a two digit number is six times the sum of its digits. If the digit in the units place is increased by 2 and the digit in the tens place is decreased by 2 then the number soformed is 4 times the sum of its digits. Find the orignal number.
Answers
Step-by-step explanation:
Given:-
A two digit number is six times the sum of its digits. If the digit in the units place is increased by 2 and the digit in the tens place is decreased by 2 then the number soformed is 4 times the sum of its digits.
To find:-
Find the orignal number ?
Solution:-
Let the digit at 10's place be X
The place value of X = 10×X=10X
Let the digit at 1's place be Y
The place value of Y = Y×1 = Y
The number = 10X+Y
If the digits reversed then the new number obtained by reversing the digits = 10Y+X
Given that
A two digit number is six times the sum of its digits.
=> 10X+Y = 6(X+Y)
=> 10X+Y = 6X+6Y
=> 10X-6X -6Y+Y = 0
=> 4X - 5Y = 0 ----------(1)
The digit in the units place is increased by 2 and the digit in the tens place is decreased by 2 then the number soformed is 4 times the sum of its digits.
10(X-2)+(Y+2) = 4(X+Y)
=>10X-20+Y+2 = 4X+4Y
=> 10X+Y-18 = 4X+4Y
=> 10X+Y-18-4X-4Y = 0
=> 6X-3Y -18 = 0
=> 3(2X-Y-6) = 0
=> 2X-Y-6 = 0
=> Y = 2X-6--------(2)
On Substituting the value of Y in (1) then
=> 4X -5(2X-6) = 0
=> 4X-10X+30 = 0
=> -6X +30 = 0
=> 6X = 30
=>X = 30/6
=> X = 5
On Substituting the value of X in (2) then
=> Y = 2(5)-6
=> Y = 10-6
=> Y = 4
Therefore, X = 5 and Y = 4
Answer:-
The required original number = 54
Check:-
The number = 54
Sum of the digits = 5+4 = 9
the number = 54=6×9
=> 6 times the sum of the digits
Digit in units place is increased by 2 = 4+2 = 6
Digit in tens place is decreased by 2 = 5-2 = 3
The number formed = 36
The sum of the digits = 3+6=9
The new number = 36=4×9
=> 4 Times the sum of the digits
Verified all the relations in the given problem.