Find only one share

$\begin{cases} \dfrac ab = \dfrac 34 & \implies b = \dfrac 43a \\ \dfrac bc = \dfrac 34 & \implies c = \dfrac 43b = \dfrac {16}9a \end{cases}$

We know that

$\begin{aligned} a+b+c=1 \\ \left(1+\frac 43 + \frac {16}9 \right)a & = 1 \\ \frac {37}9a & = 1 \\ \implies a & = \frac 9{37} \end{aligned}$

Therefore, Alex's share of $370 is $\$370 \times a = \$370 \times \dfrac 9{37} = \$\boxed{90}$ .

Lets work out everything in terms of " $b$ " (Bond's share) first,

$\dfrac{a}{b}$ $=$ $\dfrac{b}{c}$ $=$ $\dfrac{3}{4}$

$\dfrac{a}{b}$ $=$ $\dfrac{3}{4}$ $\Rightarrow a =$ $\dfrac{3}{4}$ $b$

$\dfrac{b}{c}$ $=$ $\dfrac{3}{4}$ $\Rightarrow$ $\dfrac{c}{b}$ $=$ $\dfrac{4}{3}$ $\Rightarrow c = \dfrac 43b$

The total amount to be divided is $\$370$ . So,

$\Rightarrow$ $\dfrac 34b +$ $b +$ $\dfrac 43b$ $= 370$

$\Rightarrow$ $\dfrac{(9b + 12b + 16b)}{12}$ $= 370$

$\Rightarrow$ $\dfrac{37b}{12}$ $= 370$

$\Rightarrow 37b = 4440$ ;[Multiplying $370$ with $12$ ]

$∴ b = 120$

Now we know that Bond's share is $120$

So, Alex's share $\Rightarrow a =$ $\dfrac{3}{4}$ $\times 120$ $= 90$

Therefore $a = \$90$