Problem 27 - Elementary Algebra

Question

Find two consecutive integers such that three fourths of the smaller number added to the other yields 29

Step-by-Step Solution

Answer

Answer: The two consecutive integers are 16 and 17.

Step 1: Define variables

Let the smaller integer be x. Then the consecutive integer will be x + 1.

Step 2: Set up the equation

Three fourths of the smaller number added to the other yields 29. We can write it as follows: (3/4) * x + (x + 1) = 29

Step 3: Simplify and solve for x

First, multiply both sides of the equation by 4 to get rid of the fraction: 4 * ( (3/4) * x + (x + 1) ) = 29 * 4 Simplifying the equation, we get: 3x + 4x + 4 = 116 Combine like terms: 7x + 4 = 116 Now, subtract 4 from both sides of the equation: 7x = 112 Finally, divide by 7: x = 16

Step 4: Find both integers

Now that we have found x, we can find both consecutive integers. The smaller integer is x = 16. The larger integer is x + 1 = 16 + 1 = 17. So, the two consecutive integers are 16 and 17.

Key Concepts

Understanding Algebra in Consecutive Integer Problems

Algebra is a branch of mathematics where symbols, usually letters, represent numbers in equations. This allows us to solve problems involving unknown values. In consecutive integer problems, algebra becomes quite handy, as it helps simplify and solve equations involving unknown integers.
To tackle such problems, we often begin by defining one of the unknown integers as a variable, say \( x \). Since we are dealing with consecutive integers (which follow each other without any gaps), the next integer can be expressed as \( x + 1 \).
This setup allows us to create an equation that represents the problem statement. Through algebraic manipulation, we can find the value of \( x \) and subsequently its consecutive number.

Crafting Integer Equations

Integer equations involve whole numbers and are used extensively in algebra to solve problems where numbers have to be found based on given relationships. Our goal in solving these equations is to find the unknown integers.
Start by translating the problem's words into a mathematical equation. Here, the task is to find two consecutive integers that satisfy a certain condition.

• Let the smaller integer be \( x \).
• Thus, the consecutive integer can be written as \( x + 1 \).
• The problem states that three-fourths of the smaller integer, added to the other, equals 29. This translates to: \((3/4)x + (x + 1) = 29\).

By setting up the equation correctly, we're on the path to solving for \( x \). Focus on basic manipulation like clearing fractions or combining terms to simplify the equation further.

Problem Solving Techniques for Algebraic Equations

Problem-solving in algebra involves systematic approaches to arrive at a solution. For integer equations, you'll often follow these key steps:
Define Your Variables: Start by assigning variables to the unknowns, allowing you to express other unknowns in relation to the first one.
Set Up the Equation: Translate the problem statement into an equation using the variables. Ensure all parts of the problem are represented in the equation.

• Here, we formed the equation \((3/4)x + (x + 1) = 29\).

Simplify and Solve: Use algebraic manipulation to solve for the variables. This often involves simplifying equations by combining like terms, eliminating fractions, if any, and isolating the main variable.

• We eliminate fractions by multiplying the entire equation, leading to \(3x + 4x + 4 = 116\).
• Combine and solve: \(7x = 112\), hence \(x = 16\).
• Find the consecutive integer: \(x + 1 = 17\).

By working through these steps, you not only solve the specific problem but also gain skills useful for tackling a broad range of similar problems.