Counting the Occurrences of the Digit 1 in the Tens Place from 100 to 1000: An SEO Optimized Guide

In this article, we will explore how to systematically count the number of times the digit 1 appears in the tens place within the range of numbers from 100 to 1000. This guide is intended for SEO professionals, mathematicians, and anyone interested in number analysis. Let's break down the process step by step and discuss the methods used to arrive at the answer.

Identifying the Range

The problem at hand involves identifying the frequency of the digit 1 in the tens place for all numbers from 100 to 999. It's important to note that 1000 is excluded from our analysis as it is beyond this defined range.

Breaking Down the Range

We need to represent each number in the form of xyz, where x is the hundreds digit, y is the tens digit, and z is the units digit. Our goal is to focus on the tens digit (y) and determine how many times it is 1 while satisfying the range constraint.

Determining the Hundreds Digit

The hundreds digit x can take any value from 1 to 9, as the range starts at 100. This means x has 9 possible values.

Counting the Occurrences of 1 in the Tens Place

When the tens digit y is 1, we are looking at numbers in the form x1z. For each valid value of x (which can be 1 to 9), there are 10 possible values for the units digit z (0 to 9).

Calculation

To find the total number of occurrences, we multiply the number of choices for x by the number of choices for z:

Choices for x 9

Choices for z 10

Total occurrences 9 * 10 90

This calculation confirms that the digit 1 appears in the tens place 90 times within the range of 100 to 999.

Verification Using Code

To further verify our solution, let's consider the following C code snippet:

#include stdio.h
int main() {
    int s[10];
    int i, t  0;
    for (int i  100; i  1000; i  ) {
        sprintf(s, %d, i);
        if (s[2]  '1') 
            t  ;
    }
    printf(%d, t);
    return 0;
}

This code snippet iterates through all numbers from 100 to 999, converting each number to a string and checking if the tens digit is 1. If so, a counter is incremented.

A More Detailed Analysis

Let's break down the numbers in a more granular manner. From 110 to 119, the digit 1 appears once. From 210 to 219, the digit 1 appears nine times. This pattern continues for each set of 100 numbers:

110 - 119: 1 appearance 210 - 219: 9 appearances 310 - 319: 9 appearances 410 - 419: 9 appearances 510 - 519: 9 appearances 610 - 619: 9 appearances 710 - 719: 9 appearances 810 - 819: 9 appearances 910 - 919: 9 appearances

Summing these appearances, we get:

Total appearances 1 9 * 9 82

Adding the appearance from 110 to 119 and 210 to 219 (which is 1 9 10), we get:

Total appearances 82 10 92

However, this pattern is incorrect as it includes an extra 2 due to miscounting. Therefore, the correct total is 90.

Conclusion

By breaking down the range into manageable parts and systematically counting the occurrences, we can accurately determine that the digit 1 appears 90 times in the tens place within the range of 100 to 999. This method provides a clear and concise way to solve similar problems involving digit occurrences in number analysis.

For further reading and similar problems, consider exploring number theory and digit analysis techniques. If you have any questions or need further assistance, feel free to reach out! Continue exploring and learning, as every problem is a step towards mastering mathematics and algorithmic thinking.