Question

Towns \(A, B, C,\) and \(D\) are all in the same voting district. Towns \(A\) and \(B\) have 3,000 people each who support referendum \(R\) and the referendum has an average (arithmetic mean) of 3,500 supporters in towns \(B\) and \(D\) and an average of 5,000 supporters in Towns \(A\) and \(C.\) Quantity \(\mathbf{A}\) The average number of supporters of Referendum \(R\) in Towns \(C\) and \(D\) Quantity \(\underline{B}\) The average number of supporters of Referendum \(R\) in Towns \(B\) and \(C\) A. Quantity A is greater. B. Quantity B is greater. C. The two quantities are equal, D. The relationship cannot be determined from the information given.

Short answer

A. Quantity A is greater.

Step-by-step solution

Determine the supporters in towns B and D

The average number of supporters for towns B and D is 3,500. Therefore, considering that the number of supporters in town B is 3,000, the number of supporters in town D can then be calculated by multiplying the average by 2 (because there are two towns) and then subtracting the known number of supporters in town B. This gives: \(2*3500 - 3000 = 4000\), so there are 4,000 supporters in town D.

Determine the supporters in towns A and C

The average number of supporters for towns A and C is 5,000. Since we know the number of supporters in town A is 3,000, we can get the number of supporters in town C by a similar approach to Step 1. \(2*5000 - 3000 = 7000\), thus there are 7,000 supporters in town C.

Calculate Quantity A and Quantity B

In this step, we calculate Quantity A which is the average number of supporters in towns C and D. This is easy now as we add the supporters in town C and D then divide by 2. This gives us \((7000+4000)/2 = 5500\), so Quantity A is 5500. Quantity B which is the average number of supporters in towns B and C can also be calculated in a similar way. This gives \((3000+7000)/2 = 5000\), therefore Quantity B is 5000.

Compare Quantities A and B

Now with Quantity A as 5500 and Quantity B as 5000, Quantity A is greater than Quantity B

Key concepts

Arithmetic Mean

The arithmetic mean, commonly referred to as the average, is a fundamental concept in mathematics. It helps us to determine a central value for a set of numbers. To find the arithmetic mean, you sum up all the numbers in the set, and then divide the total by the number of elements in the set. In our exercise, we see this principle applied to calculating the average number of supporters in different towns.

• Formula: If you have numbers \(x_1, x_2, \, \ldots , x_n\), the arithmetic mean is \( \frac{x_1 + x_2 + \cdots + x_n}{n} \).
• Example: For towns B and D, with a total of 7,000 supporters, the mean is 3,500, calculated as \( \frac{7000}{2} \).

This basic calculation provides a snapshot of how support for the referendum is distributed across the towns.

Average Calculation

Calculating averages is an essential skill in GRE quantitative reasoning, especially when applied to real-life scenarios like our word problem.
To determine the average number of supporters in each town, specific calculations were made by following simple steps:

• For towns B and D, with B known to have 3,000 supporters, if their average is given as 3,500, then D can be calculated using \((3500 \times 2) - 3000 = 4000\).
• Similarly, knowing that towns A and C average 5,000 supporters and A has 3,000, C can be deduced with \((5000 \times 2) - 3000 = 7000\).

These methods ensure an accurate depiction of how many people support the referendum, effectively solving part of the comparative problem presented.

Word Problems

Word problems, like the one given, are often challenging due to the complexity of translating text into mathematical expressions. This requires the ability to identify relevant information and apply arithmetic operations to find a solution.
Successful strategies for tackling word problems include:

• Reading carefully: Understand what is being asked before jumping into calculations.
• Breaking down information: Dissect the text to isolate important data, such as the number of supporters in each town.
• Step-by-step approach: As shown in the solution, solve in logical steps to avoid missing key components.

In this exercise, the word problem involves calculating average supporters in pairs of towns, requiring both arithmetic mean understanding and precise computation.

Comparison Problems

Comparison problems are prevalent in standardized tests like the GRE, where you're often tasked with comparing two quantities without direct information about both.
Approach for solving comparison problems includes:

• Calculate both quantities independently, as demonstrated by solving for Quantity A (average supporters in C and D) and Quantity B (average in B and C).
• After finding Quantity A as 5,500 and Quantity B as 5,000, comparing them reveals that Quantity A is greater.
• Remember answer choices in these problems: Often include 'A is greater', 'B is greater', 'They are equal', or 'Cannot be determined'.

This methodical comparison ensures that decisions are based on computed evidence, leading to the correct result in our scenario: Quantity A exceeds Quantity B.