A three- digit number is eleven times the two-digit formed by using the hundreds and the units digit of the three - digit number respectively, in the tens and units place of the two - digit number. If the difference between the digit in tens place and the digit in hundreds place is 1 then what is the digit in units place?

How can I frame the solution?? Can you explain me

If the digits are a,b,c then the value of the original 3-digit number is

100a+10b+c

So, now we know that

100a+10b+c = 11(10a+c)
b-a = 1
we want to find c

100a + 10(a+1) + c = 110a+11c
100a+10a+10+c = 110a+11c
10+c = 11c
c = 1

Thank you.

To solve this problem, let's break it down into steps:

Step 1: Let's assume the three-digit number is represented as ABC, where A represents the hundreds digit, B represents the tens digit, and C represents the units digit.

Step 2: According to the problem, the two-digit number formed using A and C is BC. This means the two-digit number is already given as BC. In other words, A is the same as B, and C is the same as C.

Step 3: Based on the information given, the three-digit number (ABC) is eleven times the two-digit number (BC). Mathematically, this can be represented as:

ABC = 11 * BC

Step 4: We know that A is the same as B, so we can rewrite ABC as:

ABB = 11 * BC

Step 5: Since A and B are the same, we can simplify the equation further:

BBB = 11 * BC

Step 6: We also know that the difference between the digit in the tens place (B) and the digit in the hundreds place (B) is 1. Mathematically, this can be represented as:

B - B = 1

Therefore, B - B simplifies to 0, so the equation becomes:

0 = 1

Since the equation is not true, there seems to be an error in the problem statement or in the given information. The problem cannot be solved as it stands because it is not possible for the equation to be true.

To solve this problem, let's break it down step by step:

Step 1: Understand the information given:
We are given a three-digit number that is eleven times the two-digit number formed by using the hundreds and the units digit of the three-digit number in the tens and units place of the two-digit number.

Step 2: Assign variables:
Let's assign variables to each digit of the three-digit number.
- Let the hundreds digit be represented by 'h'
- Let the tens digit be represented by 't'
- Let the units digit be represented by 'u'

Step 3: Formulate equations:
Based on the information given, we can form two equations.

Equation 1: The three-digit number is eleven times the two-digit number.
The three-digit number can be expressed as 100h + 10t + u.
The two-digit number can be expressed as 10t + u.
So, we can write the equation as:
100h + 10t + u = 11(10t + u)

Equation 2: The difference between the digit in the tens place and the digit in the hundreds place is 1.
t - h = 1

Step 4: Simplify and solve the equations:
Let's substitute the value of t - h from Equation 2 into Equation 1 to solve for h.

Substituting t - h = 1 into Equation 1, we get:
100h + 10(t - 1) + u = 110t + 11u

Simplifying this equation, we get:
100h + 10t - 10 + u = 110t + 11u
100h - 99u = 100t + 10

Step 5: Solve for h:
Rearrange the equation to isolate h:
100h = 100t + 99u - 10

Divide both sides of the equation by 100:
h = (100t + 99u - 10) / 100

Step 6: Find the value of h, t, and u:
Now, we need to find values for h, t, and u that satisfy the given conditions of the problem.

Since we are looking for a three-digit number, h cannot be zero. So, we can try values of t and u that satisfy the equation:

For example, let's choose t = 1 and u = 2.
Substituting these values into the equation, we get:
h = (100(1) + 99(2) - 10) / 100
h = (100 + 198 - 10) / 100
h = 288 / 100
h = 2.88

Since h needs to be an integer, we can round 2.88 to the nearest whole number, which is 3.

So, if t = 1, u = 2, and h = 3, we have a valid solution.

Step 7: Find the digit in the units place:
Based on the solution obtained, the digit in the units place is represented by 'u' and is equal to 2. Therefore, the digit in the units place is 2.

In conclusion, the digit in the units place is 2.