The Average of 5 Quantities is 6 GMAT Problem Solving

Question: The Average of 5 Quantities is 6. The Average of Three of Them is 8. What is the Average of the Remaining two Numbers?

1. 6.5
2. 4
3. 3.5
4. 3
5. 2.5

Correct Answer: D

Solution and Explanation:
Approach Solution 1:
The problem statement states that:
Given:

• The average of five quantities is 6.
• The average of three of them is 8.

Find out:

• The average of the remaining two numbers.

To solve this question, the candidate should know the formula for the average of two numbers. This implies that the candidate must know the formula for the average of n numbers.

If there are two numbers a and b, then the average of two numbers can be written as (a+b)/2
If there are three numbers a, b and c, then the average of the three numbers can be written as (a+b+c)/3.
This formula can be generalized for the average of n numbers.
Let x1,x2,x3,x4……..xn be the n numbers. Their average can be written as
(x1 + x2 + x3 + x4 + …. + xn)/ n —-- (1)

Here n is the number of quantities.
So Firstly, it is given to us that five quantities have an average of 6.

Let the five quantities be x1,x2, x3, x4 and x5.
Their average is 6
So using the previous formula, we get
(x1 + x2 + x3 + x4 +x5) / 5 = 6
Note that here n = 5.
We get,
(x1 + x2 + x3 + x4 +x5) = 6*5
(x1 + x2 + x3 + x4 +x5) = 30 — (2)

Now it is given that the average of three of them is 8
It is not given the average of which three is 8, so we can take any three of them.

Let the three numbers be - x1,x2, and x3.
So using the formula (1) we get,
(x1 + x2 + x3) / 3 = 8
We get
(x1 + x2 + x3) = 24 — (3)

It is asked to find the average of the remaining elements.
It is clearly visible that the remaining elements are x4 and x5
Now to find the average of x4 and x5 we will use the formula for the average of two numbers.
Average of x4 and x5 = (x4 + x5)/2 — (4)
The value of x4 + x5 is not known to us till now.
To calculate the value of (x4+x5) subtract (3) from (2)

We get,
(x1 + x2 + x3 + x4 +x5) - (x1 + x2 + x3) = 30 - 24
x4 + x5 = 6

We got the value of x4 + x5.
Putting this value in the formula (4)
We get the average of x4 and x5 = 6 / 2 = 3

Therefore, the average of the remaining two numbers = 3.

Approach Solution 2:
The problem statement informs that:
Given:

• The average of five quantities is 6.
• The average of three of them is 8.

Find out:

• The average of the remaining two numbers.

The average of five quantities is 6.
As per the formula of average, we know that:
Average of given numbers = Sum of numbers/Total number
Sum of 5 quantities = 5*6 = 30

The average of three of them is 8
As per the formula of average, we get:
Sum of 3 quantities = 3*8 = 24 (Since, Average = Sum of numbers/Total number)

Therefore, Sum of the 2 quantities = 30 - 24 = 6
Hence, the average of the 2 quantities = 6/2 = 3.

Therefore, the average of the remaining two numbers = 3.

“The Average of 5 Quantities is 6”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. This topic has been taken from the book “501 GMAT Questions”. The candidates should examine the data and calculate it properly in order to crack GMAT Problem Solving questions. The candidates can go through GMAT Quant practice papers to get familiar with several sorts of questions that will enable them to score better in the exam.

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