A two digit number is three less than seven times the sum of its digits. if the digits are reversed, the new number is 18 less than the original number. what is the new number?
let the tens digit of our number be x
let the unit digit be y
so the number is 10x + y
sum of its digit = x+y
10x + y = 7(x+y) - 3
3x - 6y = -3 , #1
the number reversed is 10y+x
10y+x = 10x+y - 18
9y - 9x = -18
y - x = -2
y = x-2
sub this into #1
3x - 6(x-2) = -3
-3x = -3-12
x = 5
y = 5-2 = 3
the original number was 53
the new number is 35
check:
is 35 less than 53 by 18 ? YEAH
7 times the sum of the digits is 56
is that less than 53 by 3 ? YEAHHHH
To solve this problem, let's start by breaking down the information given:
1. A two-digit number is three less than seven times the sum of its digits.
2. When the digits are reversed, the new number is 18 less than the original number.
Let's assume the tens digit is represented by 'x' and the units digit is represented by 'y'. Therefore, the original number would be 10x + y, and the reversed number would be 10y + x.
Now let's break down the given information into equations.
1. A two-digit number is three less than seven times the sum of its digits:
10x + y = 7(x + y) - 3
Simplifying this equation, we get:
10x + y = 7x + 7y - 3
Rearranging the terms, we get:
3x - 6y = 3
2. When the digits are reversed, the new number is 18 less than the original number:
10y + x = (10x + y) - 18
Simplifying this equation, we get:
10y + x = 10x + y - 18
Rearranging the terms, we get:
9x - 9y = 18
Now we have a system of equations:
3x - 6y = 3
9x - 9y = 18
We can solve this system of equations using any suitable method, such as substitution or elimination.
Let's use the elimination method. By multiplying the first equation by 3 and the second equation by 1, we can eliminate the 'y' term:
9x - 18y = 9
9x - 9y = 18
Now we subtract the second equation from the first:
9x - 18y - (9x - 9y) = 9 - 18
-9y + 9y = -9
The 'y' terms cancel out, leaving us with:
0 = -9
Since this equation leads to a contradiction, it means that there is no solution to the system of equations. Therefore, there is no specific new number that satisfies both conditions given in the problem.