Finding the Sixth Number Using Average and Sum Conditions

This problem involves using the properties of averages and sums to find a specific number within a set of six numbers. We are given two key pieces of information:

1. The sum of the first five numbers is 7 times the sixth number.
2. The average of all six numbers is 136.

Let's break down the problem and use these conditions to find the value of the sixth number.

Setting up Equations

Let the six numbers be represented by \(n_1, n_2, n_3, n_4, n_5,\) and \(n_6\). We can write the given information as mathematical equations.

• Condition 1: The sum of the first 5 numbers is 7 times the 6th number.
• Mathematically, this is: \(n_1 + n_2 + n_3 + n_4 + n_5 = 7 \times n_6\)
• We can simplify the sum of the first five numbers as \(Sum_{1-5}\). So, \(Sum_{1-5} = 7n_6\).
• Condition 2: The average of all 6 numbers is 136.
• The average is the sum of all numbers divided by the count of numbers.
• The sum of all 6 numbers is \(n_1 + n_2 + n_3 + n_4 + n_5 + n_6\).
• The count of numbers is 6.
• So, the average equation is: \(\frac{n_1 + n_2 + n_3 + n_4 + n_5 + n_6}{6} = 136\)

Calculating the Total Sum

From the average equation, we can find the sum of all six numbers. To do this, we multiply the average by the total count of numbers.

Sum of all 6 numbers = Average \(\times\) Number of items

Sum of all 6 numbers = \(136 \times 6\)

Let's calculate \(136 \times 6\):

So, the sum of all 6 numbers is 816.

Substituting and Solving for the Sixth Number

Now we have two main equations:

1. \(n_1 + n_2 + n_3 + n_4 + n_5 = 7n_6\)
2. \(n_1 + n_2 + n_3 + n_4 + n_5 + n_6 = 816\)

Notice that the sum of the first five numbers (\(n_1 + n_2 + n_3 + n_4 + n_5\)) appears in both equations. We can substitute the expression from equation (1) into equation (2).

Substitute \(7n_6\) for \((n_1 + n_2 + n_3 + n_4 + n_5)\) in the second equation:

Now, combine the terms involving \(n_6\):

To find the value of \(n_6\), we need to divide both sides of the equation by 8:

Let's perform the division:

So, the value of the sixth number (\(n_6\)) is 102.

Verification

Let's quickly check if this value satisfies the original conditions:

• The sixth number is 102.
• The sum of the first 5 numbers is \(7 \times 102 = 714\).
• The sum of all 6 numbers is \(714 + 102 = 816\).
• The average of all 6 numbers is \(\frac{816}{6} = 136\).

The calculated value of 102 fits both conditions perfectly.

Final Answer Summary

Based on the given conditions and calculations, the sixth number is 102.

Revision Table - Key Steps to Finding the Sixth Number

Additional Information - Average and Sum Concepts

Understanding the relationship between average, sum, and the number of items is fundamental in solving such problems.

• Average: The average (or mean) of a set of numbers is the sum of the numbers divided by the count of the numbers.
• Formula: Average = \(\frac{\text{Sum of items}}{\text{Number of items}}\)
• Sum: If you know the average and the number of items, you can find the sum by rearranging the average formula.
• Formula: Sum of items = Average \(\times\) Number of items
• In this problem, we used the average of 136 and the count of 6 numbers to find the total sum (816). Then we used the relationship between the sum of the first five numbers and the sixth number to isolate and find the value of the sixth number.