Percentages and Percent Change: Practice Questions & Study Guide
Questions covering percent conversions, percent increase and decrease, and multi-step percentage problems with real-world price or quantity contexts.
Understanding Percentages and Percent Change
A percent is simply a ratio out of 100. The ability to move fluidly between percent form, decimal form, and fraction form is essential because different question formats demand different representations. To convert a percent to a decimal, divide by 100 (so 35% = 0.35). To convert a decimal to a percent, multiply by 100. To find what percent one number is of another, divide the part by the whole and multiply by 100.
Percent change measures how much a quantity has grown or shrunk relative to its original (starting) value. The formula is: percent change = (new value − old value) / old value × 100. The direction matters: a positive result is an increase, a negative result is a decrease. Many students make the critical error of dividing by the new value instead of the old value—the denominator must always be the original amount.
Multi-step percent problems chain increases and decreases together. If a price increases by 20% and then decreases by 10%, the correct calculation is: new price = original × 1.20 × 0.90 = original × 1.08, which is an 8% net increase—not 10%, as many students assume. Each percent applies to whatever the current value is at that step, not the original. The multiplier method (multiply by 1 + r for increases, 1 − r for decreases) is the fastest and most reliable technique.
The test also covers 'percent of total' questions using two-way tables or bar charts, where you must identify the correct subgroup as both the numerator and the denominator. These are fundamentally the same as basic percent problems but require reading the table carefully before calculating.
Key Rules & Formulas
Memorize these rules — they come up directly in practice questions.
Percent of a number: P% of N = (P/100) × N
30% of 80 = 0.30 × 80 = 24
Percent change = (new − old) / old × 100; denominator is always the original
Price rises from $40 to $52: (52 − 40)/40 × 100 = 30% increase
Multiplier method: increase by r% → multiply by (1 + r/100); decrease by r% → multiply by (1 − r/100)
20% increase then 15% decrease: final = original × 1.20 × 0.85 = original × 1.02 (2% net increase)
Finding the original from a new value after a percent change: original = new value / (1 ± r/100)
After a 25% increase, a price is $75. Original = 75 / 1.25 = $60
Percent more / percent less: 'A is 20% more than B' means A = 1.20B, not A − B = 0.20
If store A's price is 15% less than store B's $200 price: A = 0.85 × 200 = $170
Percentages and Percent Change Practice Questions
Select an answer and click Check Answer to reveal the full explanation. Questions go from easiest to hardest.
A jacket originally costs $80. It is on sale for 25% off. What is the sale price of the jacket?
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Correct answer: B. $60
Explanation
Discount = 25% × $80 = $20. Sale price = $80 − $20 = $60.
A student scores 45 out of 60 on a test. What percent did the student score?
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Correct answer: C. 75%
Explanation
Percent = (45/60) × 100 = 75%.
A store increases the price of a shirt from $40 to $50. By what percent did the price increase?
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Correct answer: C. 25%
Explanation
Percent increase = (50 − 40)/40 × 100 = 10/40 × 100 = 25%.
After a 20% increase, the price of a laptop is $960. What was the original price?
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Correct answer: C. $800
Explanation
Original × 1.20 = $960. Original = $960 / 1.20 = $800.
A price increases by 30% and then decreases by 30%. What is the net percent change from the original price?
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Correct answer: B. −9%
Explanation
Multiplier: 1.30 × 0.70 = 0.91. Net change = (0.91 − 1) × 100 = −9%. The price is 9% lower than the original.
In a survey of 200 people, 35% prefer brand A and 40% prefer brand B. The remaining respondents prefer brand C. How many people prefer brand C?
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Correct answer: B. 50
Explanation
Percent for brand C = 100% − 35% − 40% = 25%. People preferring C = 25% × 200 = 50.
A population of 4,000 grows by 5% in the first year and by 8% in the second year. What is the population after 2 years?
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Correct answer: C. 4,536
Explanation
After year 1: 4,000 × 1.05 = 4,200. After year 2: 4,200 × 1.08 = 4,536.
A store offers a 15% discount on an item, and then a further 10% discount on the discounted price. What single percentage discount is equivalent to both discounts applied together?
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Correct answer: A. 23.5%
Explanation
Combined multiplier: 0.85 × 0.90 = 0.765. This means the final price is 76.5% of the original, so the total discount is 1 − 0.765 = 0.235 = 23.5%.
A salesperson earns a base salary of $2,000 per month plus a 6% commission on all sales above $10,000. If the salesperson earned $2,540 in a particular month, what were the total sales that month?
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Correct answer: B. $19,000
Explanation
Commission earned = $2,540 − $2,000 = $540. Sales above $10,000 = $540 / 0.06 = $9,000. Total sales = $10,000 + $9,000 = $19,000.
A school's enrollment increased by 12% from Year 1 to Year 2, and then decreased by 5% from Year 2 to Year 3. The enrollment in Year 3 was 532 students. What was the enrollment in Year 1?
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Correct answer: D. 500
Explanation
Year 3 = Year 1 × 1.12 × 0.95. Year 1 = 532 / (1.12 × 0.95) = 532 / 1.064 = 500.
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Common Mistakes to Avoid
These are the most frequent errors students make on Percentages and Percent Change questions. Knowing them in advance prevents costly point losses.
- !Dividing by the new value instead of the original value when computing percent change
- !Adding two percent changes directly (e.g., +20% then −20% ≠ 0%) instead of multiplying the multipliers
- !Confusing 'what percent of B is A' with 'what percent of A is B'—flipping the fraction
- !Treating 'A is 20% more than B' as 'A − B = 20' instead of 'A = 1.20B'
- !Forgetting to convert a percent back to a decimal before multiplying (e.g., multiplying by 20 instead of 0.20)
Strategy Tips: Percentages and Percent Change
Use the multiplier method for all multi-step problems—write out each multiplier as a decimal and multiply straight through to avoid the base-switching error
When working backward from a final value to the original, divide by the multiplier rather than subtracting the percent amount
If a problem asks for the percent by which A exceeds B, make sure B is in the denominator—not A
On table-based percent questions, write the specific numerator and denominator you identified before punching anything into the calculator to avoid reading the wrong cell
Other Problem Solving & Data Analysis Subtopics
Ratios, Rates, and Proportional Relationships
Questions asking you to set up proportional equations, convert units, and scale quantities in real-world contexts.
Statistics, Data Interpretation, and Distributions
Questions requiring you to calculate and interpret measures of center and spread, read graphs and tables, and understand the shape of data distributions.
Probability and Conditional Probability
Questions asking you to find simple, compound, and conditional probabilities, often from two-way frequency tables.
Statistical Inference and Study Design
Questions testing whether you can identify what a study's design allows you to validly conclude, including generalizability and causality.
Master Percentages and Percent Change
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