What is the Potato Paradox?
Okay, this is a very interesting question, and many people find it incredible the first time they hear it. Let's talk about the so-called "Potato Paradox."
What is the Potato Paradox?
The Potato Paradox isn't really a "paradox" in the true sense; it's more like a mind-bending mathematical puzzle that often makes people gasp. It reveals how unreliable our intuition can be when dealing with percentage problems.
The problem goes like this:
You have 100 kg of potatoes, and 99% of their weight is water.
You leave these potatoes out to dry for a day. Some water evaporates, and now the water content has dropped to 98%.
Question: How much do these potatoes weigh now?
What's your first guess? Many people would probably say: 99 kg? 98 kg? Something just a little less than 100 kg, right?
But the correct answer is: 50 kg.
Shocked? The weight was cut in half! Let me explain what's going on here.
Why does this happen? Where does our intuition go wrong?
Our intuition fails because we subconsciously assume that "water content dropping from 99% to 98%" is just a small "1%" change. However, we overlook a crucial point: this percentage is relative to a continuously changing total weight.
To solve this puzzle, the key is to identify the unchanging element. What doesn't decrease when the potatoes are dried in the sun?
It's the "potato solids" (which is the solid part of the potato, excluding water).
Let's break it down step by step
Don't worry, let's break it down using simple arithmetic.
1. Before Drying
- Total weight: 100 kg
- Water content: 99%
- Potato solids content: 100% - 99% = 1%
So, we can calculate the absolute weight of the "potato solids":
100 kg × 1% = 1 kg
Alright, remember this crucial number: No matter how much you dry them, this 1 kg of "potato solids" will not evaporate.
2. After Drying
Now, the potatoes have been dried for a while. What has changed?
- Water has evaporated, and the total weight has decreased (we don't know how much yet, let's call it X).
- The water content has dropped to 98%.
So, what is the percentage of "potato solids" now?
100% - 98% = 2%
See? Although the percentage of water only decreased by 1%, the percentage of "potato solids" doubled from 1% to 2%!
3. Establishing the equation and solving for the answer
Now we have a new relationship:
- Total weight after drying (X) × 2% = Weight of "potato solids"
And we know that the weight of "potato solids" has always been 1 kg. Therefore:
X * 2% = 1 kg
Solving this equation:
X = 1 kg / 2%
X = 1 kg / 0.02
X = 50 kg
There you have it, the answer. The total weight of the potatoes after drying is 50 kg.
Another perspective to fully grasp it
If you still find it a bit confusing, let me give you another analogy.
Imagine you have a "drink" that contains 1 sugar cube (potato solids) and 99 drops of water (water content). At this point, the concentration of the sugar cube is 1/100, or 1%.
Now you want to double the concentration of the sugar cube in this drink to 2%. How would you do that?
You can't add more sugar cubes, because the amount of "potato solids" is fixed. The only thing you can do is remove water.
For 1 sugar cube to have a concentration of 2% (which is 1/50), the total "stuff" (sugar cube + water) in your cup can only be 50 parts. So, you need to reduce the water from 99 drops to 49 drops.
1 sugar cube + 49 drops of water = 50 total parts
Now, the concentration of the sugar cube is 1 / 50 = 2%.
You see, to increase the sugar cube's concentration from 1% to 2%, you have to reduce the total solution by half. The Potato Paradox works on exactly the same principle.
Summary
The core of the Potato Paradox lies in:
- Identify the invariant: In this problem, it's the absolute weight of the "potato solids".
- Understand the relative relationship: When the total weight decreases, the "percentage" of the invariant part drastically increases. The "doubling" effect from 1% to 2% directly leads to the total weight being "halved".
So, it's not a logical paradox, but rather a mathematical problem that challenges our intuition. The next time someone tests you with this question, you'll be able to explain the reasoning to them!