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Chapter 1: Problem 10

How many numbers are there from 100 to 200 ? (a) 100 (b) 101 (c) 99 (d) none of these

Short Answer

Expert verified
Answer: (b) 101

Step by step solution

01

Identifying the range

First, let's identify the range of numbers we are working with. In this case, the range is from 100 to 200.
02

Subtract the lower limit

Next, we will subtract the lower limit (100) from the upper limit (200). In this case: 200 - 100 = 100
03

Add 1

Now, we need to add 1 to the difference to include both the lower and upper limits. 100 +1 = 101
04

Choose the correct answer

After following the steps, we have found there are 101 numbers from 100 to 200. Therefore, the correct answer is (b) 101.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Range Identification
When counting the number of elements between two numbers, the first step is to clearly identify the range. Let's break it down simply: when someone says "from 100 to 200," they are specifying a continuous block of numbers that starts at 100 and ends at 200. The range includes every whole number between these two, including the numbers themselves. This forms the framework for us to accurately calculate the total number of numbers.
  • Start of Range: This is the smallest number in your sequence鈥攊n our case, 100.
  • End of Range: This is the largest number in your sequence鈥攈ere it's 200.
Understanding the start and end helps you narrow down exactly which numbers need to be counted. Remember, knowing the exact bounds of your range is essential before doing any calculations.
Number Calculation
Once you know your range, the next step is figuring out how many numbers actually fall within this range. This is done by calculating the difference between the lower and upper bounds of the range.
For example, subtract the start of the range (100) from the end of the range (200).
Here's the simple math operation:
  • Upper Limit - Lower Limit = Difference
  • 200 - 100 = 100
This difference gives us the count of whole steps between the two bounds. However, this result does not yet include the starting and ending numbers themselves. It's slightly like counting the steps on a ladder without counting the bottom and top step.
This calculation sets the stage for our final step: inclusive counting.
Inclusive Counting
To truly count all numbers within a given range, including the starting and end numbers, we use inclusive counting. This simply means adding 1 to the difference we've calculated. Why do we add one?
It's straightforward: while the difference accounts for the steps between numbers, adding one ensures both end numbers (in our case, 100 and 200) are also counted.
Let's revisit our previous calculations:
  • Difference: 200 - 100 = 100
  • Inclusive Count: 100 + 1 = 101
Now, we have accounted for every single number from 100 to 200, inclusive. Without that extra one, you'd miss out on one of the numbers, which could lead to an incorrect answer. This simple addition helps to accurately reflect the full range of numbers.

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