# Applications Of Averages

The concept of averages has various applications while solving differentÂ quantitative aptitude question. Some of the important application is mentioned here along with examples to help the CAT candidates prepare well for the quantitative section.

1. Age Related Averages

In questions where one needs to assume the age of people involved, the present age is taken as a variable (say x), and the present and past ages are based on the variable x. For example,

5 years agoâ‡’ x-5

10 years from nowâ‡’ x+10

15 years agoâ‡’ x-15

Using this, one can form equations easily and solve for the present age. Take for example the following question.

Example 1:

The average age of 5 sisters is 15 years. The youngest sister is 6 years old. When she was born, the average age of the remaining sisters was N years. What is the average age of the sisters excluding the youngest sister?

Solution:

Given that average age of sisters = 15 years. Sum of all their ages= 15×5= 75 years

Sum of their ages excluding the youngest sister = 75-6 = 69 years

The average age of the remaining sisters = 69/4 = 17.25 years.

2. Time, Speed, Distance Related Averages

Average Speed = (Total Distance Traveled)/(Total time taken).

If the distance is constant, using the two speeds, say a and b, the Average speed can be calculated by finding out the harmonic mean of the speeds â€˜aâ€™ & â€˜bâ€™ (discussed in Harmonic Mean before)

Average Speed = Harmonic mean of a & b = 2ab/((a+b))

Example 2:

Athul travels from his home to office at 60 km/hr and returns back home from office at 40km/hr.If he takes 3 hrs in all find the distance in km between his home and office?

a) 48 km

b) 72 km

c) 60 km

d) 92 km

Solution:

Average speed = (2Ã—60Ã—40)/100 = 48 kmph

Speed = ( Distance Traveled)/(Time taken)

Hence, distance between home and office = 1.5 x 48= 72 kms

Application in Series of Numbers Sum of 1st n consecutive natural numbers = (n(n+1))/2

Average of 1st n consecutive natural numbers = ((n+1))/2.

3. Averages Related to Numbers

• Average of 1st 7 consecutive natural numbers = 8/2 =4
• Sum of 1st n even consecutive natural numbers = n(n+1)
• Average of 1st n even consecutive natural numbers = (n+1)
• Average of 1st 3 even natural numbers = 4
• Sum of 1st n odd consecutive natural numbers = nÂ²
• Average of 1st n consecutive odd natural numbers = n
• Average of 1st 3 odd natural numbers = 3

Example 3:

Consider a sequence of seven consecutive integers. The average of first five integers is n. The average of all seven integers is:

a) n

b) n + 1

c) k x n, where k is a function of n

d) n +(2/7).

Solution:

Let’s solve this by assumption. Let the 7 consecutive integers be 1,2,3,4,5,6,7

Average of first five integers = 3

Average of first seven integers = 4.

Alternative Method:

Let a set of any 5 integers be a, a+1, a+2, a+3, a+4

Here, the average will be the middle term i.e. a+2.

It is given that a+2= n.

Now, similarly, a set of 7 integers can be expressed asÂ a, a+1, a+2, a+3, a+4, a+5, a+6.

In this set, the average will be the middle term i.e. a+3.

The term a+3 can be written as; a+2+1.

As a+2= n,

âˆ´a+3= n+1.

So, the answer is option (b).

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