Lesson 2.3: Percentages and Percent Change

Introduction

In this lesson, students, we will dive deep into the world of percentages, an essential component of quantitative reasoning. Understanding percentages is fundamental for solving various problems on the GMAT and in real life, ranging from financial calculations to statistical analyses. By mastering this topic, you will enhance your arithmetic fluency, enabling you to tackle complex problems under time pressure.

Learning Objectives

• Understand what percentages, percent increase and decrease, and successive changes are.
• Translate "percent of" language into computation effectively.
• Identify and avoid common percentage traps including base confusion.
• Solve various percent and percent-change problems accurately.
• Avoid errors related to the base of comparison.

What is a Percentage?

A percentage is a way of expressing a number as a fraction of 100. It indicates how much of something is present compared to a whole. The symbol for percentage is "%". To calculate a percentage, you can use the formula:

$$

$\text{Percentage} = \left( \frac{\text{Part}}{\text{Whole}} $

$ight) \times 100$

$$

Example 1: Finding a Percentage

Suppose you scored 45 out of 60 on an exam. To find the percentage score, you would set it up as follows:

1. Identify the part (your score) and the whole (total possible score):

$ - Part = 45$

$ - Whole = 60$

1. Plug the values into the formula:

$$

$ \text{Percentage} = \left( \frac{45}{60} $

$ight) \times 100$

$$

1. Calculate:

• First, divide: $ \frac{45}{60} = 0.75 $
• Then, multiply: $ 0.75 \times 100 = 75\% $

Thus, your percentage score is $75\%$.

Percent Increase and Decrease

Percent Increase

Percent increase quantifies how much a value has grown relative to its original amount. The formula for percent increase is:

$$

\text{Percent Increase} = $\left($ \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}}

$ight) \times 100$

$$

Example 2: Calculating Percent Increase

Imagine a stock price increases from $50 to $65.

1. Identify the new value and the original value:

• New Value = $65
• Original Value = $50

1. Plug in the values:

$$

\text{Percent Increase} = $\left($ $\frac{65 - 50}{50}$

$ight) \times 100$

$$

1. Calculate:

• Subtract: $ 65 - 50 = 15 $
• Divide: $ \frac{15}{50} = 0.3 $
• Multiply: $ 0.3 \times 100 = 30\% $

Therefore, the percent increase is $30\%$.

Percent Decrease

Percent decrease measures how much a value has dropped compared to its original value. The formula for percent decrease is:

$$

\text{Percent Decrease} = $\left($ \frac{\text{Original Value} - \text{New Value}}{\text{Original Value}}

$ight) \times 100$

$$

Example 3: Calculating Percent Decrease

Suppose a product's price drops from $80 to $60.

1. Identify the new value and the original value:

• Original Value = $80
• New Value = $60

1. Plug the values into the formula:

$$

\text{Percent Decrease} = $\left($ $\frac{80 - 60}{80}$

$ight) \times 100$

$$

1. Calculate:

• Subtract: $ 80 - 60 = 20 $
• Divide: $ \frac{20}{80} = 0.25 $
• Multiply: $ 0.25 \times 100 = 25\% $

Hence, the percent decrease is $25\%$.

Successive Percent Changes

When you have multiple percent changes that affect a quantity, it’s crucial to apply each change step-by-step rather than simply adding or subtracting the percentages. This is because each change is calculated on the new value, not the original value.

Example 4: Successive Changes

Consider a scenario where a product’s price increases by $20\%$ and then decreases by $10\%$. Let's say the original price is $100.

1. Calculate the increase:

$$

\text{New Price After Increase} = 100 + ($0.20 \times 100$) = 100 + 20 = 120

$$

1. Now calculate the decrease on the new price:

$$

\text{New Price After Decrease} = 120 - ($0.10 \times 120$) = 120 - 12 = 108

$$

1. The final price after successive changes is $108.

Common Misconceptions

• Base Confusion: One of the most frequent errors made in percent problems is not knowing what the base is. Often, students mistakenly use the final amount instead of the original amount when computing percentage changes. Always clarify your base before performing calculations.
• Adding Percentages: It is incorrect to add percentages when computing successive changes. Each percentage should be applied to the value that has just been calculated.

Example 5: Avoiding Misconceptions

If a shirt is originally $100, and there’s a 50% discount followed by a 25% discount on the new price:

1. Calculate the first discount:

$$

\text{Price After First Discount} = 100 - ($0.50 \times 100$) = 50

$$

1. Now, apply the second discount:

$$

\text{Price After Second Discount} = 50 - ($0.25 \times 50$) = 50 - 12.5 = 37.5

$$

This results in a final price of $37.50, and not $25 as one might mistakenly calculate by simply adding the discounts.

Conclusion

In this lesson, students, we explored the critical concepts of percentages, how to calculate percent increases and decreases, and the implications of multiple successive changes. Mastering these concepts will not only help you perform better in quantitative reasoning on the GMAT but will also serve you well in everyday life situations where you encounter changes in prices, stats, and data.

Study Notes

• Percentage is expressed as a fraction of 100.
• Percent Increase Formula: $\text{Percent Increase} = \left( \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \right) \times 100$
• Percent Decrease Formula: $\text{Percent Decrease} = \left( \frac{\text{Original Value} - \text{New Value}}{\text{Original Value}} \right) \times 100$
• Always apply successive percent changes individually rather than collectively.
• Remember to clarify the base amount when calculating percentages to avoid confusion.