Find Three Consecutive Integers With A Sum Of 48.

Finding Three Consecutive Integers with a Sum of 48: A Step-by-Step Guide

Mathematics often presents problems that seem simple at first glance but require careful reasoning to solve. One such problem is finding three consecutive integers whose sum equals 48. This task not only tests basic algebraic skills but also reinforces the concept of consecutive numbers and their properties. On the flip side, in this article, we’ll explore how to approach and solve this problem, break down the mathematical reasoning behind it, and discuss its broader applications. Whether you’re a student grappling with algebra or someone curious about problem-solving techniques, this guide will provide clarity and insight.

What Are Consecutive Integers?

Before diving into the solution, let’s define the key terms. Day to day, Consecutive integers are whole numbers that follow each other in order without gaps. Take this: 5, 6, and 7 are consecutive integers, as are -3, -2, and -1. The term “consecutive” implies a sequence where each number is exactly one unit apart from its neighbor.

In algebraic problems, consecutive integers are often represented using a variable. If we let the first integer be $ n $, the next two can be expressed as $ n + 1 $ and $ n + 2 $. This notation allows us to create equations that model real-world scenarios or abstract puzzles.

Setting Up the Equation

The problem states that the sum of three consecutive integers is 48. To translate this into an equation, we assign variables to the unknown numbers. Let’s denote the first integer as $ n $. Then:

• The second integer is $ n + 1 $,
• The third integer is $ n + 2 $.

Adding these together gives the equation:
$ n + (n + 1) + (n + 2) = 48 $

This equation captures the relationship described in the problem. The next step is to simplify and solve for $ n $.

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Solving the Equation Step-by-Step

Let’s break down the simplification process:

$ n + (n + 1) + (n + 2) = 48 \ 3n + 3 = 48 $

Now, we isolate the term with 'n' by subtracting 3 from both sides of the equation:

$ 3n + 3 - 3 = 48 - 3 \ 3n = 45 $

Finally, we solve for 'n' by dividing both sides by 3:

$ \frac{3n}{3} = \frac{45}{3} \ n = 15 $

So, the first integer is 15. Now we can find the other two consecutive integers:

• Second integer: $n + 1 = 15 + 1 = 16$
• Third integer: $n + 2 = 15 + 2 = 17$

Because of this, the three consecutive integers are 15, 16, and 17. Let's verify our answer: 15 + 16 + 17 = 48. Our solution is correct!

Conclusion

Finding three consecutive integers with a sum of 48 demonstrates a fundamental principle in algebra: translating word problems into mathematical equations. In practice, by understanding the definition of consecutive integers and systematically setting up and solving the equation, we successfully arrived at the solution. This seemingly simple problem highlights the power of algebraic manipulation and reinforces the ability to logically deduce answers from given information. On top of that, the techniques employed – defining variables, translating phrases into mathematical expressions, and solving linear equations – are applicable to a wide range of mathematical challenges. Worth adding: this exercise not only provides a concrete answer but also strengthens problem-solving skills that are invaluable in various academic and real-world scenarios. The ability to break down a problem into smaller, manageable steps is a crucial skill that will serve you well in your mathematical journey.