Find the variance of $Y = X_1 X_2 / X_3$ where $X_i \sim$ Uniform(1,2) for $i = 1,2,3$ The problem is very straightforward and it says:

Let $X_1,X_2,X_3$ independent random variables with a common density $f_{x_i} \sim Uniform(1,2)$. Define $Y = X_1 X_2 / X_3$ Find the numeric value of $var(Y)$.

I propose the following transformation in order to compute the density of $Y_1$ through the margin distribution, and then compute the variance of $Y_1$:
$$ Y_1 = X_1 X_2 / X_3 $$
$$ Y_2 = X_2/X_3 $$
$$ Y_3 = X_3 $$
But I'm not getting any results. Is there an easy path to follow?
Disclaimer: I tried to solve the problem with this technique because I wanted to practice the topic of transformation of random variables.

Graphical comment: A simulation (in R) may give you some clues.
set.seed(2021);  m = 10^5
x1 = runif(m, 1,2)
x2 = runif(m, 1,2)
x3 = runif(m, 1,2)
y1 = x1*x2/x3
summary(y1);  sd(y1)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
 0.5134  1.1573  1.4795  1.5611  1.8731  3.8708 
[1] 0.5396972  # aprx SD(X1), about 2 place accuracy
y2 = x2/x3
summary(y2)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
 0.5026  0.8214  1.0018  1.0407  1.2195  1.9966 


par(mfrow=c(2,2))  # enable 4 panel plot
 hist(y1, prob=T, col="skyblue2")
 hist(y2, prob=T, col="skyblue2")
 hist(x1*x2, prob=T, col="skyblue2")
 plot(y1,y2, pch=".")
  abline(v=1:2, col="red")
par(mfrow=c(1,1))  # return to 1 panel plot

