How Many Times Does the Digit 3 Appear in 3-Digit Numbers from 1 to 500?

When analyzing the frequency of a specific digit in a sequence of numbers, it's essential to break down the problem systematically. In this article, we will determine how many times the digit 3 appears in 3-digit numbers from 1 to 500. We will explore different methods to achieve this, including arithmetic progressions and direct counting techniques.

Understanding the Range of Three-Digit Numbers

Three-digit numbers, by definition, range from 100 to 999. However, our problem is confined to the numbers from 1 to 500. This means we need to focus on the range where 3-digit numbers start and stop within this range, i.e., from 100 to 499.

Direct Counting Method

To directly count the 3-digit numbers from 100 to 499:

Step 1: Identify the Range

Step 2: Use the Formula for Counting Integers in a Range

The formula to calculate the count of integers in a range is:

Count End Number - Start Number 1

Substituting the values:

Count 499 - 100 1 400

This shows that there are 400 three-digit numbers from 100 to 499.

Arithmetic Progression Method

An alternative method to count the occurrence of the digit 3 is to use the arithmetic progression approach. This method is particularly useful for identifying numbers in a specific sequence.

Using Arithmetic Progression

Consider the sequence where each number is formed by adding 10 at each step starting from a number that contains 3 in the unit's place. This sequence can be represented as:

3, 13, 23, 33, 43, …, 493

The general term to find the last number in the sequence is given by:

an a1 (n - 1)d

Here, a1 3, d 10, and an 493

Solving for n (the number of terms), we get:

493 3 (n - 1)10

493 - 3 10(n - 1)

490 10(n - 1)

49 n - 1

n 50

This indicates that there are 50 terms in this sequence, meaning the digit 3 appears 50 times in these numbers.

Frequency of the Digit 3 from 1 to 100

To further understand the frequency of digit 3 from 1 to 100, let's break it down step by step:

Step 1: Count the Occurrence within Each Set of Ten Numbers

In each set of ten numbers (e.g., 1-10, 11-20, etc.), there is exactly one occurrence of the digit 3 (e.g., 3, 13, 23, etc.).

There are 10 such sets from 1 to 100.

Step 2: Consider the Special Set of Numbers with Only Digits 3

The set 30-39 contains 10 occurrences of the digit 3 (30, 31, 32, 33, 34, 35, 36, 37, 38, 39).

Step 3: Sum Up the Counts

Total count of digit 3 from 1 to 100:

10 (from sets of ten) 10 (from 30-39) 20

Thus, the digit 3 appears 20 times in the numbers from 1 to 100.

General Formula for Counting Digits 3

To find the frequency of the digit 3 for any range from 1 to 10^n, the formula is:

n × 10^(n-1)

For 1 to 100 (where n2):

2 × 10^(2-1) 20

This confirms our previous method and shows that the digit 3 appears 20 times in the sequence from 1 to 100.

Conclusion

In summary, we have explored multiple methods to count the digit 3 in three-digit numbers from 1 to 500. We have used direct counting, arithmetic progression, and a general formula to calculate the frequency. The results indicate a total of 400 three-digit numbers in the given range, with the digit 3 appearing 20 times from 1 to 100 and 50 times in the arithmetic sequence from 100 to 499.