Mastering Percentages and Percent Change for GRE Quantitative Reasoning

Examiner Expectations

The GRE Quantitative section tests your ability to interpret and manipulate proportional relationships, specifically how a part relates to a whole, and how quantities change relative to an initial value. Examiners assess your precision in calculation and your conceptual understanding of percentage applications in various real-world scenarios.

The Method: Deconstructing Percentage Problems

For any percentage or percent change problem, follow these steps systematically:

1. Identify the Base Value (Whole) and the Part: Clearly distinguish between the original quantity (the 'whole' or 'base' from which the percentage is calculated) and the specific portion or change (the 'part'). This is crucial for setting up the correct fraction.

2. Determine the Goal: Are you calculating a percentage of a number, a percent change, the original number after a change, or a new number after a change? This dictates the formula to use.

3. Translate to Mathematical Expression:

   • Percentage of a number: If you need \(P\%\) of \(X\), calculate \( \frac{P}{100} \times X \).
   • Percent Change: If a value changes from \(V_{initial}\) to \(V_{final}\), the percent change is \( \frac{V_{final} - V_{initial}}{V_{initial}} \times 100\% \). Remember, a decrease results in a negative change, an increase in a positive change.
   • Finding New Value after Percent Change: If \(V_{initial}\) changes by \(P\%\), the new value \(V_{final}\) is \(V_{initial} \times \left(1 + \frac{P}{100}\right)\) for an increase, or \(V_{initial} \times \left(1 - \frac{P}{100}\right)\) for a decrease.
   • Finding Original Value after Percent Change: If \(V_{final}\) is the result of \(V_{initial}\) changing by \(P\%\), then \(V_{initial} = \frac{V_{final}}{\left(1 + \frac{P}{100}\right)}\) for an increase, or \(V_{initial} = \frac{V_{final}}{\left(1 - \frac{P}{100}\right)}\) for a decrease.

4. Perform Calculations: Execute the arithmetic carefully. Simplify fractions before multiplying where possible to reduce computational burden.

5. Check Units and Context: Ensure your answer makes sense in the context of the problem and that you've applied the correct units (e.g., percentage, currency, quantity).

Fully Worked Example

A research laboratory initially had 240 petri dishes. After a successful grant application, they were able to purchase an additional 15% more petri dishes. However, during a subsequent experiment, 20% of the new total number of petri dishes were contaminated and had to be discarded. What is the final number of usable petri dishes?

1. Identify the Base Value and the Part:

   • Initial base: 240 petri dishes.
   • First change: +15%.
   • Second change: -20% of the new total.

2. Determine the Goal: Find the final number of usable petri dishes after two sequential percentage changes.

3. Translate to Mathematical Expression:

   • Step 1: Calculate the number of petri dishes after the 15% increase.
     New total \( = \text{Initial total} \times \left(1 + \frac{\text{Percent Increase}}{100}\right) \)
   • Step 2: Calculate the number of discarded petri dishes (20% of the new total).
     Discarded \( = \text{New total} \times \frac{20}{100} \)
   • Step 3: Calculate the final number of usable petri dishes.
     Final usable \( = \text{New total} - \text{Discarded} \)
     Alternatively, Final usable \( = \text{New total} \times \left(1 - \frac{\text{Percent Decrease}}{100}\right) \)

4. Perform Calculations:

   • Step 1: Calculate the new total after the 15% increase.
     \( \text{New total} = 240 \times \left(1 + \frac{15}{100}\right) = 240 \times (1 + 0.15) = 240 \times 1.15 \)
     \( 240 \times 1.15 = 240 \times \left(\frac{115}{100}\right) = 24 \times \frac{115}{10} = \frac{2760}{10} = 276 \) petri dishes.
   • Step 2: Calculate the final number of usable petri dishes after the 20% decrease.
     \( \text{Final usable} = 276 \times \left(1 - \frac{20}{100}\right) = 276 \times (1 - 0.20) = 276 \times 0.80 \)
     \( 276 \times 0.80 = 276 \times \frac{8}{10} = \frac{2208}{10} = 220.8 \) petri dishes.

5. Check Units and Context: Since petri dishes cannot be fractional, we must consider the practical implication. In a real-world scenario, you might round down to 220 usable dishes, or the problem might specify how to handle fractions. For GRE purposes, if not specified, maintain the exact decimal unless the context demands an integer (e.g., number of people). Here, the question asks for "number of usable petri dishes," implying a count. If 0.8 of a dish is unusable, it's effectively 0 dishes. So, 220 usable petri dishes. If the question implies that even a fraction of a dish counts as a "usable part" then 220.8 might be acceptable. Given the context of "discarded," it's most logical to assume only whole dishes are usable.

   Therefore, the final number of usable petri dishes is 220.

Three Mistakes That Lose Marks

1. Incorrect Base for Percent Change: A common error is calculating a percentage change based on the final value instead of the initial value, or using an intermediate value when the problem specifies otherwise. For example, if a price increases by 10% then decreases by 10%, the final price is not the original price because the 10% decrease is applied to a higher base. Always identify the correct "whole" or "initial" value.

2. Confusing "Percent Of" with "Percent Change": Students sometimes misinterpret "X is P% of Y" as "X is P% more/less than Y." These are distinct concepts. "P% of Y" is \( \frac{P}{100} \times Y \). "P% more than Y" is \( Y + \frac{P}{100} \times Y = Y \times \left(1 + \frac{P}{100}\right) \). Read the wording carefully.

3. Arithmetic Errors with Decimals/Fractions: Simple calculation mistakes, especially when converting percentages to decimals or fractions (e.g., \(15\% = 0.15\), not \(1.5\) or \(0.015\)), or when performing multiplication/division, are frequent. Double-check your arithmetic, particularly when under time pressure. Using fractions \(\left(\frac{15}{100}\right)\) can sometimes be less error-prone than decimals for complex calculations.

30-Second Recap

Percentages are parts of a whole, expressed as a fraction of 100. Percent change quantifies relative increase or decrease against an initial value. Always identify the base, the goal (percentage of, or percent change), translate to the correct formula \(\left(\frac{\text{Part}}{\text{Whole}} \times 100\%\right)\), calculate carefully, and verify context. Avoid base value errors, mixing "percent of" with "percent change," and arithmetic slips.