Hard GRE Percent Questions Practice Questions

Calculate the final value of an investment that grows by 20% in year one and then decreases by 20% in year two to see why percent changes are often counterintuitive. These Hard GRE Percent Questions frequently appear on the Quantitative Reasoning section to test your ability to handle multi-step calculations, compound interest, and weighted averages. Successfully navigating these problems requires moving beyond simple calculations and understanding how percentages interact with changing bases. For those looking to broaden their study regime, using Free GRE Practice Questions can provide a solid foundation before tackling these advanced scenarios.

Concept Explanation

Hard GRE Percent Questions involve scenarios where the base value changes through sequential increases or decreases, or where percentages are applied to unknown variables in algebraic expressions. Unlike basic percent problems that ask for a portion of a fixed number, hard-level questions often require you to set up equations such as $\text{Final} = \text{Original} \times (1 \pm r_1) \times (1 \pm r_2)$. This is common in financial contexts, such as those discussed by the Investopedia guide on compound interest, where the interest is added back to the principal before the next calculation. Another high-level concept is the "Percent Change" formula: $$\text{Percent Change} = \frac{ \text{New} - \text{Old}}{ \text{Old}} \times 100$$

On the GRE, you must also be wary of "Percent of a Percent" problems. For example, if 40% of a population are students and 20% of those students are athletes, the athletes represent $0.40 \times 0.20 = 0.08$ or 8% of the total population. For more comprehensive practice, you can explore the GRE Prep hub to see how these concepts integrate with other math topics. Mastering these requires a disciplined approach to GRE Practice Questions with Explanations to identify common traps like adding percentages instead of multiplying them.

Solved Examples

1. Sequential Percent Change: The price of a stock increased by 25% in January and then decreased by 20% in February. What was the net percent change for the two-month period?
   1. Let the initial price be $100$.
   2. After a 25% increase: $100 \times 1.25 = 125$.
   3. After a 20% decrease: $125 \times 0.80 = 100$.
   4. The net change is $\frac{100 - 100}{100} = 0\%$.
2. Percent of an Unknown: In a certain company, 60% of the employees are women. If 20% of the women and 30% of the men are managers, what percent of all employees are managers?
   1. Assume there are 100 employees. Then there are 60 women and 40 men.
   2. Female managers: $0.20 \times 60 = 12$.
   3. Male managers: $0.30 \times 40 = 12$.
   4. Total managers: $12 + 12 = 24$.
   5. As a percentage of 100, this is 24%.
3. Working Backwards: After a 15% discount, a coat costs $153. What was the original price?
   1. Let $x$ be the original price.
   2. The discounted price is $0.85x$.
   3. Set up the equation: $0.85x = 153$.
   4. Solve for $x$: $x = \frac{153}{0.85} = 180$. The original price was $180.

Practice Questions

1. A laptop's price is reduced by 20%, and then the new price is reduced by an additional 15%. What is the total percent reduction from the original price?
2. If $x$ is 25% greater than $y$, and $y$ is 20% less than $z$, then $x$ is what percent of $z$?
3. A store owner increased the price of an item by 40%. To return the item to its original price, by what percent must the new price be reduced?

4. In a group of 200 people, 40% are smokers. If 25% of the smokers quit, what percentage of the remaining group will be smokers?
5. The population of a town increased by 10% in 2020 and decreased by 10% in 2021. If the population at the end of 2021 was 9,900, what was the population at the beginning of 2020?
6. If 30% of $A$ is equal to 45% of $B$, then $B$ is what percent of $A$?
7. A solution is 20% acid and 80% water. If 2 liters of water are added to 8 liters of this solution, what is the new percentage of acid?
8. A salary is increased by $p\%$ and then decreased by $p\%$. If the final salary is 96% of the original salary, what is the value of $p$?

Answers & Explanations

1. 32%: Use an initial value of 100. First reduction: $100 \times 0.80 = 80$. Second reduction: $80 \times 0.85 = 68$. Total reduction: $100 - 68 = 32$, which is 32% of 100.
2. 100%: Let $z = 100$. Then $y = 0.80 \times 100 = 80$. Then $x = 1.25 \times 80 = 100$. Since $x = 100$ and $z = 100$, $x$ is 100% of $z$.
3. 28.57% (or $28\frac{4}{7}\%$): Let the original price be 100. New price = 140. To get back to 100, you need a reduction of 40. $\frac{40}{140} = \frac{2}{7} \approx 0.2857$.
4. 33.33%: Initially, there are $0.40 \times 200 = 80$ smokers and 120 non-smokers. If 25% of smokers quit, $0.25 \times 80 = 20$ quit. Remaining smokers = 60. New total group = $200 - 20 = 180$. Percent smokers = $\frac{60}{180} = \frac{1}{3} \approx 33.33\%$.
5. 10,000: Let $P$ be the original population. $P \times 1.10 \times 0.90 = 9900$. $P \times 0.99 = 9900$. $P = \frac{9900}{0.99} = 10,000$.
6. 66.67%: $0.30A = 0.45B$. Solve for $\frac{B}{A}$: $\frac{B}{A} = \frac{0.30}{0.45} = \frac{2}{3}$. $\frac{2}{3} \approx 66.67\%$.
7. 16%: In 8 liters of solution, acid = $0.20 \times 8 = 1.6$ liters. New total volume = $8 + 2 = 10$ liters. New percent acid = $\frac{1.6}{10} = 0.16$ or 16%.
8. 20: Final salary = $\text{Original} \times (1 + \frac{p}{100}) \times (1 - \frac{p}{100})$. This simplifies to $1 - \frac{p^2}{10000} = 0.96$. Therefore, $\frac{p^2}{10000} = 0.04$, so $p^2 = 400$, and $p = 20$.

1. If the price of an item increases by 100%, and then decreases by 50%, what is the net change?

• A50% increase
• B50% decrease
• C0% change
• D100% increase

Frequently Asked Questions

Why does a 10% increase followed by a 10% decrease not result in the original number?

This happens because the 10% decrease is applied to a larger base value created by the initial increase. For example, 10% of 110 is 11, which is more than the original 10% of 100, leading to a final value of 99.

How do I handle percent questions with unknown variables on the GRE?

The most effective strategy is to pick a smart number, usually 100, to represent the initial unknown value. This allows you to perform concrete calculations and reach a percentage that applies to any variable used in the question.

What is the difference between a percentage point and a percent?

A percentage point is the simple numerical difference between two percentages, while a percent change measures the relative growth or decay. If an interest rate moves from 5% to 10%, it is a 5 percentage point increase but a 100% increase.

Can a percent decrease be greater than 100%?

In most GRE contexts involving physical quantities or prices, a percent decrease cannot exceed 100% because a value cannot drop below zero. However, in abstract mathematical contexts or profit/loss statements, negative values can exist.

What is the fastest way to calculate a 15% tip or discount?

Calculate 10% of the total by moving the decimal one place to the left, then halve that amount to find 5%. Adding the 10% and 5% values together gives you exactly 15% without needing a calculator.