A number consists of three digits with the digit 4 in the middle. The sum of the digits is 8. If the first and third digits are interchanged, the original number is decreased the 198. Find the number.
Answers
let ones place number be y and hundred place be x.
x+4+y = 8
x+y = 4..............................(i)
again,
original number + 198 = interchanged number
100x +10*4 +y + 198 = 100y + 10*4 +x
[for example 264 it may be written as 100*2+10*6+4]
99y - 99x = 198
99(y-x) = 198
y-x = 2
multiplying both sides by -1
x-y = -2..............................(ii)
adding equation (i) & (ii), we get :
2x=2
x=1 {put this value in equation (i)}
1+y=4
y=3
The number is:
100x + 10*40 + y
= 100*1 +10*4 + 3
=100 + 40 +3
=143
Original number is 143
Lets Consider the number as
X4Y ------- [1]
When interchange,
Y4X ------- [2]
The difference between [2] - [1]
Y4X - X4Y = 198 ------- [3]
Lets split the number in Hundreds, tenths and ones
X4Y = 100X + 40 + Y ------- [4]
Y4X = 100Y + 40 + X -------- [5]
Apply value [4] and [5] in [3]
100Y + 40 + X - (100X + 40 + Y) = 198
100Y - 100X + X - Y = 198
99Y - 99X = 198
Taking out 99 as common,
99(X - Y) = 198
Y - X = 198/99
Y - X = 2 ------- [6]
Given that X + 4 + Y = 8
X + Y = 8 - 4
X + Y = 4 ------- [7]
Add [6] and [7]
2Y = 6
Y = 3
Applying Y=3 in [7]
X + 3 = 4
X = 4 - 3
X = 1
The original digit is
143
After Interchange
341
Total sum of digits
1 + 4 + 3 = 8
Difference is
341 - 143 = 198
Hope this helps