math word problem!! how old is....?

Question

the children's ages combined is three times the age of mike, who is three years younger than Shawn. The cube [^3] of Shawn's age, plus the square [^2] of Mikes age plus Laura's age totals a multiple of 100. IF ALL THE AGES ARE IN YEARS, HOW OLD IS LAURA?
(also tell the ages of the other children anyway)
showing work would help u get closer to the ten points but make sure its readable!

Answer 1

Laura is 5, Mike is 8, and Shawn is 11.

If Shawn is three years older than Mike and the three ages average out to be Mikes age, that would mean that Laura would have to be exactly 3 years younger than Mike. You can create the formula x + (x+3)^2 + (x+3+3)^3 where x is Laura's age. Just start where x = 1 and see if the answer is a multiple of 100. If not go to 2,3,4,5 and so on untill you get an answer that is a multiple of 100.

Answer 2

How many times are you going to ask this?!?!

Here are the equations:

M + S + L = 3M, or S + L = 2M or S = 2M - L
M + 3 = S
S^3 + M^2 + L is a multiple of 100 (this isn't that useful yet)

Using the valule for S (eq 1) in eq 2 we get
M + 3 = 2M - L or 3 = M - L or 3+ M = L.

So now we know that Mike is three years younger than Shawn, but also three years older than Laura.

We also know that Shawn and Laura's age must be odd. Why?
Since they are all 3 years apart, there are two possible patterns -
Odd Even Odd or Even Odd Even.
Applying the last equation to the first combination, we will end up with an odd + even + odd which adds to an even.
But if we apply it to the second combination, we will end up with an even + odd + even, which is odd and cannot end up being a multiple of 100.

The numbers that fit this pattern are 5, 8 and 11. Laura is 11 years old.

Algebraically, this is (M+3)^3 + M + (M - 3), or
(m^3 + 9m^2 + 18M + 27) + 2M -3, or
m^3 + 9m^2 + 20M + 24