Problem 50 - Algebra 1

Question

Consecutive integers are integers that follow each ther in order (for example, \(5,6,\) and 7 ). You want to find three consecutive the whose sum is 84 . a. Why does the equation \(n+(n+1)+(n+2)=84\) model the situation? b. Solve the equation in part (a). Then find the three consecutive integers.

Step-by-Step Solution

Answer

The equation \(n+(n+1)+(n+2)=84\) models the situation because it represents the sum of three consecutive integers. The solution to the equation yields \(n=27\). So, the three consecutive integers whose sum is 84 are 27, 28, and 29.

Step 1: Understand the Model

Consecutive integers continuously follow each other, therefore if one integer is represented by \(n\), the next consecutive integer can be represented by \(n+1\), and the one after that can be represented by \(n+2\). Since the problem states the sum of these three consecutive integers is 84, this situation is modeled by the equation \(n + (n+1) + (n+2) = 84\). The equation represents the sum of three consecutive integers being equal to 84.

Step 2: Simplify the Equation

We can simplify the equation by combining like terms. The left side of the equation \(n + (n+1) + (n+2)\) simplifies to \(3n + 3\). The equation then becomes \(3n + 3 = 84\).

Step 3: Solve for n

To solve for \(n\), first subtract 3 from both sides to isolate the term with \(n\) on one side of the equation. This simplifies the equation to \(3n = 81\). Now, divide both sides by 3 to solve for \(n\). This gives the value of \(n = 27\).

Step 4: Find Three Consecutive Integers

Now that we have found the value of \(n = 27\), we can substitute this value into the expressions \(n\), \(n+1\), and \(n+2\). Thus, the three consecutive integers are 27, 28, and 29.

Key Concepts

Consecutive Integers

Consecutive integers are numbers that follow one another in an unbroken sequence, with a difference of one between each number. For instance, numbers like 7, 8, and 9 are consecutive integers. Understanding consecutive integers is crucial for solving problems related to integer sums, where you need to find a sequence of such numbers.
When tackling problems involving consecutive integers, you represent the sequence mathematically starting with a variable 'n'. If 'n' is the first integer, then the next integers in the sequence are 'n+1', 'n+2', and so on. For instance, in finding three consecutive integers that sum to a specific total, you set up an equation that models the situation using 'n', 'n+1', and 'n+2'. This approach helps simplify the problem and is fundamental in solving related mathematical challenges.

Integer Sums

Integer sums involve the addition of integer numbers to find a total. In equations, integer sums can be useful in modeling real-world situations where you need to find numbers that add up to a specific value.
In problems asking for the sum of consecutive integers, it's helpful to write out what each consecutive integer represents and then sum them up. For instance, if you have integers represented by 'n', 'n+1', and 'n+2', their sum would be expressed as \(n + (n+1) + (n+2)\). This method of using integer sums is agricultural in grasping fundamental math concepts where order and sequences of numbers are essential to solve the given problems.
Understanding integer sums allows you to break down these sums into simpler parts, accurately guide solving the overall equation, and get to the root answer efficiently.

Linear Equations

Linear equations are mathematical statements of equality incorporating variables and constants. These equations form the backbone of algebraic problem-solving, particularly helpful in finding unknown values. A typical example of a linear equation is \(3n + 3 = 84\), which involves simplifying a series of terms into a statement that can be solved for 'n'.
To solve linear equations, you perform operations that simplify the equation, such as combining like terms, using addition or subtraction, and finally dividing or multiplying to isolate the variable on one side. Utilizing linear equations in problems involving consecutive integers allows you to find the values of these integers efficiently.
Mastering linear equations is highly beneficial as they often appear in various mathematical contexts, playing a crucial role in algebra but also in broader areas of mathematics and real-life applications where relationships between quantities need to be modeled plainly and precisely.