## Average

**Important terms:**

Average is defined as the ratio of sum of all terms in a group to the number of items in the group.

**Important Formulae:**

**1)**If number of quantities/items/members etc. and their total quantity are given, then we can find the average by using formula shown below. Sometimes, averages and total quantities are mentioned in the question and we have to determine number of quantities.

Hence, remember this formula can be used in number of ways to solve the numericals on ages/weights/ marks/ etc.

Average = | Sum of Quantities |

Number of Quantities |

**2)**A person travels a distance at a speed of V

_{1}km/hr and same distance at a speed of V

_{2}km/hr. His average speed in the whole journey can be determined using the formula shown below:

Average Speed = | (2 V_{1} V_{2}) |

(V_{1} + V_{2}) |

**3)**There are two batches A and B in a class. If we have to find the average of the whole class use the formula shown below:

Batch A: Number of students = a

Average of batch A = x

Batch B: Number of Students = b

Average of batch B = y

Average of whole class (Batch 1 and Batch 2) = | (ax + by) |

(a + b) |

**Quick Tips and Tricks:**

**1)**Average of n natural numbers = (n + 1) / 2

**2)**Average of even numbers = (n + 1)

**3)**If value of each term increases/decreases by x, then the average of the group also increases/decreases by a.

**4)**If we know average of two groups individually, then the average of combined group cannot be determined.

**5)**In Arithmetic Progression, if number of terms are

**i) Odd –**Average is the middle term.

**ii) Even –**Average is the average of two middle terms.

**6) Numericals on Age:**

i) When new member is added in the group/family.

Case A: If average age increases

Age of new member added = Given previous average + ( Increase in average after new member is added x Total members including new member)

Case B: If average age decreases

Age of new member added = Given previous average - ( Decrease in average after new member is added x Total members including new member)

ii) If a person is replaced by another person in a group

Case A: If average age is increased, then

Age of new member = Age of separated member + (Increase in average after new member is added x Total members including new member)

Case B: If average age is decreased, then

Age of new member added = Age of separated member – (Decrease in average after new member is added x Total members including new member)

Type 1 : Average of Numbers

**Examples:**

**Q 1.**Find the average of all numbers between 5 and 49 which are divisible by 5.

a. 20

b. 25

c. 30

d. 35

View solution

Correct Option: (b)

The numbers divisible by 5 are: 5, 10, 15, 20, 25, 30, 35, 40, 45.

Average = | Sum of Quantities | = | (5 + 10 + 15 + 20 + 25 + 30 + 35 + 40 + 45) | = | 225 | = 25 |

Number of Quantities | 9 | 9 |

**Q 2.**The average of 11 numbers is 30. If the average of first six numbers is 17.5 and that of last six is 42.5, then what is the sixth number?

a. 30

b. 36

c. 45

d. 47

View solution

Correct Option : (a) **Given:** Average of 11 numbers = 30

Step 1: Calculate total of 11 numbers by multiplying it by average value 30 = 11 x 30 = 330

Step 2: Calculate total of first six members by multiplying it by average value 17.5 = 17.5 x 6 = 105

Step 3: Calculate total of last six members by multiplying it by average value 42.5 = 42.5 x 6 = 255

Therefore, we can find sixth number by adding value of first six and last six numbers and subtracting it from the total value of 11 numbers.

Sixth number =(105 + 255)- 330 = 30

**Q 3.**The average of 15 numbers is 15. If the average of first five numbers is 14 and that of other 9 numbers is 16, then find the middle number.

a. 12

b. 11

c. 10

d. 9

View solution

Correct Option: (b)**Given:** Average of 15 numbers = 15, Average of 5 numbers = 14, Average of 9 numbers = 16

Average = | Total Numbers |

No. of Numbers |

15 = | Total Numbers |

15 |

Therefore, total numbers = 15 x 15 = 225

Middle number = (Total numbers) – [(Average of 5 num x no of num) + ( Average of 9 num x no of num)]

= (225) – [(14 x 5) + (16 x 9)]

= (225) – [214]

= 11

Therefore, the middle number is 11

**Q 4.**The average of four consecutive even numbers is 27. Find the largest of these numbers.

a. 28

b. 30

c. 32

d. 34

View solution

Correct Option: (b)

Consider the consecutive even numbers as : x, (x + 2), (x + 4) and (x+ 6)

Average = | Sum of Quantities |

Number of Quantities |

= | x + (x + 2) + (x + 4) + (x + 6) | = | (4x + 12) | = 27 |

4 | 4 |

Simplifying we get, x = 24

Therefore,

Largest number = (x + 6) = (24 + 6) = 30

Smallest number = 24