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28 Jul 2026, 00:05
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Hi Bunuel - just wanted to know if this understanding is correct,
For a 50% percent orange juice, the cost is $.6, in case of 100% the cost becomes $1.2 and in case of 33.33% the cost becomes $.40.
For a 50% percent orange juice, the cost is $.6, in case of 100% the cost becomes $1.2 and in case of 33.33% the cost becomes $.40.
28 Jul 2026, 12:52
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Hi Rishab2409,
Good news: your numbers are all correct, and the way you're thinking about it is valid here. But it's worth knowing why it works, so you don't misapply it on a similar-looking question.
Your idea is that the price tracks the orange-juice concentration:
- Day 1: 50% OJ - $0.60
- Day 2: 33.33% OJ - $0.40
- (Pure OJ, 100% - $1.20)
That's exactly the answer, so nothing is broken. Here's the reason it lines up.
Why concentration works as a stand-in
The real driver in this problem is total volume, because the number of glasses is proportional to how much orangeade you make. Since revenue is equal on both days:
price × glasses = constant, so price is inversely proportional to volume.
Now look at concentration. Concentration = (orange juice) ÷ (total volume). The catch is that the amount of orange juice is the same on both days - that's stated in the problem. With OJ fixed, concentration is just OJ ÷ volume, which means concentration is itself inversely proportional to volume.
So both "price" and "concentration" move the same way - both scale as 1/volume. That's why price ∝ concentration gives you the right number. Check it: concentration ratio = (1/3) ÷ (1/2) = 2/3, and 0.60 × 2/3 = 0.40. Same 2/3 you'd get from the volume ratio (2 parts - 3 parts).
The one thing to watch
Your shortcut leans entirely on the OJ amount being constant across days. If a future question changed the amount of orange juice each day, concentration would no longer track 1/volume, and this reasoning would give the wrong price. So the safe, always-works version is: equal revenue - price ∝ 1/volume.
So yes - your understanding is correct, just anchor it to the constant orange juice, not to concentration on its own.
Answer: D
Good news: your numbers are all correct, and the way you're thinking about it is valid here. But it's worth knowing why it works, so you don't misapply it on a similar-looking question.
Your idea is that the price tracks the orange-juice concentration:
- Day 1: 50% OJ - $0.60
- Day 2: 33.33% OJ - $0.40
- (Pure OJ, 100% - $1.20)
That's exactly the answer, so nothing is broken. Here's the reason it lines up.
Why concentration works as a stand-in
The real driver in this problem is total volume, because the number of glasses is proportional to how much orangeade you make. Since revenue is equal on both days:
price × glasses = constant, so price is inversely proportional to volume.
Now look at concentration. Concentration = (orange juice) ÷ (total volume). The catch is that the amount of orange juice is the same on both days - that's stated in the problem. With OJ fixed, concentration is just OJ ÷ volume, which means concentration is itself inversely proportional to volume.
So both "price" and "concentration" move the same way - both scale as 1/volume. That's why price ∝ concentration gives you the right number. Check it: concentration ratio = (1/3) ÷ (1/2) = 2/3, and 0.60 × 2/3 = 0.40. Same 2/3 you'd get from the volume ratio (2 parts - 3 parts).
The one thing to watch
Your shortcut leans entirely on the OJ amount being constant across days. If a future question changed the amount of orange juice each day, concentration would no longer track 1/volume, and this reasoning would give the wrong price. So the safe, always-works version is: equal revenue - price ∝ 1/volume.
So yes - your understanding is correct, just anchor it to the constant orange juice, not to concentration on its own.
Answer: D
Rishab2409
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