To find the number of integers between two numbers (excluding the endpoints), subtract the smaller number from the larger number, and then subtract 1.
Here's a breakdown with examples:
Understanding the Problem
The question asks for the count of whole numbers that lie strictly between two given numbers. This means we are not including the two given numbers themselves in our count.
Method and Explanation
- Identify the Larger and Smaller Numbers: Determine which of the two given numbers is larger and which is smaller.
- Subtract: Subtract the smaller number from the larger number.
- Subtract 1: Subtract 1 from the result obtained in step 2. This final result is the number of integers between the two original numbers.
Formula:
Number of integers between a and b (where b > a) = b - a - 1
Examples:
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Example 1: How many integers are between 5 and 10?
- Larger number: 10
- Smaller number: 5
- Calculation: 10 - 5 - 1 = 4
- Integers between 5 and 10: 6, 7, 8, 9 (a total of 4 integers)
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Example 2: How many integers are between -3 and 2?
- Larger number: 2
- Smaller number: -3
- Calculation: 2 - (-3) - 1 = 2 + 3 - 1 = 4
- Integers between -3 and 2: -2, -1, 0, 1 (a total of 4 integers)
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Example 3: How many integers are between 7.2 and 12.9?
- First, determine the integers just outside this range. The next integer above 7.2 is 8, and the integer just below 12.9 is 12.
- Now, find the number of integers between 8 and 12:
- Larger number: 12
- Smaller number: 8
- Calculation: 12 - 8 - 1 = 3
- Integers between 7.2 and 12.9: 8, 9, 10, 11, 12. The integers strictly between 7.2 and 12.9 are 8, 9, 10, 11 and 12, which yields 4 integers. The last integer is excluded since the wording means strictly between. The correction is that integers have to be computed by first finding the greatest integer smaller than upper bound and least integer larger than the lower bound. Then apply the formula of subtracting the lower bound from the upper bound and subtract 1.
- Calculation: 12 - 8 - 1 = 3
- Integers strictly between 7.2 and 12.9: 8, 9, 10, 11, 12.
Important Considerations:
- The word "between" is crucial. If the question asks for integers from one number to another (inclusive), then you would subtract and add 1 instead. For example, the number of integers from 5 to 10 (inclusive) is 10 - 5 + 1 = 6 (5, 6, 7, 8, 9, 10).
- Non-integer endpoints: If the numbers are not integers (as in Example 3), you need to consider the integers that fall strictly between the two given non-integer values. First find the nearest integers above the lower bound and below the higher bound. Then apply the formula.
In Summary:
To find the number of integers between two numbers, subtract the smaller number from the larger number and then subtract 1. Remember to adjust your approach if the problem requires including the endpoints or if the endpoints are not integers.
