Matthew, Nigel and Oscar had 84 toy soldiers altogether. Matthew gave 1/2 of his toy soldiers to Nigel. Then Nigel gave 3/7 of his soldiers to Oscar.
Finally, Oscar gave 1/5 of his soldiers to Mathew. In the end, each person had the same number of toy soldiers. How many more soldiers did Matthew have than Oscar initially?
Let x be the initial number of toy soldiers that Matthew had.
Matthew gave 1/2*x toy soldiers to Nigel, so Nigel had x/2 soldiers.
Nigel gave 3/7*(x/2) = 3/14*x toy soldiers to Oscar, so Oscar had (1-3/14)*x = 11/14*x toy soldiers.
Oscar gave 1/5*(11/14*x) = 11/70*x toy soldiers to Mathew, so Mathew had x-(11/70*x) = 59/70*x toy soldiers in the end.
Each person had 59/70*x = x/3 toy soldiers.
Thus, 59/70*x = x/3.
Multiplying both sides by 3*70 = 210, we get 177x = 70x.
Therefore, Matthew initially had 177-70 = <<177-70=107>>107 more toy soldiers than Oscar. Answer: \boxed{107}.
Let's start by setting up equations for each step of the problem.
Let's assume that initially Matthew had M toy soldiers, Nigel had N toy soldiers, and Oscar had O toy soldiers.
According to the problem, Matthew gave half of his soldiers to Nigel, so Nigel received (1/2)M soldiers. This means that Nigel's new number of soldiers is N + (1/2)M.
Then, Nigel gave 3/7 of his soldiers to Oscar, which means Oscar received (3/7)[N + (1/2)M] soldiers. This means that Oscar's new number of soldiers is O + (3/7)[N + (1/2)M].
Finally, Oscar gave 1/5 of his soldiers to Matthew, which means Matthew received (1/5)[O + (3/7)[N + (1/2)M]] soldiers. This means that Matthew's new number of soldiers is M + (1/5)[O + (3/7)[N + (1/2)M]].
According to the problem, after these exchanges, each person has the same number of toy soldiers. So we can set up an equation:
M + (1/5)[O + (3/7)[N + (1/2)M]] = N + (1/2)M = O + (3/7)[N + (1/2)M]
We can now solve this equation to find the initial number of soldiers each person had.
To simplify the equation, let's get rid of the fractions by multiplying both sides by the least common multiple of the denominators (2, 5, and 7), which is 70:
70M + 14[O + 21[N + 35M]] = 70N + 35M = 70O + 30[N + 15M]
Simplifying further:
70M + 14O + 294N + 1029M = 70N + 35M = 70O + 30N + 450M
Rearranging the terms:
70M + 1029M - 35M - 450M = 70N - 294N + 30N - 14O - 70O
654M = -234N - 84O
Now we can use the information that the total number of toy soldiers is 84 to further simplify the equation:
M + N + O = 84
Now we have a system of two equations:
654M = -234N - 84O
M + N + O = 84
To solve this system of equations, we can use substitution or elimination. Let's use elimination by multiplying the second equation by 234 to match the coefficients of M in both equations:
234M + 234N + 234O = 19776
Subtracting this from the first equation, we get:
654M - 234M = -234N - 84O - 234M
420M = -234N - 234O - 19776
Now we can substitute this back into the second equation:
420M + N + O = 84
Substituting -234N - 234O - 19776 for 420M:
-234N - 234O - 19776 + N + O = 84
Combining like terms:
-233N - 233O - 19776 = 84
Adding 19776 to both sides:
-233N - 233O = 19860
Now we can divide both sides by -233:
N + O = -19860 / -233
N + O ≈ -85.14
Since the number of toy soldiers cannot be negative, we can conclude that there is no solution to this problem. Therefore, we cannot determine how many more soldiers Matthew initially had than Oscar.