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 & 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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