The number of times the digit 5 will be written when listing the integers from 1 to 1000, is (a) 271 (b) 272 (c) 300 (d) 200

Options

A. 271
B. 272
C. 300
D. 200

Answer

To find the number of times the digit 5 appears when listing the integers from 1 to 1000, we can break it down into three parts: the units place, the tens place, and the hundreds place.

### Step-by-Step Solution:

1. **Count occurrences of 5 in the units place (1 to 999)**:
- The numbers that have 5 in the units place are: 5, 15, 25, ..., 995.
- This forms an arithmetic sequence where the first term \(a = 5\) and the common difference \(d = 10\).
- To find the number of terms \(n\), we can use the formula for the nth term of an arithmetic sequence:
\[ a_n = a + (n-1)d \]
Setting \(a_n = 995\):
\[ 995 = 5 + (n-1) \cdot 10 \\ 990 = (n-1) \cdot 10 \\ n - 1 = 99 \\ n = 100 \]
- Therefore, there are **100 occurrences of 5 in the units place**.

2. **Count occurrences of 5 in the tens place (1 to 999)**:
- The numbers that have 5 in the tens place are: 50-59, 150-159, 250-259, ..., 950-959.
- Each complete set from 50 to 59 contributes 10 occurrences of 5.
- There are 10 such sets (50-59, 150-159, ..., 950-959).
- Therefore, the total occurrences in the tens place is:
\[ 10 \text{ (from each set)} \times 10 \text{ (sets)} = 100 \]
- Thus, there are **100 occurrences of 5 in the tens place**.

3. **Count occurrences of 5 in the hundreds place (1 to 999)**:
- The only numbers that have 5 in the hundreds place are from 500 to 599.
- This gives us a total of 100 numbers (500, 501, ..., 599).
- Therefore, there are **100 occurrences of 5 in the hundreds place**.

4. **Total occurrences of 5 from 1 to 999**:
- Adding all occurrences together:
\[ 100 \text{ (units)} + 100 \text{ (tens)} + 100 \text{ (hundreds)} = 300 \]

5. **Consider the number 1000**:
- The number 1000 does not contain the digit 5, so it does not contribute to our count.

### Final Answer:
Thus, the total number of times the digit 5 is written when listing the integers from 1 to 1000 is **300**.