Question:
If the average of three consecutive even numbers is 14, then find the numbers.
Correct Answer
12, 14, and 16
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 = 14
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 = 14
Therefore, the first even number among the given three consecutive even numbers
= Number in the middle – 2
= 14 – 2 = 12
And, last even number among the given three consecutive numbers
= Number in the middle + 2
= 14 + 2 = 16
Thus, numbers are 12, 14, and 16 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 = 14
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
⇒ 14 = a+(a+2)+(a+4)/3
⇒ 14 = a+a+2+a+4/3
⇒ 14 = 3a + 6/3
After taking 3 as common from the numerator
⇒ 14 = 3 (a + 2)/ 3
⇒ 14 = a + 2
⇒ a + 2 = 14
After transposing 2 to RHS
⇒ a = 14 – 2
⇒ a = 12
Thus, first even number = 12
And, the second consecutive even number = a + 2
= 12 + 2 = 14
And, the 3rd and even number = a + 4
= 12 + 4 = 16
Therefore, the given consecutive three even numbers = 12, 14 and 16 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.
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