Question

If twice the sum of 5 consecutive odd numbers and thrice the sum of 4 consecutive even numbers is 178 and thrice the sum of 5 consecutive odd numbers and twice the sum of 4 consecutive even numbers is 177, then find the middle number when all the numbers are arranged in ascending order.

• A. 9
• B. 8
• C. 7
• D. 6

Answer 1

The correct option is B

8

Let the odd numbers be $x,x+2,x+4,x+6andx+8.$
Their Sum = $5x+20$
Let the even numbers be $y,y+2,y+4andy+6.$
Their Sum = $4y+12$
Given that:
$2(5x+20)+3(4y+12)=178$ and
$3(5x+20)+2(4y+12)=177$
Simplifying the above two equations, we get
$5x+6y=51$ ...(1)
$15x+8y=93$ ...(2)
Multiplying (1) by 3,
$15x+18y=153$ ...(3)
Subtracting (2) from (3), we get
$10y=60\Rightarrow y=6$
Substituting $y=6$ in (1), we get $x=3.$
Therefore, the odd numbers are 3, 5, 7, 9, 11
and the even numbers are 6, 8, 10, 12.


Arranging them in ascending order,
the numbers are 3,5,6,7,8,9,10,11,12.
Hence, the middle number is 8.