Problem 22 - Intermediate Algebra

Question

The sum of the digits of a three-digit number is 13 . The sum of the hundreds digit and the tens digit is 1 less than the units digit. The sum of three times the hundreds digit and four times the units digit is 26 more than twice the tens digit. Find the number.

Step-by-Step Solution

Answer

The number is 247.

Step 1: Define the digits

Let's define the hundreds digit as \(h\), the tens digit as \(t\), and the units digit as \(u\). A three-digit number is then represented as \(100h + 10t + u\).

Step 2: Formulate the first equation

According to the problem, the sum of the digits is 13. Thus, the first equation is: \[ h + t + u = 13 \]

Step 3: Formulate the second equation

The sum of the hundreds digit and the tens digit is 1 less than the units digit. So we have: \[ h + t = u - 1 \]

Step 4: Formulate the third equation

The sum of three times the hundreds digit and four times the units digit is 26 more than twice the tens digit. This gives us: \[ 3h + 4u = 2t + 26 \]

Step 5: Solve the second equation for one variable

From the second equation \(h + t = u - 1\), we can express \(u\) as: \[ u = h + t + 1 \]

Step 6: Substitute into the first equation

Substitute \(u = h + t + 1\) into the first equation \(h + t + u = 13\):\[ h + t + (h + t + 1) = 13 \]\[ 2h + 2t + 1 = 13 \]\[ 2h + 2t = 12 \]\[ h + t = 6 \]

Step 7: Solve for digits using first two equations

From \(h + t = 6\) and \(u = h + t + 1\), we know \(u = 6 + 1 = 7\). Thus, \(u = 7\).

Step 8: Substitute known digits into the third equation

Substitute \(h + t = 6\) and \(u = 7\) into the third equation \(3h + 4u = 2t + 26\): \[ 3h + 4(7) = 2t + 26 \]\[ 3h + 28 = 2t + 26 \]\[ 3h + 2 = 2t \]

Step 9: Express in terms of \(h\) or \(t\)

From \(3h + 2 = 2t\):\[ t = \frac{3h + 2}{2} \]Using \(h + t = 6\), \(t = 6 - h\).

Step 10: Solve for specific values of \(h\) and \(t\)

Since both sides are equivalent: \[ 6 - h = \frac{3h + 2}{2} \]Multiply by 2 to clear the fraction:\[ 12 - 2h = 3h + 2 \]\[ 12 - 2 = 3h + 2h \]\[ 10 = 5h \]\[ h = 2 \]Substitute \(h = 2\) into \(t = 6 - h\):\[ t = 6 - 2 = 4 \]

Step 11: Verify the solution for all equations

Plug \(h = 2\), \(t = 4\), \(u = 7\) back into the equations:1. \(h + t + u = 2 + 4 + 7 = 13\), satisfied.2. \(h + t = 2 + 4 = 6\) and \(u = 7\), hence \(u = h + t + 1\), satisfied.3. \(3h + 4u = 2t + 26\), substitute the values:\[ 3(2) + 4(7) = 2(4) + 26 \]\[ 6 + 28 = 8 + 26 \]Both sides equal 34, hence satisfied.

Key Concepts

Three-digit numbers

Understanding three-digit numbers is essential for solving problems involving digits and place values. A three-digit number is typically structured as ABC, where the A digit represents the hundreds, B represents the tens, and C represents the units. This structure can be mathematically expressed as \(100h + 10t + u\) with each letter corresponding to its respective digit's place value, allowing you to convert it into an algebraic form to solve equations.
In the example exercise, the objective is to determine the specific values for these digits under certain conditions. Recognizing the individual contributions of each digit to the total value of a number lets you precisely form equations based on given conditions, facilitating problem-solving.

Digit problems

Digit problems, especially those involving the sum or relationships between digits, are common in algebra. These problems require you to manipulate the digits using relational equations to find their exact values.
This exercise asks us to solve for three conditions or equations given:

• The sum of all digits equals a specific number
• Some specific conditions where digits relate to one another (e.g. digits differing by a specific value)
• Other complex conditions involving multiplication and addition of these digits

Such problems help in honing your algebraic skills by presenting you with multiple equations and building your ability to logically think through the relationships between numbers within defined parameters.

Solving equations step by step

Solving equations step by step engages a systematic approach to problem-solving, ensuring each stage logically progresses towards the solution. It involves several key steps:

• Define the variables clearly, representing the unknowns in a problem.
• Translate the problem statements into algebraic equations, encapsulating the relationships described in words.
• Use substitution and elimination methods to solve these equations. For example, substitute a known value from one equation into another to simplify or resolve multiple variables.
• Verify your solutions by plugging the values back into the original equations to ensure all conditions are met.

By applying a step-by-step method as demonstrated in the original solution, you systematically reduce complexity, paving the way for finding a clear and accurate solution to the problem. This structured approach is fundamental in algebra, enhancing understanding and accuracy in problem-solving.