Series and Counting Basics

A relatively popular concept on the GRE math is series – when you have a list of numbers that follow a certain pattern(e.g. consecutive numbers). Usually, the question will ask you to add up all the numbers that are part of the series.

Today, we are not going to go quite so far. Instead, we are going to focus on a step that is far more basic, yet one that often eludes students. Try the following problem:

How many integers are there in Set A, if Set A includes all the numbers 1 – 10?

Really Easy, Right?

The answer seems pretty straightforward – 10. That is, there are ten possible integers: 1, 2, 3… 9, 10.

Now, you are probably thinking that this question is really obvious. Of course, there are ten numbers, if you are counting the numbers 1 – 10. I agree, but try this almost identical question:

How many integers are there in Set A, if Set A includes all the numbers 5 – 15?

The answer is ten, right? Actually, the answer is eleven. And that’s what makes this question so tricky. In answering ten, you probably went something like this – 15 – 5 = 10; therefore, there are ten numbers.

The Twist

Whenever you are counting the total numbers in a list by subtracting, you always have to count an extra number.”For instance, with the first question, if you just subtract 10 – 1 = 9, you get one fewer number in the set than there actually is.”The reason we always add an extra one is because we want to include the number we are counting. For instance, if I have read pages 1 – 3 in a book, I would want to make sure I count page 1 as one of the pages I read.

With this in mind, now try the following problem:

Set A consists of the digits 50 – 100. How many digits are there in set A?

(A) 49

(B) 50

(C) 51

(D) 99

(E) 100


The simple formula for counting the number of elements in a consecutive sequence, like the one above, is L – F + 1. 100 – 50 + 1 = 51.

Practice Time

The next three questions pertain to a 210-page novel:

How many pages are there between 45 and 111, not including either of those pages?

(A) 45

(B) 65

(C) 66

(D) 67

(E) 76

If I am at the top of page 125 of a book, then how many pages will I have to read if I want to read to the end of page 152?

(A) 25

(B) 26

(C) 27

(D) 28

(E) 31

I am printing a book, and, for each digit I print on the page, I have to pay 5 cents (pg. 25 = 10 cents, pg. 134 = 15 cents). How much will it cost to print the page numbers, starting on page 9 and continuing for another 100 pages?

(A) 9.05

(B) 10.15

(C) 10.20

(D) 10.55

(E) 10.60


1. B

2. D

3. D