Problem 102 - Algebra for JEE ADVANCED

Question

The total number of times, the digit ' 3 ' will be written, when the integers having less than 4 digits are listed is equal to a. 300 b. 310 c. 302 d. 306

Step-by-Step Solution

Answer

The digit '3' appears 300 times.

Step 1: Understanding the Problem

We need to find out how many times the digit '3' appears when listing all integers from 0 to 999, as these are all integers with less than four digits.

Step 2: Counting in the Units Place

The digit '3' can appear in the units place in the numbers 3, 13, 23, ..., up to 993. This sequence is arithmetic with a common difference of 10. We can find the total count using the formula for arithmetic sequences: \[ n = \frac{\text{last term} - \text{first term}}{\text{common difference}} + 1 \]; here, \[ n = \frac{993 - 3}{10} + 1 = 100 \]. So, '3' appears 100 times in the units place.

Step 3: Counting in the Tens Place

Similarly, for tens place, '3' can appear in numbers like 30, 31, 32, ..., 39, then 130, 131, ..., 139, and so on till 930-939. These are 10 numbers for each complete set of hundreds, and there are 10 such sets (from 30-39 to 930-939), thus: \( 10 \times 10 = 100 \) appearances for tens place.

Step 4: Counting in the Hundreds Place

For the hundreds place, '3' appears in numbers like 300 to 399. There are 100 such numbers (since 399 - 300 + 1 = 100), hence '3' appears 100 times in the hundreds place.

Step 5: Adding Up All Countings

Add the counted occurrences for all place values to get the total: \( 100 + 100 + 100 = 300 \).

Step 6: Conclusion

The total appearance of '3' in the integers with less than 4 digits is 300 times.

Key Concepts

Arithmetic Sequences

An arithmetic sequence is a number sequence where each term after the first is generated by adding a constant called the "common difference" to the previous term. In our problem of counting the digit '3', arithmetic sequences help us systematically track where '3' appears across series of numbers.
For example, when counting appearances of '3' in the units place, we start with numbers like 3, 13, 23, and so on. Each number is 10 more than the previous one, making it an arithmetic sequence. The common difference here is 10.
To find out how many numbers are in this sequence, we use the formula for the count of terms in an arithmetic sequence: \[ n = \frac{\text{last term} - \text{first term}}{\text{common difference}} + 1 \], where "n" is the total number of terms, allowing us to determine that '3' appears 100 times in the units place.

Units Place Counting

The units place in a number determines the digit that comes last. For example, in the number 473, '3' is in the units place. When counting how many times '3' appears in the units place from numbers 0 to 999, we need to find every number ending in 3.
These numbers form an arithmetic sequence: 3, 13, 23, 33,..., 993. Each number is achieved by adding 10 to the previous, such as 3 → 13 → 23, and so on.
The number of times '3' appears in the units place is calculated by finding how many terms are in this sequence using the arithmetic sequence formula. In this case, there are 100 numbers where 3 appears as the last digit from 0 to 999.

Tens Place Counting

In tens place counting, we look at the digit just before the last one. This is the "tens" position. For numbers like 33, 43, 133, and so on, the '3' is in the tens place.
The counting involves sequences like 30 to 39, 130 to 139, continuing to 930 to 939. Each range from something like 30 to 39 contains 10 numbers, each sequence of which places '3' in the tens place 10 times. Because this pattern repeats every hundred numbers (e.g., 100s, 200s, ... 900s), we have 10 such sets.
By multiplying the number of appearances per set by the number of sets, i.e., 10 appearances × 10 sets, we end up with the '3' appearing 100 times in the tens place.

Hundreds Place Counting

The hundreds place is the third digit from the right in a number. For instance, in 345, the '3' is in the hundreds place.
To count the occurrences of '3' in the hundreds place, we identify numbers that range from 300 to 399. In this case, every number from 300, 301, ..., to 399 has '3' in the hundreds position.
This range from 300 to 399 has exactly 100 numbers, because 399 - 300 + 1 equals 100. Therefore, '3' appears 100 times in the hundreds place across all valid numbers under 1000.