The number xy is such that three times the sum of it's digits is less than the value of the number by 8. When the digits are reversed the value of the number increases by 9.find the number
3(x+y) = 10x+y - 8
10y+x = 10x+y + 9
so now just solve as usual.
3(x+y) + 8 = 10 x + y
10 y + x = 9 + 10 x + y
3 x + 3 y = 10 x + y - 8
9 y - 9 x = 9
- 7 x + 2 y = -8
-9 x + 9 y = 9
-63 x + 18 y = -72
-18 x + 18 y = 18
--------------------------- subtract
-45 x +0 = -90
x = 2
-7 (2) + 2 y = -8
-14 + 2y = -8
2 y = 6
y = 3
so at last
23
Let's break down the problem step by step to find the value of the number.
Step 1: Understanding the problem
We are given a two-digit number, xy, where x is the tens digit and y is the units digit. We are also given two conditions:
1. Three times the sum of its digits is less than the value of the number by 8. This can be expressed as 3(x + y) = 10x + y - 8.
2. When the digits are reversed, the value of the number increases by 9. This can be expressed as 10y + x = 10x + y + 9.
Step 2: Solving the equations
Let's solve these two equations to find the values of x and y.
Equation 1: 3(x + y) = 10x + y - 8
Expanding this equation, we get:
3x + 3y = 10x + y - 8
Bringing similar terms together, we have:
3x - 10x + 3y - 1y = -8
Simplifying further:
-7x + 2y = -8 --> equation A
Equation 2: 10y + x = 10x + y + 9
Expanding this equation, we get:
10y - y = 10x - x + 9
Simplifying, we have:
9y = 9x + 9
Dividing both sides by 9, we get:
y = x + 1 --> equation B
Step 3: Substituting the value of y in equation A
Now let's substitute the value of y (from equation B) into equation A to solve for x.
-7x + 2(x + 1) = -8
-7x + 2x + 2 = -8
Combining like terms, we have:
-5x + 2 = -8
Subtracting 2 from both sides, we get:
-5x = -10
Dividing both sides by -5, we have:
x = 2
Step 4: Finding the value of y
Now that we have found x, we can substitute its value into equation B to find y.
y = x + 1
y = 2 + 1
y = 3
Therefore, the value of the number xy is 23.