Understanding the Average of Positive Numbers

The problem involves the concept of average, specifically the average of five positive numbers. The average is calculated by dividing the sum of the numbers by the count of the numbers.

We are given that the average of five positive numbers is 56. Let the five positive numbers be \(n_1, n_2, n_3, n_4,\) and \(n_5\).

The formula for the average is:

$$\text{Average} = \frac{\text{Sum of numbers}}{\text{Count of numbers}}$$

In this case, the sum of the five numbers is \(n_1 + n_2 + n_3 + n_4 + n_5\), and the count is 5. The average is given as 56.

So, we can write the equation:

$$\frac{n_1 + n_2 + n_3 + n_4 + n_5}{5} = 56$$

Calculating the Total Sum of Five Positive Numbers

From the average formula, we can find the sum of the five numbers:

$$\text{Sum of numbers} = \text{Average} \times \text{Count of numbers}$$

Sum of the five numbers \( = 56 \times 5 \)

Sum of the five numbers \( = 280 \)

So, \(n_1 + n_2 + n_3 + n_4 + n_5 = 280\).

Relating the First Number to the Sum of the Last Four Numbers

The question gives a relationship between the first number and the sum of the last four numbers. It states that the first number (\(n_1\)) is three-fourth of the sum of the last four numbers (\(n_2 + n_3 + n_4 + n_5\)).

Let \(S_4\) be the sum of the last four numbers. So, \(S_4 = n_2 + n_3 + n_4 + n_5\).

According to the problem statement:

$$n_1 = \frac{3}{4} \times (n_2 + n_3 + n_4 + n_5)$$

Substituting \(S_4\):

$$n_1 = \frac{3}{4} S_4$$

Finding the Sum of the Last Four Numbers

We know that the total sum of the five numbers is 280:

$$n_1 + (n_2 + n_3 + n_4 + n_5) = 280$$

Substituting \(S_4 = n_2 + n_3 + n_4 + n_5\), we get:

$$n_1 + S_4 = 280$$

Now, substitute the relationship \(n_1 = \frac{3}{4} S_4\) into this equation:

$$\frac{3}{4} S_4 + S_4 = 280$$

Combine the terms involving \(S_4\):

$$\left(\frac{3}{4} + 1\right) S_4 = 280$$

$$\left(\frac{3}{4} + \frac{4}{4}\right) S_4 = 280$$

$$\frac{7}{4} S_4 = 280$$

To find \(S_4\), multiply both sides by \(\frac{4}{7}\):

$$S_4 = 280 \times \frac{4}{7}$$

$$S_4 = \frac{280}{7} \times 4$$

$$S_4 = 40 \times 4$$

$$S_4 = 160$$

So, the sum of the last four numbers is 160.

Calculating the Average of the Last Four Numbers

The question asks for the average of the last four numbers. The last four numbers are \(n_2, n_3, n_4,\) and \(n_5\), and their sum is \(S_4 = 160\). The count of these numbers is 4.

The average of the last four numbers is:

$$\text{Average of last four numbers} = \frac{\text{Sum of last four numbers}}{\text{Count of last four numbers}}$$

$$\text{Average of last four numbers} = \frac{S_4}{4}$$

$$\text{Average of last four numbers} = \frac{160}{4}$$

$$\text{Average of last four numbers} = 40$$

Therefore, the average of the last four numbers is 40.

Let's check the options:

• Option 1: 50
• Option 2: 35
• Option 3: 30
• Option 4: 40

Our calculated average matches Option 4.

Step-by-Step Solution Summary

1. Write down the given information: Average of 5 numbers = 56.
2. Calculate the total sum of the 5 numbers: Sum = Average × Count = \(56 \times 5 = 280\).
3. Identify the relationship between the first number (\(n_1\)) and the sum of the last four numbers (\(S_4\)): \(n_1 = \frac{3}{4} S_4\).
4. Write the total sum in terms of \(n_1\) and \(S_4\): \(n_1 + S_4 = 280\).
5. Substitute the relationship from step 3 into the equation from step 4: \(\frac{3}{4} S_4 + S_4 = 280\).
6. Solve for \(S_4\): \(\frac{7}{4} S_4 = 280 \implies S_4 = 280 \times \frac{4}{7} = 160\).
7. Calculate the average of the last four numbers: Average = \(\frac{S_4}{4} = \frac{160}{4} = 40\).

Revision Table: Average and Sum Concepts

Additional Information: Solving Average Word Problems

Solving average word problems often involves translating the given information into algebraic equations. Key steps usually include:

• Identifying the total sum from the given average and count.
• Using any given relationships between numbers to form additional equations.
• Substituting one equation into another to solve for unknown quantities (like the sum of a subset of numbers).
• Finally, calculating the requested average using the sum and count of the relevant numbers.

Understanding how the sum, average, and count relate to each other is fundamental to solving these types of quantitative aptitude problems.