Problem: Finding the Product of the Largest and Smallest Among Four Consecutive Odd Numbers

Let's break down the process of solving a mathematical problem involving the average and product of four consecutive odd numbers. This problem is a great example of how to apply basic arithmetic concepts to find the solution. Follow along as we walk through the steps.

Understanding the Problem

Suppose we have four consecutive odd numbers. Let's denote these numbers as follows:

The smallest number: ( x )

The second number: ( x 2 )

The third number: ( x 4 )

The largest number: ( x 6 )

Our goal is to find the average of these four numbers and then calculate the product of the largest and smallest of them.

Step 1: Calculate the Average

The average of the four numbers can be calculated using the formula:

( text{Average} frac{x (x 2) (x 4) (x 6)}{4} )

Let's simplify the numerator:

( frac{4x 12}{4} )

This simplifies to:

( x 3 )

We know that the average is given as 60:

( x 3 60 )

Step 2: Solve for ( x )

Subtract 3 from both sides:

( x 60 - 3 )

( x 57 )

Now that we have the value of ( x ), we can determine the four consecutive odd numbers:

The smallest number: ( 57 )

The second number: ( 59 )

The third number: ( 61 )

The largest number: ( 63 )

Step 3: Calculate the Product of the Largest and Smallest Numbers

Finally, we need to find the product of the largest and smallest numbers:

( 57 times 63 3591 )

The product of the largest and smallest numbers among these four consecutive odd numbers is 3591.

Conclusion

In this problem, we used basic arithmetic and algebraic simplification to find the product of the largest and smallest consecutive odd numbers given their average. This method can be applied to similar problems to help solve more complex mathematical scenarios.