A Classroom Episode
The bell had just rung when Mr. Sharma walked into the classroom with a strange smile.
He asked a question.
Imagine we have 100 kg of fresh mushrooms. These mushrooms are 99% water. We leave them out in the sun for a day. The next morning, they are 98% water. What do you think their new weight is?”
A few students looked at one another. Then Rahul raised his hand with confidence.
“Sir, it lost 1% water. So, 1% of 100 kg is 1 kg. The new weight must be 99 kg.”
Several students nodded immediately.
Dilip felt something different. Water is not 100%. It's 99%. The loss of 1% is in water content. So only 0.99 kg is lost. And new weight should be 99.01 kg. This looked more convincing. Several students including Rahul agreed on this.
Mr. Sharma did not reject or accept the answer. He waited as few students were still working. They were looking enthusiastic.
Suddenly, Neha raised her hand and said, "Sir, the new weight should be 50 kg."
Many students laughed. How come a reduction of 1% bring down the weight from 100 kg to 50 kg!
The teacher smiled and said, “Good. Let us test the answers.”
He called Neha to the board and asked her to defend her answer.
The Trap of the Obvious Answer
Neha being a shy student came to the board but was nervous to start. Sensing that, Mr. Sharma turned to the class and asked, “What are mushrooms made of?”
“Water and solid matter,” came the reply.
“Excellent. Before drying, if the mushrooms weigh 100 kg and are 99% water, how much water is there?”
“99 kg,” said the class.
“And how much solid matter?”
“1 kg.”
Mr. Sharma nodded. “Now listen carefully. The sun can evaporate water. Can it evaporate the solid mushroom pulp too?”
“No,” said Neha quickly. “Only the water leaves.”
“Exactly,” he said. “So, what happens to the solid matter?”
“It stays 1 kg,” she answered and continued.
"Dried Mushroom has 98% water content so this 1 kg solid matter makes remaining 2% of it. Since 1 kg is 2% of the mushroom now, the weight of the mushroom is now 50 kg as 2% of 50 kg is 1 kg."
The class was stunned finding it hard to believe. There must be something wrong. It can't be true, some of the students were whispering.
The teacher suggested students to check this answer by finding the water content of this 50 kg mushroom. They realized that in this, 1 kg is solid part which remained as it was and remaining 49 kg is water after rest of the water got evaporated. This is exactly 98% of 50 kg. That means, water content in mushroom now is indeed 98%.
Now the room became quieter.
Dilip was wondering what went wrong with his approach. So, he asked the teacher to help him understand his mistake. He was knowing that 1% reduction was in water only. So, he had reduced 1% of 99 from 100 kg to get 99.01 kg.
The class felt that Rahul answer was clearly wrong as he had reduced 1% of the total weight. But Dilip had reduced 1% of water only as asked in the question. What went wrong with it?
Mr. Sharma wrote the numbers on the board.
Total weight = 99.01 kg
Solid matter = 1 kg
Water = 98.01 kg
This was as per the answer given by Dilip.
Then he asked, “What percentage of 99.01 kg is 98.01 kg?”
The students began calculating. It was roughly 98.99%.
The class understood and applauded Neha for her clear and correct thinking. Neha felt confident and said, "the main point to understand here is that if something has very large share then a small reduction in its share in terms of ratio or percentage will require significant reduction in its quantity."
Imagine a class of 100 has 99% girls. Means there are 99 girls and 1 boy. Suppose only number of girls can be changed not boys, and we want class composition as 98% girls and remaining 2 % boys. Reducing number of girls will help get this composition only when that 1 boy forms 2% of the class which means 50 students in the class. The common concept about ratio, proportion, percent etc. get challenged quite often when composition is heavily disproportionate.
The class had experienced this.
Rahul first answer had been based on additive intuition: if the water percentage drops by 1%, then the weight should also drop by 1%.
Dilip was more careful as he noticed the change was only in one component. But he missed the point that the change in percent was not in that component but was in the overall composition of mushroom.
Applying first principle or earth element of PanchTatva Learning is quite helpful here.
The Next Exercise: 50% Water
Just as the students began relaxing, Mr. Sharma picked up the chalk and gave a twist.
“Now for one more challenge,” he said. “Suppose the mushrooms dry further until they are 50% water. What is their total weight now?”
This time nobody shouted an answer.
They had learned caution.
Ananya spoke first. “If water is 50%, then solids are also 50%.”
“Correct,” said Mr. Sharma.
“And the solid matter is still 1 kg,” she continued.
“So, 50% of the total weight is 1 kg.”
Mr. Sharma nodded.
Rahul was already working it out.
“If 50% is 1 kg, then 100% must be 2 kg.”
Mr. Sharma pointed at him. “Exactly. Two kilograms.”
That answer landed even harder than the first one.
From 100 kg to 50 kg at 98% water.
From 50 kg to 2 kg at 50% water.
The students could finally feel the nonlinearity in their bones.
A percentage does not behave like a linear step when the total itself is changing.
That was the hidden structure.
Mr. Sharma wanted to take students to some practical implementation of such concepts. So, he gave another version of same problem and wanted students to use that in making a decision.
The Businessman’s Version of the Same Problem
“Let us use the same idea in a business problem,” he said.
A businessman buys 200 kg of fresh mushrooms at ₹120 per kg.
On testing, it is found that these mushrooms are 90% water. He wants to dry them and sell in packs of 100 grams of dried mushrooms. For long shelf life, mushrooms should be dried to 10% water content.
He sends them for drying, and the moisture drops to 10% water.
The drying facility charges ₹3,000 for one lot of mushrooms drying irrespective of the lot quantity.
If he wants a 20% profit on his total cost, at what price per pack of 100 g should he sell the dried mushrooms?
This time the students did not rush.
They had learned to look for the invariant first.
If the mushrooms are 200 kg and 90% of that is water, then the solid matter is 10% of 200 kg.
That means the solid mass is 20 kg.
That 20 kg does not disappear.
It remains the same after drying.
Now the mushrooms are 10% water, so they are 90% solid.
So, 90% of the dried weight must equal 20 kg.
That gives the dried weight as 20 ÷ 0.9 = 22.222... kg.
Then they calculated the cost.
Purchase cost = 200 × ₹120 = ₹24,000
Drying fee = ₹3,000
Total cost = ₹27,000
For a 20% profit, the selling price must be:
₹27,000 × 1.2 = ₹32,400
Now divide by the dried weight:
₹32,400 ÷ (20 ÷ 0.9) = ₹1,458 per kg
The class stared at the number.
₹1,458 per kg.
The mushrooms began at ₹120 per kg, but after drying, the price had to jump dramatically just to make a modest profit.
Now the business logic made sense.
The weight had collapsed, the solids had remained, and the price per kilogram had to rise. So, a 100 g pack of dried mushroom should be priced at ₹1,45.8 to generate 20% profit.
Going Deeper
When the bell rang, Mr. Sharma left few questions for the students to ponder:
What if the businessman wants to decrease the price of 100 g pack without changing the profit percent. Purchase cost and drying cost per lot too can't be changed.
If his purchase cost and drier cost remains same, what is the minimum selling price he has to keep to avoid a loss.
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