Problem 38 - Elementary and Intermediate Algebra

Question

Find a three-digit positive integer such that the sum of all three digits is \(14,\) the tens digit is 2 more than the ones digit, and if the digits are reversed, the number is unchanged.

Step-by-Step Solution

Answer

The number is 464.

Step 1: Identify the Conditions

Summarize the given information: 1) The sum of the digits is 14. 2) The tens digit is 2 more than the ones digit. 3) The number is a palindrome (same when reversed).

Step 2: Define Variables

Let the three-digit number be represented as \(ABC\), where \(A\) is the hundreds digit, \(B\) is the tens digit, and \(C\) is the ones digit.

Step 3: Palindrome Property

Since the number is a palindrome, \(A = C\). Therefore, we can rewrite the number as \(ACA\).

Step 4: Equation from Digit Sum

Given that the sum of the digits is 14, we can write equation \(A + B + A = 14 \Rightarrow 2A + B = 14\).

Step 5: Equation from Tens Digit

Given that the tens digit (B) is 2 more than the ones digit (C), and considering \(A = C\), write the equation \(B = A + 2\).

Step 6: Solve Equations

Substitute \(B = A + 2\) into the first equation: \2A + (A + 2) = 14 \Rightarrow 3A + 2 = 14 \Rightarrow 3A = 12 \Rightarrow A = 4.\

Step 7: Find the Digits

Since \(A = 4\), we have \(C = 4\) and \(B = A + 2 = 6\). Thus, the number is \464\.

Key Concepts

palindrome number

In mathematics, a palindrome number is a number that remains the same when its digits are reversed. These numbers are symmetrical and they read the same forwards and backwards. For example, 121, 1331, and 12321 are palindromic numbers. For a three-digit number like the one in our problem:

• The first digit (hundreds place) is the same as the last digit (units place).
• This symmetry makes it easier to write equations and solve the problem.

In our problem, the number, when reversed, remains the same, indicating it is a palindrome. Hence, if the number is represented as \(ABC\), we have \(A = C\). This way, our number can be rewritten as \(ACA\).

digit sum

The digit sum is the sum of all the individual digits in a number. This is an important concept in solving many numerical problems. In our specific problem, the sum of the digits of the three-digit number is given to be 14.

• If our number is \(ACA\), then the equation for the sum of the digits is \(A + C + A\).
• Since \(A = C\) for a palindrome, the equation becomes \(2A + B = 14\), simplifying our calculations.

By identifying and using the digit sum property, we can more easily identify the specific values of the digits that satisfy the given conditions.

algebraic equations

Algebraic equations are foundational in solving many mathematical problems, including finding specific numbers that satisfy certain conditions. In this problem, we use algebraic equations to encode the information about the digit-sum and the relationship between the digits.
Here's how we formulate and solve the equations:

• First, knowing the number is palindromic, we wrote it as \(ACA\) and noted that \(A = C\).
• We then set up the equation for the sum of the digits: \(2A + B = 14\).
• Next, we used the condition that the tens digit (B) is 2 more than the units digit (A), leading to \(B = A + 2\).

Combining these equations:
1. Substitute \(B = A + 2\) into \(2A + B = 14\) results in
\[2A + (A + 2) = 14 \] \[3A + 2 = 14 \] \[3A = 12 \] \[A = 4 \]
2. With \(A = 4\), we get \(C = 4\) and \(B = 4 + 2 = 6\). Thus, the number is 464.