Finding the Highest of Three Consecutive Numbers

Let's break down this problem step by step to find the highest of the three consecutive numbers that add up to 126.

Consecutive numbers are numbers that follow each other in order, with a difference of 1 between each number. For example, 5, 6, and 7 are consecutive numbers.

Setting Up the Equation for Consecutive Numbers

We need to represent the three consecutive numbers using algebra. Let the first number be represented by x.

• The first number is x.
• Since the numbers are consecutive, the second number will be one more than the first, so it is x + 1.
• The third number will be one more than the second, so it is (x + 1) + 1, which simplifies to x + 2.

The question states that the sum of these three consecutive numbers is 126. So, we can write an equation:

First number + Second number + Third number = 126

Substituting our algebraic expressions:

\(x + (x + 1) + (x + 2) = 126\)

Solving for the First Consecutive Number

Now, we need to solve this equation for x.

Combine the x terms and the constant terms:

\(x + x + 1 + x + 2 = 126\)

\(3x + 3 = 126\)

Subtract 3 from both sides of the equation to isolate the term with x:

\(3x + 3 - 3 = 126 - 3\)

\(3x = 123\)

Divide both sides by 3 to find the value of x:

\( \frac{3x}{3} = \frac{123}{3} \)

\(x = 41\)

So, the first number is 41.

Finding All Three Consecutive Numbers

Now that we know x = 41, we can find the other two consecutive numbers:

• First number: x = 41
• Second number: x + 1 = 41 + 1 = 42
• Third number: x + 2 = 41 + 2 = 43

The three consecutive numbers are 41, 42, and 43.

Identifying the Highest Number

The question asks for the highest number among these three consecutive numbers. Comparing 41, 42, and 43, the highest number is 43.

The sum of 41, 42, and 43 is indeed 126.

Therefore, the highest number is 43.

Revision Table: Key Concepts

Additional Information on Consecutive Integers

Consecutive integers can be positive, negative, or zero. The method used above works for any set of consecutive integers. If the problem involved an even sum or an odd sum, representing the numbers differently (e.g., x-1, x, x+1 or x, x+2, x+4 for consecutive even/odd) can sometimes simplify the algebra, but the x, x+1, x+2 method always works for any three consecutive integers.

For example, if the sum of three consecutive integers was 0, using x, x+1, x+2:

\(x + (x+1) + (x+2) = 0\)

\(3x + 3 = 0\)

\(3x = -3\)

\(x = -1\)

The numbers would be -1, 0, and 1, which are consecutive and sum to 0.