If you write down all the numbers from 1 to 100, how many times will the digit 4 be written in total (counting all positions in each number)?

Introduction / Context:
This counting question focuses on digit frequency in a range of numbers. You must count how many times the digit 4 appears when writing all numbers from 1 to 100. This checks attention to detail and understanding of place value, as the digit can appear in the tens place or the units place.

Given Data / Assumptions:
We consider all integers from 1 to 100 inclusive.We must count every occurrence of the digit 4 in every position.This includes occurrences in both the tens place and the units place.

Concept / Approach:
The most efficient strategy is to count occurrences separately in the tens place and the units place. For each place, we identify how many numbers in the range have a 4 in that specific position. Because some numbers, such as 44, contain the digit twice, we must ensure these are counted correctly in each place.

Step-by-Step Solution:
Step 1: Count the numbers between 1 and 100 that have 4 in the tens place.Step 2: The tens digit is 4 for numbers from 40 to 49 inclusive. There are 10 such numbers: 40, 41, 42, 43, 44, 45, 46, 47, 48 and 49.Step 3: Therefore, there are 10 occurrences of the digit 4 in the tens place.Step 4: Next, count numbers with 4 in the units place.Step 5: These are 4, 14, 24, 34, 44, 54, 64, 74, 84 and 94. Again, there are 10 such numbers.Step 6: Each of these contributes one occurrence of 4 in the units place, so there are 10 occurrences in the units place.Step 7: The number 44 has a 4 in both tens and units places, and we have correctly counted it twice, once in each place.Step 8: Total occurrences of the digit 4 = 10 (tens) + 10 (units) = 20.

Verification / Alternative check:
We can list all numbers containing the digit 4 and check the total occurrences. In the tens place they are 40 to 49 (10 occurrences). In the units place they are all numbers ending in 4, a set of 10 numbers. Counting each 4 explicitly yields the same total of 20, confirming our calculation.

Why Other Options Are Wrong:
The value 11 would result from incorrectly counting only the numbers that contain a 4 at least once and not counting multiple occurrences. The values 14 and 17 are intermediate and might arise from partial counting of tens or units contributions. None of these reflect the full breakdown into tens and units positions.

Common Pitfalls:
Many learners forget that 44 has two occurrences of the digit 4 and treat it as a single occurrence. Others may count either only the tens place or only the units place, but not both, leading to undercounting. Breaking the problem into clear place value counts helps avoid these errors.

Final Answer:
The digit 4 is written a total of 20 times when listing all numbers from 1 to 100.