Raju, Sam and Tristan had a total of $435. Raju spent $\frac{4}{5}$ of his money, Sam spent $\frac{2}{3}$ and Tristan spent $\frac{3}{4}$ 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? Ans : ___

1. Express the remaining money of each person in terms of units

Let Sam's remaining money be $1$ unit ($1u$).
Since Raju had $\char36{}55$ more than Sam in the end, Raju's remaining money is $1u + 55$.
Since Tristan had $\char36{}10$ more than Sam in the end, Tristan's remaining money is $1u + 10$.

2. Relate the remaining money to the total money at first using fractions

Raju spent $\frac{4}{5}$, so he had $\frac{1}{5}$ left. His total money at first was $5 \times (1u + 55) = 5u + 275$.
Sam spent $\frac{2}{3}$, so he had $\frac{1}{3}$ left. His total money at first was $3 \times 1u = 3u$.
Tristan spent $\frac{3}{4}$, so he had $\frac{1}{4}$ left. His total money at first was $4 \times (1u + 10) = 4u + 40$.

3. Form an equation for the total money at first and solve for 1 unit

Total money at first = $(5u + 275) + 3u + (4u + 40) = 12u + 315$.
We are given the total is $\char36{}435$.
$12u + 315 = 435$
$12u = 435 - 315 = 120$
$1u = 10$

4. Calculate Tristan's money at first

Tristan's money at first = $4u + 40$
$= 4(10) + 40$
$= 40 + 40 = 80$