a two digit is such that the sum of the ones and tens digit is ten. If the digits are reversed the new number formed exceeds the original by 54. Find the number

The digits are a and b.

The original number:

10 a + b

The reversed number:

a + 10 b

If the digits are reversed the new number formed exceeds the original by 54 means:

a + 10 b - ( 10 a + b ) = 54

a + 10 b - 10 a - b = 54

- 9 a + 9 b = 54

Now you must solve system:

a + 10 b = 10

- 9 a + 9 b = 54
____________

The solutions are:

a = 2 , b = 8

The original number:

10 a + b = 10 ∙ 2 + 8 = 20 + 8 = 28

The reversed number:

a + 10 b = 2 + 10 ∙ 8 = 2 + 80 = 82

82 - 28 = 54

28

Let's say the original two-digit number is represented by AB, where A is the tens digit, and B is the ones digit.

According to the problem, the sum of the digits is 10, so we have the equation:
A + B = 10 (equation 1)

When the digits are reversed, the new number formed exceeds the original by 54. This can be written as:
10B + A = AB + 54

Since we know that A + B = 10, we can substitute 10 - A for B in the equation above:
10(10 - A) + A = A(10 - A) + 54

Simplifying this equation:
100 - 10A + A = 10A - A^2 + 54
100 - 9A = 10A - A^2 + 54

Rearranging this equation:
A^2 - 19A + 54 - 46 = 0
A^2 - 19A + 8 = 0

To solve this quadratic equation, we can factor it:
(A - 1)(A - 8) = 0

So, either A - 1 = 0, or A - 8 = 0

If A - 1 = 0, then A = 1, and substituting this back into equation 1 gives:
1 + B = 10
B = 10 - 1
B = 9

Therefore, the number AB = 19.

If A - 8 = 0, then A = 8, and substituting this back into equation 1 gives:
8 + B = 10
B = 10 - 8
B = 2

Therefore, the number AB = 82.

So, the two possible numbers that satisfy the given conditions are 19 and 82.

To solve this problem, we need to set up and solve a system of equations.

Let's let the tens digit be represented by "T" and the ones digit be represented by "O."

From the given information, we know that the sum of the ones and tens digit is ten:

T + O = 10 Equation 1

We also know that if the digits are reversed, the new number formed exceeds the original by 54:

(10 * O + T) - (10 * T + O) = 54

By simplifying this equation, we get:

10 * O + T - 10 * T - O = 54
9 * O - 9 * T = 54
O - T = 6 Equation 2

We now have a system of two equations (Equations 1 and 2) with two variables (T and O). We can solve this system of equations using substitution or elimination.

Let's use substitution method. Solve Equation 1 for O in terms of T:

O = 10 - T

Substituting this value of O into Equation 2, we get:

(10 - T) - T = 6
10 - 2T = 6
-2T = 6 - 10
-2T = -4
T = -4 / -2
T = 2

Now that we have the value of T (2), we can substitute it back into Equation 1 to find O:

2 + O = 10
O = 10 - 2
O = 8

Therefore, the tens digit (T) is 2 and the ones digit (O) is 8. So, the number is 28.