The sum of the digits of a two-digit counting number is 5. When the digits are reserved, the new number is 27 greater than the original number. What are the two numbers?

Let's define the two-digit number as \( 10a + b \), where \( a \) is the tens digit and \( b \) is the units digit. 

We are given two conditions:

1. The sum of the digits is 5:
   \[
   a + b = 5
   \]

2. When the digits are reversed, the new number is 27 greater than the original number:
   \[
   10b + a = 10a + b + 27
   \]

### Step 1: Simplify the second equation
Start by simplifying the second equation:
\[
10b + a = 10a + b + 27
\]
Subtract \( b \) and \( a \) from both sides:
\[
9b - 9a = 27
\]
Now, divide through by 9:
\[
b - a = 3
\]

### Step 2: Solve the system of equations
Now we have a system of two equations:
1. \( a + b = 5 \)
2. \( b - a = 3 \)

Add the two equations together to eliminate \( a \):
\[
(a + b) + (b - a) = 5 + 3
\]
\[
2b = 8
\]
\[
b = 4
\]

### Step 3: Find \( a \)
Substitute \( b = 4 \) into the first equation:
\[
a + 4 = 5
\]
\[
a = 1
\]

### Final Answer:
The original number is \( 10a + b = 10(1) + 4 = 14 \), and the reversed number is \( 10b + a = 10(4) + 1 = 41 \).

Thus, the two numbers are \( 14 \) and \( 41 \).

Would you like further details on any step?

Here are five related questions to explore:
1. What happens if the sum of the digits is 7 instead of 5?
2. How does the equation change if the new number is 35 greater than the original?
3. What if the digits differ by 4 instead of 3?
4. Can you generalize the method for any two-digit number where the digits are reversed?
5. What if the number is three digits instead of two?

**Tip:** When working with digit-related problems, translating the digits into algebraic terms (like \( 10a + b \)) can make the relationships between numbers easier to work with.

The sum of the digits of a two-digit counting number is 5. When the digits are reversed, the new number is 27 greater than the original number. What are the two numbers?

Learn how to solve the problem where the sum of the digits of a two-digit number is 5, and the reversed number is 27 greater than the original. This solution covers algebraic techniques involving systems of linear equations and digit manipulation. Ideal for students in Grades 6-8.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation