Modelling the idea of quantity X be more frequent than quantity Y

Question

A recipe for fruit punch calls for 3 liters of orange juice and 2 liters of pineapple juice. Jo creates a drink mixing 4 liters of orange juice with 3 liters of pineapple juice.

Which recipe will have the stronger pineapple taste?

Me and my friend are arguing what would be the correct quantity can be derived from the given which would indicate the drink with greater pineapple taste.

Me: The pineapple taste is propotional to the total percent of pineapple in the mixture. So, for instance, in the first case with 3 liters orange juice and 2 liters of pineapple juice,and, 4 liters of orange and 3 liters of pineapple juice, we'd have that the mixture of 4 liters of orange and 3 liters of pineapple juice has more pineapple taste since it has a higher of pineapple juice (~40%) vs the lesser pineapple tasting juice having ~20%

Friend: The total pineapple taste would be propotional to the amount of pineapple juice per unit orange juice, hence the second juice would taste more of pineapple (3/4>2/3). Here are two pictures my friend made (which is also the reason I got convinced):

My friends arguements sounds right, but also seems wrong at the same time to me.

For instance, if we were to replace drinks with balls, and say orange juice is analogue to red balls , and pineapple juice is analogue to black balls. And ask the question, "which bag has more ball of red color?" similar situation would arise. Seems to me that this issue is indeed independent of cooking.

Could someone shed some light on this?

Answer 1

The two models are equivalent.

• If we measure the proportion of pineapple in the total, it is $\frac{2}{2+3} = 0.4$ for the first mix and $\frac{3}{3+4} \approx 0.42$ for the second mix.

• If we measure the fraction pineapple/orange, it is $\frac23 \approx 0.67$ for the first mix and $\frac34 = 0.75$ for the second mix.

Both answers say that the second mix has more pineapple.

In general, these methods will always give the same answers: $$ \frac{a}{a+b} > \frac{c}{c+d} \color{red}{\iff} \frac{a+b}{a} < \frac{c+d}{c} \iff 1+\frac ba < 1 + \frac dc \iff \frac ba < \frac dc \color{red}{\iff} \frac ab > \frac cd. $$ (The two implications highlighted in red are taking the reciprocal of both sides, so they reverse the direction of the inequality.)