there is two digit figure in which one digit is exactly half the other. If the position of both fonts or digits is interchanged, the resultant figure when added with the origional figure sum to 99. what is the origional figure?
Let the first digit be x and the second be y.
The number is xy(note that they're not multiplied together)
xy+yx = 99
let y=2x
Trial and error from here
36+63=99
:)
Let's represent the original two-digit number as 10x + y, where x and y are the digits in the tens and units place, respectively.
According to the problem, one digit is exactly half the other. So we have two possibilities:
1. x = 2y
2. y = 2x
Let's start with the first possibility, x = 2y.
We know that the position of both digits is interchanged in the second number. So the second number can be represented as 10y + x.
The sum of the original number (10x + y) and the second number (10y + x) is given as 99.
So, we have the equation: (10x + y) + (10y + x) = 99
Simplifying the equation, we get: 11x + 11y = 99
Dividing both sides of the equation by 11, we have: x + y = 9
Since x and y are digits, the possible values for (x, y) could be (1, 8), (2, 7), (3, 6), (4, 5), or (5, 4).
Checking the conditions, in (1, 8), x = 2y is not satisfied.
In the remaining sets, only (4, 5) satisfies the condition x = 2y.
Therefore, the original figure is 45.
To solve this problem, let's break it down step by step:
Step 1: Let's assume the original two-digit number is represented as 10x + y, where x is the tens digit, and y is the units digit. Since it is mentioned that one digit is exactly half the other, we can represent this condition as x = (1/2)y.
Step 2: If we interchange the position of the digits, the resulting number would be 10y + x.
Step 3: According to the problem, if we add the original number with the number obtained by interchanging the digits, the sum will be 99. So, we can set up the equation 10x + y + 10y + x = 99.
Step 4: Simplify the equation by combining like terms: 11x + 11y = 99.
Step 5: Divide both sides of the equation by 11 to solve for x + y: x + y = 9.
Step 6: Since we know from Step 1 that x = (1/2)y, we can substitute the value of x into the equation from Step 5: (1/2)y + y = 9.
Step 7: Simplify the equation by combining like terms: (3/2)y = 9.
Step 8: Solve for y by multiplying both sides of the equation by (2/3): y = (9 * 2) / 3 = 6.
Step 9: Substitute the value of y into the expression x = (1/2)y to solve for x: x = (1/2) * 6 = 3.
Therefore, the original number is 10x + y = 10(3) + 6 = 30 + 6 = 36.