What are the numbers if the average of 4 consecutive even numbers is 11?

Understand the Problem

The question is asking to find four consecutive even numbers whose average is 11. To find these numbers, we first understand that the average of a set of numbers is calculated by dividing the total sum of the numbers by the count of the numbers. Since the average is 11, we can determine the total sum of the four numbers, and then deduce the consecutive even numbers from there.

Answer

The four consecutive even numbers are $8, 10, 12, 14$.

Steps to Solve

Find the Total Sum of the Four Numbers

Since the average of the four consecutive even numbers is 11, we can find the total sum by multiplying the average by the number of numbers:

$$ \text{Total Sum} = \text{Average} \times \text{Count} = 11 \times 4 = 44 $$

Set Up the Equation for Consecutive Even Numbers

Let the first even number be $x$. The next three consecutive even numbers can be expressed as $x + 2$, $x + 4$, and $x + 6$.

So, we can set up the equation:

$$ x + (x + 2) + (x + 4) + (x + 6) = 44 $$

Combine Like Terms

Now, combine the terms on the left side of the equation:

$$ 4x + 12 = 44 $$

Solve for x

To find $x$, subtract 12 from both sides of the equation:

$$ 4x = 44 - 12 $$

This simplifies to:

$$ 4x = 32 $$

Now, divide both sides by 4:

$$ x = 8 $$

Find the Consecutive Even Numbers

Now that we have $x$, we can find the four consecutive even numbers:

The first number: $x = 8$

Second number: $x + 2 = 10$

Third number: $x + 4 = 12$

Fourth number: $x + 6 = 14$

So, the four consecutive even numbers are $8, 10, 12, 14$.

The four consecutive even numbers are $8, 10, 12, 14$.

More Information

The average of four consecutive even numbers being 11 means their total sum adds up to 44. When we break down the average into individual numbers, we can easily derive the set of numbers using basic algebra techniques.

Tips

A common mistake is to miscalculate the total sum when using the average. It’s important to remember that the sum is found by multiplying the average by the count.
Another mistake might be not correctly setting up the expression for consecutive even numbers. Always ensure you maintain the relationship (i.e., an even number plus 2 for each next number).

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