Question : The average of 8 numbers is 44. The average of the first three numbers is 50 and the average of the next two numbers is 52. If the sixth number is 6 and 8 less than the seventh and eighth numbers respectively, then what is the value of the eighth number?
Option 1: 36
Option 2: 32
Option 3: 40
Option 4: 56
Correct Answer: 36
Solution :
We know that,
Average = $\frac{\text{Sum of all the observations}}{{\text{Total number of observations}}}$
Let the eight numbers as $a$, $b$, $c$, $d$, $e$, $f$, $g$, and $h$ respectively.
From the problem,
The average of 8 numbers is 44.
$⇒\frac{a+b+c+d+e+f+g+h}{8} = 44$ ____(1)
The average of the first three numbers is 50.
$⇒\frac{a+b+c}{3} = 50$ ____(2)
The average of the next two numbers is 52.
$⇒\frac{d+e}{2} = 52$ ____(3)
The sixth number is 6 less than the seventh number.
$⇒f = g - 6$ ____(4)
The sixth number is 8 less than the eighth number.
$⇒f = h - 8$ ____(6)
From equation 2,
$⇒a+b+c = 150$
From equation (3),
$d+e = 104$
Substituting $a+b+c$ and $d+e$ into equation (1),
$⇒f+g+h = 44 \times8 - 150 - 104 = 98$
From equations (4) and (5),
$⇒g = f + 6$ and $h = f + 8$
Substituting $g$ and $h$ into the equation $f+g+h = 98$,
$⇒3f + 14 = 98⇒f = \frac{98 - 14}{3} = 28$
Substituting $f = 28$ into the equation $h = f + 8$,
$⇒h = 28 + 8 = 36$
Hence, the correct answer is 36.
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