in a two digit number, the sum of the digit is 8. the different between this number and the number with the digit reversed is 54, what is the number.
n m = 10 n + m
n+m = 8
m n = n + 10 m
n + 10 m - (10 n + m) = 54
-9 n + 9 m = 54
-n + m = 6
but
+n + m = 8
---------- add
0 +2m = 14
m = 7
n = 1
so
17
original:
unit digit -- x
tens digit -- 8-x
the number: 10(8-x) + x
the number reversed: 10x + (8-x)
10x + 8-x - (10(8-x) + x) = 54
10x + 8 -x - 80 + 10x - x = 54
18x = 126
x = 7
unit digit = 7
tens digit = 1
original number 17
number reversed 71
or
original number is 71
number reversed is 17
in both cases the difference between them is 54
To solve this problem, let's break it down step by step.
Let's assume the two-digit number consists of a tens digit (T) and a units digit (U).
According to the first condition, the sum of the digits is 8. Therefore, we can write the equation:
T + U = 8
According to the second condition, the difference between the number and the number with the digits reversed is 54. Let's break it down further:
The number can be written as 10T + U (since the tens digit is in the tens place and the units digit is in the units place).
The number with the digits reversed can be written as 10U + T.
The difference between the two is:
(10T + U) - (10U + T) = 54
Now, let's simplify the equation:
10T + U - 10U - T = 54
9T - 9U = 54
Dividing both sides by 9:
T - U = 6
Now we have a system of two equations:
Equation 1: T + U = 8
Equation 2: T - U = 6
To solve this system, we can add the two equations together:
(T + U) + (T - U) = 8 + 6
2T = 14
Dividing both sides by 2:
T = 7
Now we can substitute the value of T in either of the two equations to find U:
7 + U = 8
U = 8 - 7
U = 1
So the tens digit (T) is 7 and the units digit (U) is 1. Therefore, the two-digit number is 71.