a two digit number is such that the sum of its digits is 11. the number is 27 greater tha the number obtained by interchanging the digits. find the number

a = first digit

b = second digit

Two digit number is formed when ten's digit which is multiplied by 10 and add to unit place digit is multiplied by 1.

In this case your number is 10 a + b

The number obtained by interchanging the digits is 10 b - a

The sum of digits is 11 mean:

a + b = 11

The number is 27 greater than the number obtained by interchanging the digits mean:

10 a + b = 10 b + a + 27

Now:

10 a + b = 10 b + a + 27

Subtract b to both sides

10 a + b - b = 10 b + a + 27 - b

10 a = 9 b + a + 27

Subtract a to both sides

10 a - a = 9 b + a + 27 - a

9 a = 9 b + 27

Divide both sides by 9

a = b + 3

Replace this in equation:

a + b = 11

b + 3 + b = 11

2 b + 3 = 11

Subtract 3 to both sides

2 b + 3 - 3 = 11 - 3

2 b = 8

Divide both sides by 2

b = 4

a = b + 3 = 4 + 3 = 7

a = 7

Your number is 74

Proof:

a + b = 7 + 4 = 11

74 = 47 + 27

Well, well, well! We've got ourselves a tricky math problem here. Let's put on our thinking clown shoes and tackle it!

Let's call the tens digit of our mystery number "x" and the units digit "y". According to the problem, we know that x + y = 11.

Now, the number is 27 greater than the number obtained by swapping the digits. If we swap the digits, we get 10y + x.

So, we can write the equation as:
10x + y = (10y + x) + 27

Let's simplify that:
9x - 9y = 27

Well, look at that! We can divide both sides of the equation by 9:
x - y = 3

Now we have a system of equations:
x + y = 11
x - y = 3

We can add these two equations together to get rid of the "y":
2x = 14

Divide both sides by 2:
x = 7

Plugging the value of x into one of the equations, we find:
7 + y = 11
y = 4

So, the two-digit number that satisfies the conditions is 74.

Hope that brought a smile to your face!

Let's denote the tens digit of the two-digit number as 'x' and the units digit as 'y'.

According to the given information, we have two equations:

1) The sum of the digits is 11:
x + y = 11

2) The number is 27 greater than the number obtained by interchanging the digits:
10x + y = 10y + x + 27

To solve these equations, we can use the substitution method.

Let's solve equation 1) for x:
x = 11 - y

Substitute this value of x into equation 2):
10(11 - y) + y = 10y + (11 - y) + 27

Simplify the equation:
110 - 10y + y = 10y + 11 - y + 27

Combine like terms:
110 - 11 = 10y + 10y + 27 + y - y

Simplify further:
99 = 21y + 27

Subtract 27 from both sides:
99 - 27 = 21y

72 = 21y

Divide both sides by 21:
y = 72 / 21
y = 3

Substitute the value of y back into equation 1):
x + 3 = 11

Subtract 3 from both sides:
x = 11 - 3
x = 8

Therefore, the two-digit number is 83.

To find the two-digit number that satisfies the given conditions, we can follow these steps:

1. Let's assume the tens digit of the unknown number is denoted by 'x' and the ones digit is denoted by 'y'. Therefore, the number can be written as 10x + y.

2. We know that the sum of the digits is 11. So, we can write the equation: x + y = 11.

3. According to the given condition, the number is 27 greater than the number obtained by interchanging the digits (which would be 10y + x). This can be written as: 10x + y = 10y + x + 27.

Now, we have a system of two equations:

Equation 1: x + y = 11
Equation 2: 10x + y = 10y + x + 27

To solve this system, we can use the method of elimination or substitution. Let's use substitution:

From Equation 1, we can express x in terms of y: x = 11 - y.

Substituting this expression for x into Equation 2:

10(11 - y) + y = 10y + (11 - y) + 27
110 - 10y + y = 10y + 11 - y + 27
110 - 9y = 11y + 38

Let's simplify the equation:

110 - 9y = 11y + 38
110 - 38 = 11y + 9y
72 = 20y
y = 72/20
y = 3.6

Since y represents the ones digit, it cannot be a decimal. Therefore, we made an error during the calculation.

Let's try another approach. Since the digits must be integers, we can guess and check different values for y.

Let's try y = 1:

x + 1 = 11 (Equation 1)
10x + 1 = x + 27 (Equation 2)

Solving Equation 1 for x:
x = 11 - 1
x = 10

Substituting these values into Equation 2:
10(10) + 1 = 10 + 27
101 = 37

The equation is not satisfied for y = 1. Let's try y = 2:

x + 2 = 11 (Equation 1)
10x + 2 = x + 27 (Equation 2)

Solving Equation 1 for x:
x = 11 - 2
x = 9

Substituting these values into Equation 2:
10(9) + 2 = 9 + 27
92 = 36

Again, the equation is not satisfied for y = 2. Let's try y = 3:

x + 3 = 11 (Equation 1)
10x + 3 = x + 27 (Equation 2)

Solving Equation 1 for x:
x = 11 - 3
x = 8

Substituting these values into Equation 2:
10(8) + 3 = 8 + 27
83 = 35

Once again, the equation is not satisfied for y = 3.

Since none of the values we tested for y produced a valid solution, it appears that there is no two-digit number that satisfies the given conditions.