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Maths Question 18 – AP-EAMCET 2026

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The number of times the digit 5 occurs in all the numbers from 1 to 1000 is

To count the occurrences of a specific digit across a range of numbers, it's often helpful to consider its presence in each place value (units, tens, hundreds, etc.) independently.

Step 1: Understand the Counting Method for Digits✦ Active

To count the total occurrences of a specific digit (say, '5') in a range of numbers, we analyze each place value (units, tens, hundreds) separately. This method correctly accounts for numbers that contain the digit multiple times (e.g., 551 has two '5's). We consider the range from 1 to 999 and then check 1000 separately.

💡 Teacher's Secret Hint

Remember to count each instance of the digit, not just the number containing the digit. For example, 55 counts as two '5's.

Step 2: Count Occurrences in the Units Place○ Expand

The digit '5' appears in the units place for numbers like 5, 15, 25, ..., 95, 105, ..., 995. In every block of 100 numbers (e.g., 1-100, 101-200), '5' appears 10 times in the units place. Since there are 10 such blocks from 1 to 1000 (i.e., numbers ending in '5' in 1-100, 101-200, ..., 901-1000), the total occurrences in the units place is:

10 times/hundred×10 hundreds=100 times
Step 3: Count Occurrences in the Tens Place○ Expand

The digit '5' appears in the tens place for numbers like 50-59, 150-159, 250-259, ..., 950-959. In every block of 100 numbers, '5' appears 10 times in the tens place (e.g., 50, 51, ..., 59). Again, there are 10 such blocks from 1 to 1000. The total occurrences in the tens place is:

10 times/hundred×10 hundreds=100 times
Step 4: Count Occurrences in the Hundreds Place○ Expand

The digit '5' appears in the hundreds place for numbers like 500-599. This range consists of 100 numbers (500, 501, ..., 599), where '5' is the hundreds digit. The digit '5' does not appear in the hundreds place for any other block of 100 numbers within 1 to 1000. The number 1000 also does not contain the digit '5'.

1 hundred-block×100 times/block=100 times
Step 5: Calculate the Total Occurrences○ Expand

Sum the occurrences from each place value:

Total occurrences=(Units Place)+(Tens Place)+(Hundreds Place)
Total occurrences=100+100+100=300 times
💡 Teacher's Secret Hint

This systematic approach avoids missing any occurrences, especially in numbers like 550 or 505.

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