Problem 73 - Algebra and Trigonometry

Question

Number Problem The sum of the digits of a two-digit number is \(7 .\) When the digits are reversed, the number is increased by \(27 .\) Find the number.

Step-by-Step Solution

Answer

The number is 25.

Step 1: Define Variables for Digits

Let's define \( x \) as the tens digit and \( y \) as the units digit of the two-digit number. The number can be expressed as \( 10x + y \).

Step 2: Equation for Sum of Digits

According to the problem, the sum of the digits is 7. Hence, we have the equation: \( x + y = 7 \).

Step 3: Equation for Reversed Digits

When the digits are reversed, the new number becomes \( 10y + x \). It's given that reversing the digits increases the original number by 27, leading to the equation: \( 10y + x = 10x + y + 27 \).

Step 4: Simplify the Reversal Equation

Simplify the reversal equation: \[ 10y + x = 10x + y + 27 \] \[ 10y - y = 10x - x + 27 \] \[ 9y = 9x + 27 \] \[ y = x + 3 \].

Step 5: Solve the System of Equations

We now have two equations: \( x + y = 7 \) and \( y = x + 3 \). Substitute \( y = x + 3 \) in the first equation: \[ x + (x + 3) = 7 \] \[ 2x + 3 = 7 \] \[ 2x = 4 \] \[ x = 2 \]. Then, substitute \( x = 2 \) in \( y = x + 3 \): \[ y = 2 + 3 \] \[ y = 5 \].

Step 6: Verify Solution

The original number is \( 10x + y = 10(2) + 5 = 25 \). Reversing the digits gives \( 10y + x = 10(5) + 2 = 52 \). The difference is \( 52 - 25 = 27 \), which matches the problem statement.

Key Concepts

Two-Digit Numbers

Understanding the structure of numbers is crucial, especially when dealing with two-digit numbers. A two-digit number always comprises two significant parts: the tens digit and the units digit.
For instance, in the number 45, the digit '4' represents the tens, while '5' represents the units.
The value of the number is determined by this formula: \(10 \times \text{tens digit} + \text{units digit}\). Hence, for 45, it's \(10 \times 4 + 5 = 45\). This formula is essential as it helps in breaking down and understanding the number's composition more deeply.
By knowing and using this formula, one can disassemble any two-digit number, making it easier to solve problems involving operations on these digits.

Digit Reversal

Digit reversal is an interesting and common concept in algebra word problems.
Reversing the digits of a two-digit number simply means switching the positions of the tens and units digit.
For example, if we have 34, reversing the digits will give us 43.
This process can be expressed differently. The original number can be written as \(10x + y\), where \(x\) is the tens digit and \(y\) is the units digit. After reversal, the number turns into \(10y + x\).
Understanding this transformation is key in solving problems where you need to find relationships between original and reversed numbers, like determining differences, sums, or increases in value.

System of Equations

In algebra, a system of equations is a set of two or more equations with the same variables.
To solve these, you need to find the value of the variables that satisfy all the equations simultaneously.
In our problem, we have two equations derived from the conditions given:

• The sum of the digits: \(x + y = 7\)
• Digit reversal relationship: \(y = x + 3\)

Solving this system requires substitution or elimination methods. We initially substitute the expression for \(y\) from the second equation into the first, leading to a simplified path to find \(x\). Once \(x\) is found, it can easily be used to find \(y\). Using such techniques helps algebra students approach complex problems systematically.

Sum of Digits

The sum of digits is a concept where you add the tens digit and units digit of a number together.
It's often used as a condition in word problems to establish relationships between digits of numbers.
Consider the number 53. Its tens digit is 5 and the units digit is 3, thus the sum of its digits is \(5 + 3 = 8\).
In the given problem, the sum of the digits equals 7. This condition is represented by the equation \(x + y = 7\).
Understanding and setting up such equations is fundamental, as it provides necessary constraints that aid in solving the problem and finding the actual numbers involved. It's a powerful tool in translating verbal problem statements into mathematical expressions.