A number consists of two digits whose sum is 9.if 9 is subtracted from the number , the digits interchange their places.find the number please
let the unit digit be x and the tens digit by y
We are told x+y = 9 **
So the original number is 10y+x
we are told: 10y+x - 9 = 10x + y
9x - 9y = -9
x-y = -1 ***
add ** and ***
2x = 8
x = 4 , then in ** y = 5
so the number is
10y+x
= 10(5) + 4 = 54
check: take 9 from 54 to get 45, which is the original number reversed.
We could have found the answer by using common sense:
we were told the sum of digits was 9
only possible cases :
90 --- take 9 away: 80 <----- not the number reversed
81 --- take 9 away: 72 <----- not the number reversed
72 --- take 9 away: 63 <----- not the number reversed
63 --- take 9 away: 80 <----- not the number reversed
54 --- take 9 away: 45 <----- Ahhhh, the number reversed
To find the number, we need to set up a system of equations based on the given conditions.
Let's assume that the digit in the tens place is 'x' and the digit in the ones place is 'y'.
According to the first condition, the sum of the digits is 9. So, we can write the equation as:
x + y = 9 --- Equation 1
According to the second condition, if 9 is subtracted from the number, the digits interchange their places. In other words, the tens digit becomes the ones digit, and the ones digit becomes the tens digit. So, the new number can be expressed as 10y + x. If we subtract 9 from this, it should be equal to the original number. Hence, we can write the equation as:
10y + x - 9 = 10x + y --- Equation 2
Now, we can solve this system of equations to find the values of 'x' and 'y'.
Let's rearrange Equation 2 to make 'x' and 'y' terms on one side:
10y - y - x + 9 = 10x - x
Simplifying:
9y - x + 9 = 9x
Rearranging again:
9y - 9x = x + 9
Now, we can substitute the value of 'x' from Equation 1 into the above equation:
9y - 9(9 - y) = (9 - y) + 9
Simplifying:
9y - 81 + 9y = 18 - y
18y - 81 = 18 - y
Adding 'y' to both sides:
18y - y - 81 = 18
Simplifying:
17y - 81 = 18
Adding 81 to both sides:
17y = 99
Dividing by 17:
y = 99 / 17
y = 5.82
Since 'y' represents a digit, it must be a whole number. However, 5.82 is not a whole number, so we made an error somewhere in our calculations. Let's analyze the problem again:
We assumed that the digit in the tens place is 'x' and the digit in the ones place is 'y'. Since the sum of the digits is 9, 'x' must be less than 9, as 'y' cannot be negative. Also, if we subtract 9 from a two-digit number, the result will always be a two-digit number. Therefore, the number we are looking for must be greater than or equal to 19.
Let's try different values for 'x' and see if we can find a valid solution:
If 'x' is 1, then 'y' would be 9 - 1 = 8. So the number would be 18. If we subtract 9 from 18, we get 9, which interchanges the digits.
Therefore, the number we were looking for is 18.
Please note that it is always important to double-check and analyze the problem again if the initial solution does not make sense. Sometimes small errors in calculations or assumptions can lead to incorrect answers.