Lalu works hard but no profit
Lalu had a bakery in the small Indian town of Bahadurganj which had a very low rate of literacy. It was a small bakery where he worked with his son Bholu, and one servant. He barely made a living. His wife Laxmi worked in Thakur Raghuveer Singh’s household and had become friendly with their 12 year old granddaughter Mani. One Saturday when Laxmi came to work, she looked tired as if she did not sleep the night before.
Mani to her mom: Mom, looks like Laxmi aunty is not well.
Mom: Laxmi, are you alright?
Laxmi said with tears in her eyes: I should not be bringing my troubles here to work. I will be alright.
Mani: Laxmi aunty, you have to tell us. Maybe we can help. Otherwise, how will your troubles end on their own?
Laxmi: Beti (daughter), you don’t have to worry.
Mani: You called me your beti. Now you have to tell me. If your son Bholu had asked you, would have told him. You have to tell your beti Mani too.
Laxmi: Lalu and Bholu have been working hard. They are tired when they come home. Still, Lalu says that they are not making any money. We don’t know what to do.
Mani and her mom visited the bakery
Laxmi kept working. Lalu’s bakery was nearby and Mani asked her mom if they could go there just for a few minutes. She was curious what the problem was. They went there because Mani’s mom thought that it would be a good walk for the two of them.
When Mani reached Lalu’s bakery, they were making their Lalu special biscuits. They had their own secret recipe. The dough was spread out with a roller on a 1 meter by 1 meter square tray. The servant had a cutter with which he was cutting circles of 10 centimeter as shown in the picture below. He would get 10 circular cuts in one row because the dough plate was 1 meter by 1 meter (equals 100 cm x 100 cm). After he cut one row, he would do another so that he would make 100 biscuits from the dough on one tray. He would then throw away the remaining dough because it had become too dry.
Mani: Mom. I think I know how to help Lalu uncle. Trust me. Just call him, please.
Mom: Are you serious?
Mani: There is nothing to lose in just talking.
Mom asked Lalu to come and talk to them. Lalu came over.
Mani: Lalu uncle, why does he throw away the dough after cutting out for the 100 biscuits.
Lalu: The dough becomes too dry. It is not possible to rework it properly. Reworked dough gives very bad biscuits. I also don’t want to make square biscuits. One time we made a few square biscuits but I did not like that shape.
Mani: Lalu uncle. I know how to make you a rich man.
I know that you sell this biscuits for Rs 3 each. That would mean Rs. 300 from one tray of dough. How much does the material for this much dough cost?
Lalu: We make 5 such trays every day. The total cost of the material per day is Rs. 1000. So one tray would cost Rs. 200 but there are other costs like the rent for the shop, electricity, water and salary for the servant. The material used to be cheaper but with inflation we cannot make a living. The shop is losing money even though Bholu and I are working hard.
Mani: If you could make more biscuits, could you sell them?
Lalu: Yes, yes. We have many customers who want to place larger orders but more biscuits sold will not make up for the loss.
I will come back in two hours and show you what I am talking about. At home Mani showed mom her ideas and together they did the needed calculations. Mani came back to the shop as promised.
You throw away 21.4% of the dough
Mani: I calculated that your one biscuit is 78.6 cm2. One tray of the dough is 100 cm x 100 cm or 10,000 cm2 out of which you use only 7,860 cm2. So you are throwing away 21.4 % of the dough. You could improve on this by cutting the circles slightly differently but not by much. You will decrease your cost if you did not throw out the 21.4 %.
Lalu: But how? I don’t want to sell bad biscuits by reworking the dough. I also don’t like making square Lalu special biscuits.
Mani: What if you changed the shape of the biscuit a little bit. So that it is a hexagon instead of a circle. It would still look similar. It could be a hexagon that has the same corner to corner length as the diameter of a circle.
Lalu: What are you saying?
Mani: Here I draw a circle and inside that is the hexagon. From corner to corner the hexagon is as big as the circle but actually it uses only 82.7% of the dough compared to the circle. You would save 17.3% of the material and no one would notice the size difference.
Lalu: It would be a little bit deceptive but the Lalu special would have a new design. I would not have to raise the price, and I could even make a little bit of profit rather than lose money. You know, it would be good if I did not have to throw away the 21% of the dough as we do now.
Mani: You are going to have to give me a free biscuit before I tell you that I have solved that problem too.
Lalu came with a biscuit and said: But how?
Stacked hexagons are like a bee-hive
Mani: Think of a bee-hive. It is made of stacked hexagons with no wasted space in between. You could use the hexagon cutter to cut them without leaving any wasted dough. Here is the picture of how to cut with the new cutter I am suggesting. From 1 meter2 dough, one row could be cut into 11 hexagons and you could make 13 such rows. This would give 143 biscuits with very little dough being wasted.
Lalu: I think this will work. From a supply of Rs. 200, I used to make Rs. 300 worth of biscuits. Now I can make about Rs. 429 worth. If this works, I could make up to 10 such trays every day. Then, every day Bholu and I could take home some money instead of our long faces. Thanks Mani beti. I don’t know why we did not figure this out ourselves.
Mani: You should have sent Bholu to school. Then he would have figured it out.
Months went by. Laxmi seemed to be in a better mood now. One day, she came with a box of biscuits. She said that Lalu was happy now, and had given her these for Mani in appreciation of her help.
Why did Mani suggest hexagons? Would this have worked with pentagons, heptagons or even with octagons? Draw some pictures to test out this idea. No solution provided for this challenge.