Question:
If the average of five consecutive even numbers is 16, then find the smallest and the greatest numbers.
Correct Answer
12 and 20
Solution And Explanation
Solution
Some Important Terms Used in the given Question
Consecutive Numbers: Numbers in the 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 the 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.
Now, Solution of the given Question
Let first even number = x.
Therefore, the second consecutive even number = x + 2
And, the third consecutive even number = x + 4
And, the fourth consecutive even number = x + 6
And, the fifth consecutive even number = x + 8
Now,
the formula to find the
Average = Sum of observations/Numbers of observations
Therefore, the average of the given five consecutive even numbers
= Sum of the five consecutive even numbers/5
⇒ 16 = x+(x+2)+(x+4)+(x+6)+(x+8)/5
⇒ 16 = x+x+2+x+4+x+6+x+8/5
⇒ 16 = 5x + 20/5
After taking 5 as common from the numerator
⇒ 16 = 5 (x + 4)/ 5
⇒ 16 = x + 4
⇒ x + 4 = 16
After transposing 4 to RHS
⇒ x = 16 – 4
⇒ x = 12
Thus, first and the smallest even number = 12
And, the 5th and the greatest even number = x + 8
= 12 + 8 = 20
∴ The 5th and the greatest even number = 20
Therefore, the smallest and the greatest even numbers among the given consecutive five even numbers = 12 and 20 Answer
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