Question

How many numbers are there from 67 through to 99?

Short answer

There are 33 numbers from 67 to 99.

Step-by-step solution

Identify the range of numbers

The problem asks how many numbers exist from 67 to 99. First, note that the starting number is 67 and the ending number is 99.

Write down the formula for counting numbers

To find out how many numbers there are within a range, use the formula: \[ \text{Number of elements} = (\text{Last number} - \text{First number}) + 1 \]

Substitute the values into the formula

Here, the first number is 67 and the last number is 99. Substitute these values into the formula: \[ (\text{99} - \text{67}) + 1 \]

Perform the calculation

Now, do the subtraction first: \[ 99 - 67 = 32 \] Then add 1: \[ 32 + 1 = 33 \]

State the answer

The total number of integers from 67 to 99 is 33.

Key concepts

counting numbers formula

When solving problems that ask how many numbers exist within a specific range, you're dealing with counting numbers. The counting numbers formula is a straightforward way to get the answer. This formula states: \[ \text{Number of elements} = (\text{Last number} - \text{First number}) + 1 \] It helps us find the total count of integers in a given range. For example, in our problem, we calculated the number of integers from 67 to 99.
Consider the steps:

• First number (67)
• Last number (99)

By substituting these values into our formula, we found that the total number of integers is 33.
Understanding this formula is crucial as it is often used in GMAT math problems related to counting and sequencing.

range calculation

Another critical concept in this problem is range calculation. The range of numbers between two values can be easily found by using basic arithmetic. Here, our range starts at 67 and ends at 99. To find out how many numbers lie in this range, follow these steps:

• Subtract the starting number from the ending number: \[ 99 - 67 \]
• Add 1 to your result: \[ (99 - 67) + 1 \]

This method works because you're effectively including both endpoints of the range in your count.
This concept helps simplify many GMAT math problems, making it a vital tool in your math toolkit.

basic arithmetic

The solution to this exercise is rooted in basic arithmetic, which involves operations like addition, subtraction, multiplication, and division. In this problem, we mainly used subtraction and addition:

• Subtraction: Removing the starting number (67) from the ending number (99)
• Addition: Adding 1 to include both ends of the range

So from our steps:
\[ 99 - 67 = 32 \]
And then adding 1:
\[ 32 + 1 = 33 \]
Mastering these fundamental math skills is essential as they recur in various problems on the GMAT.
Strengthening these basics will make tackling more advanced problems much easier.