Problem 57 - Algebra and Trigonometry

Question

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 original number is 25.

Step 1: Define Variables

Let the original number be represented as \(10a + b\), where \(a\) is the tens digit and \(b\) is the units digit of the number.

Step 2: Set Up Equations

According to the problem, the sum of the digits is 7, so we can write the equation \(a + b = 7\).

Step 3: Create Reversed Number Equation

When the digits are reversed, the number becomes \(10b + a\). The problem states that this reversed number is 27 more than the original number, so the equation is \(10b + a = 10a + b + 27\).

Step 4: Simplify and Solve Equations

Using \(10b + a = 10a + b + 27\), simplify to get \(9b - 9a = 27\), which simplifies to \(b - a = 3\).

Step 5: Solve System of Equations

Now, solve the system of equations: \(a + b = 7\) and \(b - a = 3\):1. From \(b - a = 3\), we have \(b = a + 3\).2. Substitute \(b = a + 3\) into \(a + b = 7\): \(a + (a + 3) = 7\)3. Simplify to get \(2a + 3 = 7\).4. Solve for \(a\): \(2a = 4\) gives \(a = 2\).5. Substitute back to find \(b\): \(b = 2 + 3 = 5\).

Step 6: Conclusion

Using \(a = 2\) and \(b = 5\), the original number is \(10a + b = 10(2) + 5 = 25\).

Key Concepts

Sum of Digits

In the given problem, our task starts with understanding the sum of digits of a two-digit number. This concept involves adding the digits that form a number. Imagine any number you know. If it's 25, you would take 2 and 5 as digits, and their sum equals 7.

For the problem, we are given that the two digits of a number add up to 7. This helps us form the equation: \(a + b = 7\), where \(a\) is the tens digit, and \(b\) is the units digit. This is a classic way to break down numbers and is the first step to finding how digits relate in different parts of math problems.

The sum of digits concept is useful in many problems where you want to find relationships between the digits without knowing the whole number. It's a simple yet powerful tool!

Reversing Digits

Reversing the digits of a number can change the number itself significantly. In this problem, the number is reversed, increasing its value. Let's consider how this changes things:

- For a number "\(10a + b\)", reversing its digits gives us "\(10b + a\)".
- The problem states that reversing the number makes it bigger by 27.

Thus, this leads to another equation: \(10b + a = 10a + b + 27\). It uses the fact that a change occurs in the number's value when you switch the position of digits.

Understanding reversing of digits is crucial when dealing with palindromes or other problems where digit order affects mathematical properties. It's simple with two-digit numbers, though larger numbers can get a bit more complex.

System of Equations

Having set up our main equations, the problem moves us into solving a system of equations. It involves finding solutions for two or more equations that are true at the same time.

For our scenario, the equations are:

• \(a + b = 7\)
• \(b - a = 3\)

These equations arise from the digits' sum and the effect of reversing the digits. To solve a system of equations, we find values of variables that satisfy all equations together.

This is a critical algebra skill, which can be tackled by methods like substitution or elimination. Here, substitution is used to find \(a = 2\) and \(b = 5\). Practicing with small numbers, like a two-digit challenge, builds the foundation to handle larger systems in advanced algebra.

Algebraic Expressions

Algebraic expressions represent numbers in terms of variables. They help us build equations and solve problems without specific numerical inputs initially.

In our exercise, the expressions are:

• \(10a + b\) for the original number
• \(10b + a\) for the reversed number

These expressions define the number based on its digits, \(a\) and \(b\).

Algebraic expressions make it easier to generalize. Instead of sticking to concrete numbers, we use expressions to work through logical relationships and achieve a solution. They function as building blocks of equations allowing flexibility in experimentation and verification.

Learning this concept opens doors to solving complex equations and ultimately, understanding the language of algebra, essential in fields ranging from science to engineering!