Lesson 6.3: Percent, Ratio, Rate, and Proportion

Introduction

In this lesson, we will explore four fundamental concepts of arithmetic: percent, ratio, rate, and proportion. These concepts are vital for solving many quantitative reasoning problems found on the GRE. By the end of this lesson, you should be able to:

Calculate percent change, percent of, and successive percentages.
Set up and solve ratios and proportions effectively.
Solve rate problems, including work and distance.
Solve percent and percent-change problems with confidence.
Translate ratio and proportion relationships into equations.

Let’s begin by understanding what these terms mean and how they can be applied in real-world scenarios.

Percent

Percent is a way of expressing a number as a fraction of 100. It is denoted by the symbol "%". For example, 45% means 45 out of every 100, or $\frac{45}{100}$.

1.1 Percent Calculation

To calculate a percentage of a number, you can use the formula:

$$\text{Percent of a number} = \frac{p}{100} \times n$$

where $p$ is the percentage and $n$ is the number.

Example 1: Calculate 20% of 50

To find 20% of 50:

Substitute into the formula: $\frac{20}{100} \times 50$
Calculate: $\frac{20 \times 50}{100} = \frac{1000}{100} = 10$

Thus, 20% of 50 is 10.

1.2 Percent Change

Percent change measures how much a quantity has increased or decreased relative to its original value. The formula for percent change is:

$$\text{Percent Change} = \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100$$

Example 2: Calculate the percent change when a price changes from $45 to $60

Identify the original value (\$45) and new value (\$60).
Apply the formula: $\frac{60 - 45}{45} \times 100$
Calculate: $\frac{15}{45} \times 100 = \frac{15 \times 100}{45} = \frac{1500}{45} \approx 33.33\%$

The percent change is approximately 33.33% increase.

1.3 Successive Percentages

When applying multiple percentage changes in succession, the overall effect can be calculated by sequentially applying each percentage change, rather than simply adding them.

Example 3: Calculate an increase of 10% followed by an increase of 20% on a value of 100

First Increase: $\text{New value} = 100 + \frac{10}{100} \times 100 = 100 + 10 = 110$
Second Increase: $\text{New value} = 110 + \frac{20}{100} \times 110 = 110 + 22 = 132$

Thus, after both increases, the final value is 132.

Ratio

A ratio compares two quantities and shows how much of one there is compared to another. Ratios can be expressed in three ways: as fractions, with a colon, or in words. For example, the ratio of 2 to 3 can be expressed as $\frac{2}{3}$, $2:3, or "2 to 3".

2.1 Setting Up Ratios

Ratios can be set up from word problems, equations, or relationships between two quantities.

Example 4: If there are 10 apples and 15 oranges, what is the ratio of apples to oranges?

The ratio of apples to oranges is:

$$\text{Ratio} = \frac{\text{Number of Apples}}{\text{Number of Oranges}} = \frac{10}{15} = \frac{2}{3}$$

Thus, the ratio of apples to oranges is 2:3.

2.2 Solving Ratios

Once ratios are set up, solving for unknowns can be done using cross-multiplication.

Example 5: If the ratio of cats to dogs is 4:5 and there are 20 cats, how many dogs are there?

Let $d$ represent the number of dogs:

$$\frac{4}{5} = \frac{20}{d}$$

Cross-multiply: $4d = 100$

Thus, $d = \frac{100}{4} = 25$.

Therefore, there are 25 dogs.

Rate

Rate is a specific type of ratio that compares two different units, such as distance and time or money and hours worked. Common examples include speed and unit price.

3.1 Work Rate Problems

In work problems, you often need to find how long it takes to complete a task based on the rate of work.

Example 6: If 3 workers can finish a job in 4 hours, how long would it take 6 workers to finish the same job?

The rate of work for 3 workers is:

$$\text{Rate} = \frac{1 \text{ job}}{4 \text{ hours}}$$

Thus, the rate for 3 workers is $\frac{1}{4}$ jobs per hour. The rate for 1 worker is $\frac{1}{4} \div 3 = \frac{1}{12}$ jobs per hour.

Going from 1 worker to 6 workers, the new rate is:

$$6 \times \frac{1}{12} = \frac{6}{12} = \frac{1}{2} \text{ jobs per hour}$$

Since the workers can complete $\frac{1}{2}$ of the job in one hour, it will take them 2 hours to finish the job.

3.2 Distance Rate Problems

Distance problems relate speed, time, and distance together using the formula:

$$\text{Distance} = \text{Rate} \times \text{Time}$$

Example 7: If a car travels at 60 miles per hour for 3 hours, how far does it travel?

Using the formula:

$$\text{Distance} = 60 \text{ miles/hour} \times 3 \text{ hours} = 180 \text{ miles}$$

Thus, the car travels 180 miles.

Proportion

A proportion states that two ratios are equal. This can be expressed as:

$$\frac{a}{b} = \frac{c}{d}$$

Indicating that $a$ is to $b$ as $c$ is to $d$. Proportions can be solved using cross-multiplication.

4.1 Solving Proportions

When given a proportion, you can find the unknown by applying the cross-multiplication method.

Example 8: Solve for $x$ in the proportion $\frac{2}{3} = \frac{x}{12}$

Cross-multiply:

$$2 \times 12 = 3 \times x$$

Thus, $24 = 3x$ implies $x = \frac{24}{3} = 8$.

Conclusion

In this lesson, we covered the key concepts of percent, ratio, rate, and proportion. You learned how to perform calculations involving percentages, set up and solve ratios, solve rate problems, and understand the importance of proportions. Mastering these concepts will significantly aid you in your quantitative reasoning on the GRE.

Study Notes

Percent: Represents a fraction of 100, used for comparison and analysis.
Percent Change: Indicates the relative increase or decrease of a quantity; utilize the formula provided.
Rate: The comparison of quantities with different units, includes speed and efficiency.
Ratio: Compares two quantities and can be expressed in various ways; solving involves cross-multiplication when necessary.
Proportion: An equation stating that two ratios are equivalent; cross-multiplication is key for solving related problems.