ggplot2 How to align graphs with equal y scale as percentage

Question

I'm trying to obtain 2 aligned bar plots with percentage scales of 2 different factors. The y scales calculated as percent are different. I would like to have the same y scales for both plots, for example from 0 to 40% in both. I've tried ylim() which doesn't work on the percentage scale. Example below

library(ggplot2)
library(scales)
data("diamonds")

First bar-plot for cut

  p<- ggplot(diamonds, aes(x = cut)) +  
    geom_bar(aes(y = (..count..)/sum(..count..), fill=cut)) + 
    scale_y_continuous(labels = percent) +
    geom_text(aes(y = ((..count..)/sum(..count..)), label = 
    scales::percent((..count..)/sum(..count..))), 
            stat = "count", vjust = -0.25) +
    ggtitle("Cut") + theme(plot.title = element_text(hjust = 0.5, size=14, 
    face="bold")) + 
    xlab("Cut") +
    ylab("Percent") +
    theme(legend.position="bottom")

Second bar-plot for clarity

p1<- ggplot(diamonds, aes(x = clarity)) +  
geom_bar(aes(y = (..count..)/sum(..count..), fill=clarity)) + 
scale_y_continuous(labels = percent) +
geom_text(aes(y = ((..count..)/sum(..count..)), label = 
scales::percent((..count..)/sum(..count..))), 
        stat = "count", vjust = -0.25) +
ggtitle("Clarity") + theme(plot.title = element_text(hjust = 0.5, size=14, 
face="bold")) + 
xlab("Clarity") +
ylab("Percent") +
theme(legend.position="bottom")

Arranging bar-plot with different scales

  grid.arrange(p,p1, ncol = 2)  

different scales but I would like for example both at 40% top

If scales weren't percentages I would do this:

p<- ggplot(diamonds, aes(x = cut)) +  
  geom_bar(aes(y = (..count..)/sum(..count..), fill=cut)) + 
  scale_y_continuous(labels = percent) +
  geom_text(aes(y = ((..count..)/sum(..count..)), label = scales::percent((..count..)/sum(..count..))), 
            stat = "count", vjust = -0.25) +
  ggtitle("Cut") + theme(plot.title = element_text(hjust = 0.5, size=14, face="bold")) + 
  xlab("Cut") +
  ylab("Percent") +
  ylim(0, 40)
  theme(legend.position="bottom")

But here, of course, it doesn't work and returns this:

Answer 1

Ok I found a way, here the code for Cut for a % scale limited to 60%

p<- ggplot(diamonds, aes(x = cut)) +  
  geom_bar(aes(y = (..count..)/sum(..count..), fill=cut)) + 
  geom_text(aes(y = ((..count..)/sum(..count..)), label = scales::percent((..count..)/sum(..count..))), 
            stat = "count", vjust = -0.25) +
  ggtitle("Diamonds Cut") + theme(plot.title = element_text(hjust = 0.5, size=14, face="bold")) + 
  scale_y_continuous(labels = scales::percent, limits=c(0,0.6)) + labs(y="Percent")
xlab("Cut") +
  theme(legend.position="bottom")
p