The price of artisanal ice cream is $60\%$ labor cost and $40\%$ ingredient cost. Production costs then experience increases: $32\%$ for labor, $52\%$ for ingredients. Write the law that expresses the price a of ice cream after the increases as a function of the price $b$ of ice cream before the increases and determine what kind of proportionality exists between $a$ and $b$. What is the total percentage increase in the price of ice cream?
My attempts: Let us call $p_1$ the initial price. It we know to be composed of $p_1=0.6x+0.4y$ where $x$ is the cost of manpower and $y$ is the cost of ingredients. Assuming $x=y=1$ we have $p_1=1$. Calling $p_2$ the price after increases, we can say that $$p_2=1.32\cdot 0.6x+1.52\cdot 0.4y=0.792x+0.608y$$ By posing again in this case $x=y=1$ we have that $p_2=1.4$. Thus $p_2=1.4\cdot 1 p_1$ that is, a direct proportionality. Trivially, the total percentage increase in the price of ice cream therefore turns out to be $40\%=0.4$.
Is there an alternative proof with the complete steps?
I'm a bit confused by your $x$ and $y$. The cost of Labour is $x = 0.6p_1$ and the cost of ingredients is $y = 0.4p_1$.
You should have $$ p_1 = \underbrace{0.4 p_1}_{\text{labour}} + \underbrace{0.6 p_1}_{\text{ingerdients}} $$ Then, $$ p_2 = (1.52)(0.4 p_1) + (1.32)(0.6p_1) = 1.4 p_1. $$