GRE Exam: A Simple Guide To Number Properties - Evenly Spaced Sets

An evenly spaced set is any series of numbers in which the spacing between consecutive terms is constant.

The most basic example of an evenly spaced set is consecutive integers. In the set 1, 2, 3, 4, 5, the spacing between successive terms is 1.

Other examples are:
2, 4, 6, 8 . . .
10, 15, 20 . . .
3, 8 13, 18 . . .

Note that you can describe the first example as consecutive even integers and the second example as consecutive multiples of 5. Though the items in the third set are not multiples of the same number, the set is still evenly spaced since the increment between successive terms is 5.

Properties of Evenly Spaced Sets
The GRE will expect you to know certain properties of evenly spaced sets.

Property 1: All the terms in an evenly spaced set can be expressed using one of the terms in the set.

If the sum of three consecutive multiples of 4 is 60, what is the value of the smallest term?

SOLUTION: Approaching this algebraically, you can let a = the smallest term, b = the middle term, and c = the largest term. Therefore, a + b + c = 60. But notice that you have additional information about these variables! Since you are dealing with consecutive multiples of 4, you can let b = a + 4 and c = b + 4 = a + 8.

Thus using substitution, you can arrive at one equation with one variable:

Property 2: The average (arithmetic mean) of an evenly spaced set = the median of the set = the average of the endpoints.
 

What is the average of x, x + 4, x + 8, x + 12, and x + 16?

SOLUTION: Recognize that each term is 4 greater than the previous term. You thus have an evenly spaced set. To find the average, you need the median, which in this case, is x + 8.

What is the average of the integers from 15–21, inclusive?

SOLUTION: Since average = median for an evenly spaced set, you could list out all the terms and find the median. But a faster approach would be to take the average of the endpoints: . The average is 18.

Property 3: Use the following formula to determine the number of terms in an evenly spaced set:

Example: How many integers are there from 6–10, inclusive?

SOLUTION: Note that inclusive means that you will include both endpoints of the set when determining the answer. You might be tempted to simply take the difference of 10 and 6 and arrive at 4 as your answer, but this would omit one of the items. If you list out the numbers, you will see that there are five (6, 7, 8, 9, and 10).

Using the preceding formula, you arrive at:

How many multiples of 15 are there from 40–160, inclusive?

SOLUTION: Before using the formula, note that the endpoints of this set are not multiples of 15. To use the formula, you need the endpoints to have the property that the formula specifies (in this case, multiples of 15). Your endpoints will thus be 45 (the smallest multiple of 15 that is greater than 40) and 150 (the greatest multiple of 15 that is less than 160). Now you know that your endpoints are 45 and 150, and the increment between each term in the set is 15.

Plug these values into the formula:

Property 4: To determine the sum of the values in an evenly spaced set, use the average formula: A × N = S.
Properties 2 and 3 specified how to determine the average and number of items in a set, so using these properties, you can determine the sum of a set.

What is the sum of the even integers from 4–180, inclusive?

SOLUTION: Based on Property 4, you should multiply the average and the number of items in the set. From Property 2, you know that the average of an evenly spaced set equals the average of the endpoints: . Using Property 3, you can determine the number of items:


SOLUTION: A × N = S, so 92 × 89 = S = 8,188.

Quantitative Comparison Strategy: Evenly Spaced Sets
Many Quantitative Comparison questions testing evenly spaced sets will give you large sets of values.

As is always the case with Quantitative Comparison questions, you should minimize calculations by comparing the quantities and identifying what terms they have in common:

SOLUTION: Instead of calculating, identify similarities between the two columns.

Quantity A can be rewritten as: (1 + 2 + 3 . . . 6) + (7 + 8 . . . 28).
Quantity B can be rewritten as: (7 + 8 . . . 28) + 29.

SOLUTION: The two quantities share the sum of the terms from 7–28, inclusive. You can thus subtract this sum from both quantities. The new comparison is:


SOLUTION: The value in Quantity A = 21. 21 < 29, so Quantity B is greater.