Problem 22 Solve using the five "Steps for ... [FREE SOLUTION]

Chapter 3: Problem 22

Solve using the five "Steps for Solving Applied Problems." Find three consecutive odd integers such that four times the smallest is 56 more than the sum of the other two.

Short Answer

Expert verified

The three consecutive odd integers are 31, 33, and 35. We found these by setting up and solving the equation \(4x = 2x + 62\) for the smallest integer, x, which is equal to 31. The check confirms that these integers satisfy the given conditions.

Step by step solution

Understand the problem

We need to find three consecutive odd integers such that four times the smallest is 56 more than the sum of the other two integers.

Translate the problem into equations

Let's call the smallest integer x. Since they are consecutive odd integers, the next integer will be x + 2 and the third integer will be x + 4. According to the problem, four times the smallest is 56 more than the sum of the other two. We can write this as: \(4x = (x + 2) + (x + 4) + 56\)

Decoding the equation

Now, let's simplify and solve the equation. \[ \begin{aligned} 4x &= (x + 2) + (x + 4) + 56 \\ 4x &= 2x + 6 + 56 \\ 4x &= 2x + 62 \end{aligned} \]

Solving the equation

To solve for x, let's subtract 2x from both sides of the equation. \[ \begin{aligned} 4x - 2x &= 2x - 2x + 62 \\ 2x &= 62 \end{aligned} \] Now, we can divide both sides by 2 to find x. \[ \begin{aligned} \frac{2x}{2} &= \frac{62}{2} \\ x &= 31 \end{aligned} \]

Find the consecutive odd integers and check

Now that we have the value of x, we can determine the three consecutive odd integers. Smallest integer: \(x = 31\) Second integer: \(x + 2 = 31 + 2 = 33\) Third integer: \(x + 4 = 31 + 4 = 35\) We found the three consecutive odd integers: 31, 33, and 35. Now, let's check to see if our solution is correct: Four times the smallest integer is 4*31 = 124. The sum of the other two integers is 33+35 = 68. Four times the smallest integer is indeed 56 more than the sum of the other two integers: \(124 = 68 + 56\) Therefore, our solution is correct. The three consecutive odd integers are 31, 33, and 35.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Consecutive Integers

In math, when we talk about consecutive integers, we're looking at numbers that follow one another in order. For example, 1, 2, and 3 are consecutive integers. For odd integers like 1, 3, and 5, each number increases by 2 to remain odd. When solving problems involving consecutive integers, distinguishing between even and odd is crucial. In this exercise, we're dealing with odd integers. If we say the first number is \(x\), then in the sequence of consecutive odd integers, the next one will be \(x + 2\), and the next \(x + 4\). This pattern keeps each integer odd and orderly.

Problem-Solving Steps

Solving math problems systematically requires clear steps. Let's walk through the typical process:

Understand the Problem: Carefully read and clarify the question. Identify what's given and what needs to be found.
Translate the Information Into Equations: Use algebra to represent the problem. Assign variables for unknowns and write down the relationships between them.
Decode and Simplify the Equation: Work with the equation by gathering like terms and simplifying. This often involves arithmetic operations to make solving easier.
Solve the Equation: Perform calculations to find the value of the variable. This often involves isolating the variable and using inverse operations, such as division or subtraction.
Verify the Solution: Check your answer by substituting it back into the context of the problem. Ensure the solution satisfies the original conditions.

Following these problem-solving steps can help break down even complex problems into manageable tasks.

Applied Mathematics

Applied mathematics involves using mathematical methods to solve real-world problems. In this context, we are applying algebraic techniques to solve for numbers described in the problem scenario. For instance, understanding how consecutive integers relate gives meaning to a pure algebraic exercise by anchoring it in everyday tasks. When you practice applied mathematics, you're not just solving numbers; you're modeling and mimicking real-world situations that the mathematical model can explain or predict. This blending of math into real situations helps deepen our comprehension beyond abstract numbers and operations.

Equation Translation

Equation translation is a vital skill in algebra. It involves taking a statement or problem described in words and converting it into a mathematical equation. This is like speaking a new language where sentences are written in numbers and symbols. In the original exercise, translating the phrase, "four times the smallest is 56 more than the sum of the other two," into \(4x = (x + 2) + (x + 4) + 56\) illustrates this. The challenge is to ensure each part of the sentence is captured correctly as a mathematical expression. Mastering equation translation allows students to tackle a wide array of problems by applying a standard approach.

91影视

Solve using the five "Steps for Solving Applied Problems." Find three consecutive odd integers such that four times the smallest is 56 more than the sum of the other two.

Short Answer

Step by step solution

Understand the problem

Translate the problem into equations

Decoding the equation

Solving the equation

Find the consecutive odd integers and check

Key Concepts

Consecutive Integers

Problem-Solving Steps

Applied Mathematics

Equation Translation

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