Let the smallest odd number be
. Then, the three consecutive odd numbers will be
. The sum of the three consecutive odd numbers is:
x + (x + 2) + (x + 4) = 3x + 6
Let the smallest even number be
. Then, the four consecutive even numbers will be
. The sum of the four consecutive even numbers is:
y + (y + 2) + (y + 4) + (y + 6) = 4y + 12
We are told that the sum of the four consecutive even numbers is 107 more than the sum of the three consecutive odd numbers. Therefore, we can write the equation:
4y + 12 = (3x + 6) + 107
Simplifying:
4y + 12 = 3x + 113
4y = 3x + 101 {(Equation 1)}
We are also told that the sum of the smallest odd number and the smallest even number is 55. Therefore:
x + y = 55 {(Equation 2)}
Now, solve these two equations simultaneously. From Equation 2:
y = 55 - x
Substitute this into Equation 1:
4(55 - x) = 3x + 101
220 - 4x = 3x + 101
220 - 101 = 3x + 4x
119 = 7x
x = 119 / 7 = 17
Substitute
into Equation 2:
17 + y = 55
y = 55 - 17 = 38
Thus, the smallest even number is
.