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Let's solve the problem step by step.

### Step 1: Define the number
Let the three-digit number be represented as \( 100h + 10t + u \), where:
- \( h \) = hundreds digit,
- \( t \) = tens digit,
- \( u \) = units digit.

### Step 2: Translate the given conditions
1. **The sum of the digits is 19**:
   \[
   h + t + u = 19
   \]

2. **When the tens and units digits are interchanged, the number is diminished by 27**:
   The new number becomes \( 100h + 10u + t \). The equation is:
   \[
   100h + 10t + u - (100h + 10u + t) = 27
   \]
   Simplify:
   \[
   9t - 9u = 27 \quad \Rightarrow \quad t - u = 3
   \]

3. **When the hundreds and tens digits are interchanged, the number is increased by 180**:
   The new number becomes \( 100t + 10h + u \). The equation is:
   \[
   100t + 10h + u - (100h + 10t + u) = 180
   \]
   Simplify:
   \[
   90t - 90h = 180 \quad \Rightarrow \quad t - h = 2
   \]

### Step 3: Solve the system of equations
We now have three equations:
1. \( h + t + u = 19 \)
2. \( t - u = 3 \)
3. \( t - h = 2 \)

#### From equation (3): \( t = h + 2 \)
Substitute \( t = h + 2 \) into equation (2):
\[
h + 2 - u = 3 \quad \Rightarrow \quad u = h - 1
\]

#### Substitute \( t = h + 2 \) and \( u = h - 1 \) into equation (1):
\[
h + (h + 2) + (h - 1) = 19
\]
Simplify:
\[
3h + 1 = 19 \quad \Rightarrow \quad 3h = 18 \quad \Rightarrow \quad h = 6
\]

#### Find \( t \) and \( u \):
\[
t = h + 2 = 6 + 2 = 8
\]
\[
u = h - 1 = 6 - 1 = 5
\]

### Step 4: Verify the solution
The number is \( 100h + 10t + u = 100(6) + 10(8) + 5 = 685 \).

- Sum of digits: \( 6 + 8 + 5 = 19 \) (satisfied).
- Interchanging \( t \) and \( u \): \( 685 - 658 = 27 \) (satisfied).
- Interchanging \( h \) and \( t \): \( 865 - 685 = 180 \) (satisfied).

### Final Answer:
The number is **685**.

Do you have any questions or want further clarification? Here are five related problems to consider:
1. What if the sum of the digits was 21 instead of 19?
2. How would the solution change if the increase when interchanging \( h \) and \( t \) was 190?
3. Can this problem be solved using matrices for the system of equations?
4. What happens if the digits \( t \) and \( u \) are equal?
5. How can such problems be represented and solved programmatically?

**Tip:** Always break multi-condition problems into manageable equations and solve systematically.

The sum of the digits of a three-place number is 19. If the tens and units digits are interchanged the number is diminished by 27, and if the hundreds and tens digits are interchanged the number is increased by 180. What is the number?

Learn how to solve a problem involving a three-digit number with specific conditions on the sum and interchange of digits. This step-by-step solution uses algebraic equations and logical reasoning, suitable for students in Grades 8-10.

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