a three digit number has ten digit two greater than the units digit and the hundreds digit one greater than the tens digit. the sum of the tens and the hundreds is three times the units digit. what is the number
xyz is the number.
y=2+z
x=y+1
x+y=3z
putting them in matrix form
0,1,-1,2
1,-1,0,1
1,1,-3,0
subtract 2 from 3
0,2,-3,-1
multiply 1 by 2 and subtract from above
0,2,-2,4
0,0,-1,-5
or z=5
then from 1), y=7 and you can solve for x.
Let's break down the given information step by step:
1. The ten's digit is one less than the hundreds digit. Let's call the ten's digit 'x', so the hundreds digit would be 'x + 1'.
2. The unit's digit is two less than the ten's digit. So, the unit's digit would be 'x - 2'.
3. The sum of the tens and the hundreds is three times the units digit. We can represent this as: 'x + (x + 1) = 3(x - 2)'.
Now, let's solve this equation step by step:
2x + 1 = 3x - 6 (distributing 3 to x - 2)
x = 7 (subtracting 2x and adding 6 to both sides)
So, the tens digit (x) is 7. Therefore, the hundreds digit would be 7 + 1 = 8, and the unit's digit would be 7 - 2 = 5.
Therefore, the number is 785.
To solve this problem, we can break it down into smaller steps:
Step 1: Understand the problem.
- We are given a three-digit number.
- The tens digit is one less than the hundreds digit.
- The units digit is two less than the tens digit.
- The sum of the tens and the hundreds is three times the units digit.
Step 2: Assign variables.
- Let's assume the hundreds digit is represented by the variable "h".
- The tens digit can be represented by the variable "t".
- The units digit can be represented by the variable "u".
Step 3: Translate the given information into equations.
- The tens digit is one less than the hundreds digit: t = h - 1.
- The units digit is two less than the tens digit: u = t - 2.
- The sum of the tens and the hundreds is three times the units digit: t + h = 3u.
Step 4: Substitute variables and solve the equations.
- Substitute t in terms of h from the first equation: h - 1 + h = 3u.
- Simplify the equation: 2h - 1 = 3u.
- Substitute u in terms of t from the second equation: 2h - 1 = 3(t - 2).
- Solve this equation: 2h - 1 = 3t - 6.
Step 5: Solve for the variables.
- Simplify the equation further: 2h - 1 = 3t - 6 → 2h - 3t = -5.
- To find a solution, we can assign values to h or t and see if we can find consistent values for all three variables.
Let's start with h = 4:
- Substitute h = 4 in the equation 2h - 3t = -5.
- We get 2(4) - 3t = -5 → 8 - 3t = -5.
- Then, solve for t: -3t = -5 - 8 → -3t = -13 → t = 13/3 (not an integer).
Since t is not an integer, we can't have h = 4. Let's try h = 5:
- Substitute h = 5 in the equation 2h - 3t = -5.
- We get 2(5) - 3t = -5 → 10 - 3t = -5.
- Solve for t: -3t = -5 - 10 → -3t = -15 → t = 15/3 = 5.
Now that we have t = 5, we can substitute it back into the equation t = h - 1 to find the value of h:
- 5 = h - 1 → h = 5 + 1 → h = 6.
Finally, we can substitute the values of t and h into the equation u = t - 2:
- u = 5 - 2 → u = 3.
So, the three-digit number is 653.