcus scored \( 36 \) out of \( 45 \) on a quiz. What percent did he score?

Question 3 of 5

A \( 20\% \) tip on a restaurant meal is \$12. What was the total bill before the tip?

Question 4 of 5

The price of a bicycle increased from \$200 to \$250. What is the percent increase?

Question 5 of 5

A \$50 jacket is \( 30\% \) off. After the discount, \( 10\% \) sales tax is added. What is the final price?