mcqs Math Average If the average of three consecutive even numbers is 16, then find the numbers. 161

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Question: If the average of three consecutive even numbers is 16, then find the numbers.

(A)  14, 16, and 18
(B)  12, 14, and 16
(C)  14, 16, and 18
(D)  10, 12, and 14

Correct Answer 14, 16, and 18

Solution And Explanation

Solution

Shortcut Trick to find the numbers if the average of three consecutive numbers are given

Here, given

The average of three consecutive even numbers = 16

This means the number lies in the middle of the given three consecutive numbers will be the average

Therefore, number lies in middle of the given three consecutive even numbers = 16

Therefore, the first even number among the given three consecutive even numbers

= Number in the middle – 2

= 16 – 2 = 14

And, last even number among the given three consecutive numbers

= Number in the middle + 2

= 16 + 2 = 18

Thus, numbers are 14, 16, and 18 Answer

Procedure to find the numbers if the average of three consecutive even numbers are given

It is necessary to know the procedure, as it applies to solving all similar types of given problems, as shortcut tricks are not applicable everywhere.

Given, the average of three consecutive even numbers = 16

Thus, the smallest and the greatest number among the given three even numbers = ?

Let first even number = a

Therefore, the second consecutive even number = a + 2

And, the third consecutive even number = a + 4

Now,

Formula to find the

Average = ^{Sum of observations}/_{Numbers of observations}

Therefore, the average of the given three consecutive even numbers

= ^{Sum of the three consecutive even numbers}/₃

⇒ 16 = ^{a+(a+2)+(a+4)}/₃

⇒ 16 = ^a+a+2+a+4/₃

⇒ 16 = ^{3a + 6}/₃

After taking 3 as common from the numerator

⇒ 16 = ^{3 (a + 2)}/₃

⇒ 16 = a + 2

⇒ a + 2 = 16

After transposing 2 to RHS

⇒ a = 16 – 2

⇒ a = 14

Thus, first even number = 14

And, the second consecutive even number = a + 2

= 14 + 2 = 16

And, the 3rd and even number = a + 4

= 14 + 4 = 18

Therefore, the given consecutive three even numbers = 14, 16 and 18 Answer

Some Important Terms Used in the given Question

Consecutive Numbers: Numbers in continuation are called Consecutive Numbers, for example, 1, 2, 3, 4, . . . etc.

Even Number:
Numbers that can be divided exactly by 2, i.e. leaving no remainder after dividing by 2, are known as Even Numbers

Consecutive Even Numbers:
Even numbers in continuation are called Consecutive Even Numbers, such as 2, 4, 6, 8, . . . . etc.

How to get next consecutive even numbers to the given even number

Here, it can be observed easily, that there is a difference of 2 between two consecutive even numbers. This means the next consecutive even number can be obtained by the addition of 2 in an even number.

Example:

If 2 is the first even number, then second consecutive even number to 2

= 2 + 2 = 4

Similarly, the third consecutive even number to 2

= 2 + 2 + 2 = 2 + 4 = 6

And, the fourth consecutive even number to 2

= 2 + 2 + 2 + 2 = 2 + 6 = 8 and so on.

In short, next consecutive even number can be obtained by adding 2 in an even number.