Raju, Sam and Tristan had a total of $435. Raju spent 4/5 of his money, Sam spent 2/3 and Tristan spent 3/4 of his money. In the end, Raju had $55 more than Sam and Tristan had $10 more than Sam. How much money did Tristan have at first?
Let's assume Raju had R dollars, Sam had S dollars, and Tristan had T dollars at first.
R + S + T = $435 ...(1) (since the total money they had is $435)
R - 4/5R = 2/3S + $55 ...(2) (since Raju spent 4/5 of his money and had $55 more than Sam)
T - 3/4T = 2/3S + $10 ...(3) (since Tristan spent 3/4 of his money and had $10 more than Sam)
Simplify equations (2) and (3):
1/5R = 2/3S + $55
1/4T = 2/3S + $10
Simplify further by multiplying both equations by 15 to eliminate the fractions:
3R = 10S + $825 ...(4)
15T = 10S + $150 ...(5)
Now, subtract equation (4) from equation (5):
15T - 3R = 10S + $150 - (10S + $825)
15T - 3R = $975 - $825
15T - 3R = $150 ...(6)
We have three equations, (1), (4), and (6). Let's solve them simultaneously.
From equation (1):
435 - S - T = R ...(7)
Substitute equation (7) into equation (4):
3R = 10S + $825
3(435 - S - T) = 10S + $825
1305 - 3S - 3T = 10S + $825
-3T - 13S = $825 - $1305
-3T - 13S = -$480
3T + 13S = $480 ...(8)
Now, solve equations (6) and (8) simultaneously:
15T - 3R = $150
3T + 13S = $480
From equation (8):
3T = $480 - 13S ...(9)
Substitute equation (9) into equation (6):
15($480 - 13S) - 3R = $150
7200 - 195S - 3R = $150
-3R - 195S = $150 - $7200
-3R - 195S = -$7050 ...(10)
Multiply equation (7) by 3:
3(435 - S - T) = 3R
1305 - 3S - 3T = 3R
-3R = -1305 + 3S + 3T
-3R = 3S + 3T - 1305
-3R = 3(S + T - 435)
-3R - 3(S + T) = -1305
-3(R + S + T) = -1305
3(R + S + T) = 1305
3($435) = 1305
$1305 = $1305
I'm sorry for this redundant computation. The result is simply an identity equation, meaning the result is always true regardless of the values of R, S, and T. It doesn't help us obtain specific values of R, S, and T.
Therefore, the problem lacks information to find the amount of money Tristan had at first.