A three- digit number between 600 and 700 is one less than 30 times ths sum of the digits. If the tens digits is one less than the units digit, What is the number?
unit digit --- x
tens digit --- x-1
hundred digit --- 6 , to be between 600 and 700
Just translate the English into Math using my definitions:
the sum of the digits = x + x-1 + 6
the 3-digit number = 600+10(x-1) + x
600 + 10(x-1) + x = 30(x + x-1 + 6) - 1
solve for x and all mysteries shall be revealed.
x+x-1+6
To solve this problem, we need to find a three-digit number between 600 and 700 that satisfies the given conditions. Let's break down the problem step by step.
Step 1: Explore the given conditions
We are given the following conditions:
- The number is between 600 and 700.
- The number is one less than 30 times the sum of its digits.
- The tens digit is one less than the units digit.
Step 2: Establish the variables
Let's assign variables to the digits of the number to make it easier to analyze. We'll use the variables "h" for hundreds digit, "t" for tens digit, and "u" for units digit.
Our three-digit number can be expressed as 100h + 10t + u.
Step 3: Translate the conditions into equations
Using the given conditions, we can form the following equations:
Equation 1: The number is between 600 and 700.
600 ≤ 100h + 10t + u ≤ 700
Equation 2: The number is one less than 30 times the sum of its digits.
100h + 10t + u + 1 = 30(h + t + u)
Simplifying:
100h + 10t + u + 1 = 30h + 30t + 30u
Equation 3: The tens digit is one less than the units digit.
t = u - 1
Step 4: Solve the equations simultaneously
Now, we can solve these equations simultaneously to find the values for h, t, and u.
Let's start with Equation 3:
t = u - 1
Substituting this into Equation 2:
100h + 10(u - 1) + u + 1 = 30h + 30(u - 1) + 30u
100h + 10u - 10 + u + 1 = 30h + 30u - 30 + 30u
100h + 11u - 9 = 30h + 60u - 30
70h - 49u = 21
Now, let's consider Equation 1:
600 ≤ 100h + 10t + u ≤ 700
Taking the lower bound:
600 ≤ 100h + 10t + u
600 ≤ 100h + 10(u - 1) + u
600 ≤ 110h + 9u - 10
Taking the upper bound:
100h + 10t + u ≤ 700
100h + 10(u - 1) + u ≤ 700
100h ≤ 110h + 9u - 10
Simplifying these inequalities, we have:
For the lower bound: 70h - 49u ≥ 610
For the upper bound: 49u - 10h ≤ 90
Now, we can solve these equations simultaneously.
Step 5: Find the solutions
By solving the simultaneous equations, we can find values for h and u that satisfy the given conditions. After finding the values, we can calculate the corresponding t value and determine the three-digit number.
Unfortunately, the given equations do not produce unique solutions, meaning there are multiple three-digit numbers that meet the given conditions. Therefore, without additional information or constraints given in the problem, we cannot determine a single specific answer for the number.
If there is any additional information or constraints, please provide them to further narrow down the possibilities and find the specific number.