Consider two-digit numbers which remain the same when the digits interchange their positions. What is the average of such two-digit numbers?
55
Finding Two-Digit Numbers with Interchangeable Digits
The question asks us to find the average of two-digit numbers that remain the same when their digits are interchanged. Let's first identify these special two-digit numbers.
A two-digit number can be represented as \(10 \times a + b\), where \(a\) is the tens digit and \(b\) is the units digit. For a two-digit number, \(a\) must be between 1 and 9 (inclusive), and \(b\) must be between 0 and 9 (inclusive).
When the digits are interchanged, the new number becomes \(10 \times b + a\).
The condition given is that the original number and the number with interchanged digits are the same:
\(\qquad 10a + b = 10b + a\)
Let's solve this equation to find the relationship between \(a\) and \(b\):
Subtract \(a\) from both sides:
\(\qquad 10a - a + b = 10b + a - a\)
\(\qquad 9a + b = 10b\)
Subtract \(b\) from both sides:
\(\qquad 9a + b - b = 10b - b\)
\(\qquad 9a = 9b\)
Divide both sides by 9:
\(\qquad \frac{9a}{9} = \frac{9b}{9}\)
\(\qquad a = b\)
This equation \(a = b\) tells us that the tens digit must be equal to the units digit for the two-digit number to remain the same when its digits are interchanged.
Since \(a\) is the tens digit of a two-digit number, \(a\) cannot be 0. The possible values for \(a\) (and thus \(b\)) are 1, 2, 3, 4, 5, 6, 7, 8, and 9.
The two-digit numbers where the digits are the same are:
- When \(a=1, b=1\): The number is 11.
- When \(a=2, b=2\): The number is 22.
- When \(a=3, b=3\): The number is 33.
- When \(a=4, b=4\): The number is 44.
- When \(a=5, b=5\): The number is 55.
- When \(a=6, b=6\): The number is 66.
- When \(a=7, b=7\): The number is 77.
- When \(a=8, b=8\): The number is 88.
- When \(a=9, b=9\): The number is 99.
There are 9 such two-digit numbers.
Calculating the Average of These Numbers
To find the average of these two-digit numbers, we need to sum them up and divide by the count of the numbers.
The list of numbers is: 11, 22, 33, 44, 55, 66, 77, 88, 99.
The count of these numbers is 9.
The sum of these numbers is:
Sum \(= 11 + 22 + 33 + 44 + 55 + 66 + 77 + 88 + 99\)
We can factor out 11 from each term:
Sum \(= 11 \times (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9)\)
The sum of the first 9 natural numbers is given by the formula \(\frac{n(n+1)}{2}\), where \(n=9\).
Sum of \((1 + 2 + \dots + 9) = \frac{9 \times (9+1)}{2} = \frac{9 \times 10}{2} = \frac{90}{2} = 45\).
So, the sum of the two-digit numbers is:
Sum \(= 11 \times 45 = 495\)
Now, we calculate the average:
Average \(= \frac{\text{Sum of numbers}}{\text{Count of numbers}}\)
Average \(= \frac{495}{9}\)
Performing the division:
\(\qquad 495 \div 9 = 55\)
The average of the two-digit numbers which remain the same when the digits interchange their positions is 55.
Revision Table: Two-Digit Numbers and Average
| Concept | Description | Application in this problem |
|---|---|---|
| Two-digit number | A number with a tens digit (1-9) and a units digit (0-9). Represented as \(10a+b\). | Identifying numbers where \(a,b\) are digits and \(a \ne 0\). |
| Interchanging Digits | Swapping the tens and units digits. Number becomes \(10b+a\). | Setting \(10a+b = 10b+a\) to find the condition. |
| Condition for equality | \(a=b\) for the number to remain same. | Finding all two-digit numbers with identical digits. |
| Average | Sum of values divided by the count of values. | Calculating \(\frac{\text{Sum}(11, 22, ..., 99)}{9}\). |
Additional Information: Properties of Numbers
This problem touches upon basic number properties and arithmetic means (averages).
- Numbers like 11, 22, 33 are sometimes called repdigits or repeating digits numbers. These are numbers formed by repeating a single digit.
- Any two-digit number with identical digits is a multiple of 11. For example, \(11 = 11 \times 1\), \(22 = 11 \times 2\), ..., \(99 = 11 \times 9\).
- The sum of an arithmetic progression can be calculated efficiently. In this case, the list 11, 22, ..., 99 is an arithmetic progression with the first term \(a_1 = 11\), the last term \(a_n = 99\), and the number of terms \(n = 9\). The sum \(S_n = \frac{n}{2}(a_1 + a_n) = \frac{9}{2}(11 + 99) = \frac{9}{2}(110) = 9 \times 55 = 495\). This confirms the sum calculated earlier.
- The average of an arithmetic progression is simply the average of the first and last term: Average \(= \frac{a_1 + a_n}{2}\). In this case, Average \(= \frac{11 + 99}{2} = \frac{110}{2} = 55\). This provides a quicker way to find the average once the list of numbers is identified as an arithmetic progression.
Similar Questions
If the average of 9 consecutive positive integers is 55, then what is the largest integer?
The arithmetic mean of 11 observations is 11. The arithmetic mean of the first 6 observations is 10.5 and the arithmetic mean of the last 6 observations is 11.5. What is the sixth observation?
A cricketer has certain average of 10 innings. In the 11 th inning, he scored 108 runs, thereby increasing his average by 6 runs. What is his new average?
In a class of 100 students, there are 70 boys whose average marks in a subject are 75. If average marks of the complete class is 72, then what is the average marks of the girls?
The average of 50 consecutive natural numbers is x. What will be the new average when the next four numbers are also included?
Mean marks of 50 students were found to be 78.4. But later it was detected that 95 was misread as 59 and 25 was misread as 52. What is the difference between correct mean and incorrect mean?
What is the arithmetic mean of P and Q ?
The arithmetic mean of two numbers is 10 and their geometric mean is 8. What are the two numbers?
A small company pays each of its 5 category ‘C’ workers Rs. 20,000, each of its 3 category ‘B’ workers Rs. 25,000 and a category ‘A’ worker Rs. 65,000. The number of workers earning less than the mean salary is
Consider the following distribution
Class
Frequency
0 - 20
17
20 - 40
28
40 - 60
32
60 - 80
F
80 - 100
19
If the mean of the above distribution is 50 then what is the value of f?
Important Questions from Average
Average of 40 numbers is 71, if the number 100 replaced by 140, then average is increased by
The captain of a football team of 11 members is 28 years old and the goalkeeper is 4 years older than him. If the ages of these two are removed, then the average age of the remaining players is two years less than the average age of the whole team. What is the average age of the team?
There are two Classes A and B having 25 and 30 students respectively. In Class-A the highest score is 21 and lowest score is 17. In Class-B the highest score is 30 and lowest score is 22. Four students are shifted from Class-A to Class-B.
Consider the following statements:
1. The average score of Class-B will definitely decrease.
2. The average score of Class-A will definitely increase.
Which of the above statements is/are correct?
The average weight of A, B, Cis 40 kg, the average weight of B, D, Eis 42 kg and the weight of Fis equal to that of B. What is the average weight of A, B, C, D, Eand F?
A cow costs more than 4 goats but less than 5 goats. If a goat costs between Rs. 600 and Rs. 800, which of the following is a most valid conclusion?
