Question

Quantity \(\mathbf{A}\) The greatest number of consecutive nonnegative integers which have a sum less than 22 Quantity_B 6 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

Maximal sum of consecutive numbers less than 22 is achieved for n=6 (the integers from 0 to 6 sum up to 21 which is less than 22). Thus, Quantity A is equal to Quantity B, and hence the correct answer is 'C. The two quantities are equal'.

Step-by-step solution

Determine Quantity A

To determine quantity A, we have to add up consecutive nonnegative integers starting from 0 (i.e., 0, 1, 2, 3, ....), until the sum becomes just less than 22. This essentially means summing up the first 'n' integers, given by the formula \(\frac{n(n+1)}{2}\), where 'n' is the number of terms. To find 'n', you can continue adding consecutive integers starting from zero until the sum becomes less than 22. The number of terms added will then give 'n'.

Evaluate Quantity B

Quantity B is already given as 6.

Compare Quantity A and Quantity B

After finding the value of Quantity A (which is the maximum number of consecutive integers that can be added up to a sum less than 22) and having Quantity B as 6, you need to compare these two quantities.

Key concepts

Consecutive Integers

Consecutive integers are numbers that follow each other in order, without any gaps between them. For example, 0, 1, 2, 3, and so on are consecutive integers. These numbers are crucial to understanding many mathematical questions, including the one discussed here.
In problems involving consecutive integers, it's essential to recognize that each number is one unit more than the previous one. This property allows us to use specific formulas to solve problems, especially those requiring summation, like calculating how many consecutive numbers sum up to a particular value. By understanding consecutive integers, you are better equipped to tackle various mathematical challenges, especially in quantitative comparison tasks, which are common in the GRE.

Sum of Integers

The sum of consecutive integers is a frequent topic in quantitative reasoning problems. It involves adding a sequence of numbers without skipping any. The formula for the sum of the first n consecutive integers is \( \frac{n(n+1)}{2} \). This formula is derived by recognizing that each number adds to the sum starting from 0 up to n.

• For example, to find the sum of the first 4 integers, plug the value of n=4 into the formula: \( \frac{4(4+1)}{2} = 10 \).
• Understanding this formula is crucial in determining questions like how many consecutive numbers sum up to less than a given number. In this problem, by finding when the sum is just under 22, you find the maximum number of consecutive integers you can use.
• Such insights illuminate the pathways to mastering quantitative reasoning skills.

Quantitative Reasoning

Quantitative reasoning involves the application of mathematical concepts to solve real-world problems. It requires a logical approach to compare quantities, determine relations, and decide the most efficient way to solve a problem.
In the context of the GRE, mastering quantitative reasoning means understanding how to use mathematical operations and principles effectively. For example, in this problem, you need to calculate the sum of consecutive integers and then determine how many such integers fit a certain criterion – that they sum up to less than 22. This type of reasoning is crucial for making quick and accurate comparisons, like the ones tested in GRE's quantitative comparison questions.
By practicing these skills, you can enhance your problem-solving abilities, making it easier to navigate complex mathematical concepts.

GRE Problem-Solving

GRE problem-solving skills require a strategic approach. Problems like the one presented are common and involve understanding specific mathematical concepts and applying them appropriately.
To tackle GRE problem-solving effectively, consider these steps:

• Identify the problem type: Know whether it's about algebra, geometry, statistics, etc.
• Recognize patterns or formulas: Like using the sum of integers formula when dealing with consecutive numbers.
• Compare quantities logically: Often, you'll be asked to decide if one quantity is greater than another, as in this problem, where Quantity A needs comparison to the number 6.
• Practice: Familiarize yourself with various problem types, enhancing your ability to think under timed conditions.

Focusing on these strategies can aid significantly in achieving a higher GRE quantitative score. By honing these skills, students not only improve their math abilities but also their overall GRE test performance.