Quantity A |
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ | Quantity B |

$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ | ||

The number of integers between $100$ and $500$ that are multiples of $11$ | $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ | $36$ |

- Quantity A is greater.
- Quantity B is greater.
- The two quantities are equal.
- The relationship cannot be determined from the information given.

So, you were trying to be a good test taker and practice for the GRE with PowerPrep online. Buuuut then you had some questions about the quant section—specifically question 2 of the second Quantitative section of Practice Test 1. Those questions testing our knowledge of **Integers** can be kind of tricky, but never fear, PrepScholar has got your back!

## Survey the Question

Let’s search the problem for clues as to what it will be testing, as this will help shift our minds to think about what type of math knowledge we’ll use to solve this question. Pay attention to any words that sound math-specific and anything special about what the numbers look like, and mark them on your paper.

The question asks if numbers are **multiples** of other numbers, which is an indication that it likely tests our **Integers** math skill. Let’s keep what we’ve learned about this skill at the tip of our minds as we approach this question.

## What Do We Know?

Let’s carefully read through the question and make a list of the things that we know.

- We want to compare the value $36$ to the number of integers between $100$ and $500$ that are multiples by $11$

## Develop a Plan

We know that numbers that are multiples of $11$ are divisible by $11$. Hmmm…this question seems more tedious than difficult. Of course, given unlimited time, we could just plug in every number between $100$ and $500$ into our calculator to see if it is divisible by $11$. That’ll take too long though, so let’s try to find a more clever way to solve this question.

Might as well start by dividing $100$ and $500$ by $11$ to see what we get, since we want to know how many numbers between them are divisible by $11$:

$100/11$ | $=$ | $9.09$ |

$ $ | $ $ | |

$500/11$ | $=$ | $45.45$ |

So here we get $9.09$ and $45.45$. Hmm…we do know that if an integer is divisible by $11$, then our calculator should give us an answer without any numbers after the decimal place. We want numbers divisible by $11$ between $100$ and $500$. Given what we just found, it makes since that **any integer between $100$ and $500$ that is divisible by $11$ will give us an integer between $9.09$ and $45.45$ when we divide it by $11$**. Ha! That’s it! Alright, let’s count how many integers there are between $9.09$ and $45.45$, and that’ll be our answer.

## Solve the Question

So between $9.09$ and $45.45$, the integers will go from $10$ to $45$. To count the number of integers between two integers, we can just subtract them and add $1$:

$$45-10+1 = 36$$

Of course, we need to remember to add this extra $1$ back in. If we had forgotten to do this, we would have missed this question. Let’s say on test day we can’t remember whether or not we’re supposed to add $1$ to the difference between these two integers. *The next best thing to knowing something is knowing how to figure it out*, so we could just quickly do a test to see. The goal should be to choose two integers where we can calculate the answer quickly, then use that to see if we need to add $1$ back in. For example, let’s choose $1$ and $3$. We know that between $1$ and $3$, including both $1$ and $3$, we have three integers. We also know that $3-1=2$, so if we just subtract them without adding $1$ back in we’ll get the wrong answer. We can then extrapolate this thought experiment to $10$ and $45$, showing us that we need to add $1$ to their difference.

We’re humans and we sometimes forget things, so we should always feel free to devise a test if we’re not sure about something. The test should be quick and easy though, as we are being timed on these questions.

So Quantity B $(36)$ is the same as Quantity A. **The correct answer is C, the two quantities are equal**.

## What Did We Learn

**Whenever we want to count the number of integers between two integers (inclusive of both integers), we can find their difference and then add $1$ **.

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