One of the digits of a two-digit number is four times the other digit. If we interchange the digits, the sum of the new number formed and the original number is 55. Find the original number

Let's denote the original two-digit number as \( 10x + y \), where \( x \) is the tens digit and \( y \) is the units digit.

### Step 1: Formulating the First Equation
According to the problem, one of the digits is four times the other. We have two cases:
1. \( x = 4y \)
2. \( y = 4x \)

Let's handle both cases.

### Step 2: Formulating the Second Equation
When the digits are interchanged, the new number becomes \( 10y + x \). The sum of the new number and the original number is given as 55. Therefore, the equation is:
\[
(10x + y) + (10y + x) = 55
\]
Simplifying this equation:
\[
11x + 11y = 55
\]
Dividing both sides by 11:
\[
x + y = 5
\]

### Step 3: Solving for Both Cases

#### Case 1: \( x = 4y \)
Substitute \( x = 4y \) into the equation \( x + y = 5 \):
\[
4y + y = 5
\]
\[
5y = 5
\]
\[
y = 1
\]
Now, substitute \( y = 1 \) back into \( x = 4y \):
\[
x = 4 \times 1 = 4
\]
Thus, the original number is \( 10x + y = 10 \times 4 + 1 = 41 \).

#### Case 2: \( y = 4x \)
Substitute \( y = 4x \) into the equation \( x + y = 5 \):
\[
x + 4x = 5
\]
\[
5x = 5
\]
\[
x = 1
\]
Now, substitute \( x = 1 \) back into \( y = 4x \):
\[
y = 4 \times 1 = 4
\]
Thus, the original number is \( 10x + y = 10 \times 1 + 4 = 14 \).

### Step 4: Conclusion
There are two possible original numbers: **41** and **14**. Both satisfy the conditions of the problem.

Would you like further clarification on any step?

Here are 5 related questions:
1. How would the problem change if the sum of the numbers was 66 instead of 55?
2. Can we apply the same approach to a three-digit number with similar conditions?
3. What if the digits differ by 3 instead of one being a multiple of the other?
4. How can this method be adapted to non-decimal (e.g., binary or hexadecimal) systems?
5. What other conditions could be added to further limit the possible original numbers?

**Tip**: When faced with digit-related problems, try setting up equations based on place values (e.g., tens and units digits) for clarity.

One of the digits of a two-digit number is four times the other digit. If we interchange the digits, the sum of the new number formed and the original number is 55. Find the original number.

Solve a problem where one digit of a two-digit number is four times the other. Interchanging the digits and summing the numbers gives 55. Step-by-step solution using algebra and linear equations. Suitable for Grades 6-8.

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