How many times does the number 5 occur in the range of numbers from 1 to 100? A. 21 B. 22 C. 20 D. 19
C
Counting the Digit 5 in Numbers from 1 to 100
Let's find out how many times the digit '5' appears in the numbers starting from 1 up to 100. To do this, we can look at the positions where the digit '5' can occur: in the units place and in the tens place.
Counting '5' in the Units Place
First, consider the numbers from 1 to 100 where the units digit is 5. These are:
- 5
- 15
- 25
- 35
- 45
- 55
- 65
- 75
- 85
- 95
There are 10 numbers that have '5' in the units place.
Counting '5' in the Tens Place
Next, let's consider the numbers from 1 to 100 where the tens digit is 5. These are the numbers in the fifties:
- 50
- 51
- 52
- 53
- 54
- 55
- 56
- 57
- 58
- 59
There are 10 numbers that have '5' in the tens place.
Calculating the Total Count
Now, we need to find the total number of times the digit '5' occurs. We counted the numbers ending in 5 and the numbers in the fifties. Notice that the number 55 was included in both lists. The number 55 has the digit '5' in both the tens place and the units place.
When we counted the units place, we added one occurrence for the '5' in 55's units place. When we counted the tens place, we added one occurrence for the '5' in 55's tens place. So, by adding the counts from the two lists, we correctly account for both '5's in the number 55.
Total count = (Count of '5' in units place) + (Count of '5' in tens place)
Total count = $10 + 10 = 20$
The digit '5' appears 20 times in the numbers from 1 to 100.
Summary of Occurrences
| Position | Numbers | Count |
|---|---|---|
| Units Place | 5, 15, 25, 35, 45, 55, 65, 75, 85, 95 | 10 |
| Tens Place | 50, 51, 52, 53, 54, 55, 56, 57, 58, 59 | 10 |
| Total Occurrences | 20 |
Therefore, the digit 5 occurs 20 times in the range of numbers from 1 to 100.
Revision Table - Counting Digits
| Concept | Explanation | Example (for digit 5 in 1-100) |
|---|---|---|
| Units Place Count | Numbers ending with the target digit. | 5, 15, ..., 95 (10 numbers) |
| Tens Place Count | Numbers where the tens digit is the target digit. | 50, 51, ..., 59 (10 numbers) |
| Double Counting | Numbers with the target digit in both units and tens place (like 55). These are counted once in each category. | 55 is counted when checking units place and when checking tens place, correctly adding 2 to the total. |
| Total Count | Sum of counts from each position. | 10 (units) + 10 (tens) = 20 |
Additional Information - Counting Digits in Ranges
This method of counting digits by position (units, tens, hundreds, etc.) is useful for any digit and any range of numbers. For larger ranges, you would extend this approach.
- Digit 0: Counting 0 is slightly different because numbers don't start with 0 (except for 0 itself, which is often excluded in ranges like 1-100). In 1-100, 0 appears in 10, 20, ..., 90 (9 times) and 100 (twice). Total 11 times.
- Digits 1-9: For any digit from 1 to 9, the method of counting occurrences in the units, tens, hundreds places, etc., and summing them up works directly. For the range 1-100, each digit from 1 to 9 appears 20 times (10 in units place like 5, 15,...; 10 in tens place like 50, 51,...).
- Generalization: For a range like 1 to 10^n, each digit from 1 to 9 appears $n \times 10^{n-1}$ times. Digit 0 appears slightly fewer times depending on how the range is defined (e.g., includes 0 or not).
Important Questions from Integers
Find the value of \(\sqrt{2025}\) .
A prime number
A. is not a positive integer.
B. has no divisor at all.
C. has only 1 and itself as divisors.
D. has more than two divisors.
__________ are twin prime number.
A. (4, 9)
B. (2, 3)
C. (4, 6)
D. (3, 5)A factory produced 18,58,509 cassettes in the month of January, 7623 more cassettes in the of February and owing to short supply of electricity produced 25,838 less cassettes in March than in February. Find the total production in all?
A. 55,57,312
B. 59,83,245
C. 55,64,935
D. 56,08,988Divide 3740 in three parts in such a way that half of the first part, one-third of the second part and one-sixth of the third part are equal.
A. 700, 1000, 2040
B. 340, 1360, 2040
C. 680, 1020, 2040
D. 500, 1200, 2040
