Problem 26 - Elementary Algebra

Question

Find two consecutive integers such that two thirds of the smaller number added to the other yields \(11 .\)

Step-by-Step Solution

Answer

Question: Find two consecutive integers such that two thirds of the smaller number added to the other yields 11. Answer: The two consecutive integers are 6 and 7.

Step 1: Assign Variables

Let's denote the smaller integer by \(x\). Since the problem states that the other integer is consecutive, we will denote it as \(x+1\).

Step 2: Formulate Equation

Now we'll create an equation from the given information: "two thirds of the smaller number added to the other yields 11". This can be written as: \(\frac{2}{3}x + (x+1) = 11\).

Step 3: Combine Like Terms

Now, we'll simplify the equation by combining like terms, i.e. the \(x\) terms: \(\frac{2}{3}x + x = 11-1 \Rightarrow \frac{2}{3}x + \frac{3}{3}x = 10.\)

Step 4: Solve for x

Now we'll solve for \(x\) by adding the two fractions: \(\frac{2}{3}x + \frac{3}{3}x = \frac{5}{3}x = 10.\) To find the value of \(x\), we'll multiply both sides of the equation by 3: \(5x = 30 \Rightarrow x = 6.\)

Step 5: Find Consecutive Integer

Now that we have found the smaller integer \(x=6\), we will find the consecutive integer by adding 1 to it: \(x+1 = 6+1 = 7.\) So, the two consecutive integers are 6 and 7.

Key Concepts

Equation Solving

Understanding how to solve an equation is a fundamental skill in algebra. In this example, we were required to find two consecutive integers, and the task was presented in the form of an equation. The steps below outline how to tackle such problems effectively.

• First understand what is asked: Identify the unknowns, which in this case are the two integers.
• Assign variables: Start by assigning a variable to one of the unknowns, often the smaller integer, which we called \(x\).
• Construct the equation: Translate the word problem into a mathematical equation. Here, this involves expressing relationships using operations like addition and multiplication.
• Simplify: Combine like terms and simplify the equation as much as possible, which helps in isolating the variable.
• Solve: Use algebraic operations, like addition, subtraction, multiplication, or division, to solve for the unknown variable. In our example, solving involved transforming fractions and using multiplication to isolate \(x\).

Solving equations involves logical steps, each building on the last. Practice can help improve efficiency and accuracy in solving these types of problems.

Algebraic Expressions

An algebraic expression is a combination of constants, variables, and arithmetic operations. In this exercise, we used expressions to represent the relationship between the two consecutive integers.

• Using Variables: Algebra often requires replacing unknown numbers with symbols or letters, such as \(x\) here.
• Incorporating Operations: The expression \(\frac{2}{3}x + (x + 1)\) includes fractions, addition, and variables to encode the problem statement.
• Simplification: By combining like terms, such as \(\frac{2}{3}x\) and \(x\), we transform the expression into a more manageable form.

Understanding algebraic expressions allows for translating real-world problems into a mathematical context, facilitating easier manipulation and solution-finding. When simplified, they reveal the concise meaning and solution pathways that can be tackled through algebraic methods.

Integer Properties

Integers have unique properties, especially when discussing consecutive numbers. These properties assist in predicting patterns and formulating equations to solve problems.

• Definition: Consecutive integers are numbers that follow each other in order without any gaps, such as 6 and 7.
• Relations: When handling concerns with these numbers, it's important to realize that for any integer \(x\), the next consecutive integer is \(x+1\).
• Algebraic Representation: By using a variable to denote an integer, one can easily express consecutive integers. This is beneficial in creating linear equations.

Recognizing integer properties speeds up problem-solving. For example, knowing two numbers in sequence always differ by 1 can simplify many problems, offering a straightforward basis for translating word problems into equations. These core concepts are key tools for algebra students.