This is a narrative of a classroom episode in which students could see how mathematics can help someone avoid some common mistakes and take a right decision. A simple question involving ratios and simultaneous linear equations could do this with right learning strategy.
Students were given a question to work on, and the discussion build on that as given below.
A class was working on the following problem:
A vendor makes two types of milkshakes using milk and syrup. The ratio of these two ingredients in two types of Shakes is given as
- ShakeX, milk : syrup = 2 : 3
- ShakeO, milk : syrup = 3 : 4
He has 23 liters of milk and 31 liters of syrup.
How much of each shake should he prepare so that everything is used?
The discussion went like:
We assume he prepares (x) liters of ShakeX and (y) liters of ShakeO,
It was noted:
- Each shake consumes some milk and some syrup
- Total milk used must become 23 liters
- Total syrup used must become 31 liters
If both totals are expressed in terms of (x) and (y) and equated to the respective values, then the values of x and y satisfying both the equations simultaneously will give the required quantities. So, the two equations representing the use of Milk and Syrup are to be formed and solved as simultaneous equations to get the answer.
Some students quickly wrote:
2x + 3y = 23 for milk and 3x + 4y = 31 for syrup
They solved and got:
x = 1 and y = 7
The solution looked neat, so they raised their hands and announced the answer.
They were then asked:
Does this answer actually use all the milk and syrup?
These students confidently said Yes, but the teacher asked them to check the quantities of milk and syrup used for these values of x and y and verify with the quantity available.
Some students simply substituted the values of x and y into the equations and found that those equations were being satisfied. So, they were happy with the answers. But how can 1 liter of ShakeX and 7 liters of ShakeO consume 23 liters of milk and 31 liters of syrup. Looks too unrealistic and seriously wrong.
The teacher asked the students to see the question again and find the quantity of milk and syrup needed in 1 liter of ShakeX. They are consumed in the ratio of 2:3. That means 2/5 liter of milk and 3/5 liter of syrup make one liter of ShakeX. Students realized the blunder they had done. They understood that the original equations were wrong.
The class revisited the meaning of ratio.
2 : 3 does not mean 2 liters and 3 liters.
It means 2 parts out of 5.
So, in 1 liter of ShakeX:
- milk = (2/5)
- syrup = (3/5)
For ShakeO (3 : 4):
- milk = (3/7)
- syrup = (4/7)
The equations were rebuilt:
(2/5)x + (3/7)y = 23 and
(3/5)x + (4/7)y = 31
Solving gave:
x = 5, and y = 49
A check confirmed that both milk and syrup were used exactly.
At this point, the solution seemed complete.
But the teacher wanted to create some ripples (water element of PanchTatva Learning Strategy).
A new situation was introduced:
The vendor can now spend some money to buy a little extra milk or syrup. Which one should he buy to increase his profit?
Well, Milk. That was the automatic choice of some of the students. Why? Because quantity of milk available is lesser than syrup so it's better to buy the one which is lesser.
Some students felt that only procuring more milk will not help. Because both the ingredients are completely used here. To produce anything more than what is being done, both the ingredients are required.
That's good. So let us bring profit data now, said the teacher.
ShakeX gives profit of Rs. 50 per liter and ShakeO Rs. 30 per liter.
What is the profit at present level of production of two types of Shakes?
(5 x 50) + (49 x 30) = 250 + 1470 = 1720.
Rs. 1720 is the profit at present level and both the resources Milk and Syrup are fully used.
Is this the best profit with present data? Can the vendor earn more profit with same resources and products?
Students were convinced that present decision is good as it utilizes the resources to the maximum. But some of them were not sure whether it gives best possible profit or not.
Teacher asked them as which shake the vendor would produce more if some more milk and syrup are provided.
Students knew that ShakeX is more profitable than ShakeO, so it is better idea to produce that to get more profit.
So, why not apply the same logic right in the beginning! Use resources to produce ShakeX as much as possible.
Since 3/5 liter of syrup is used in 1 liter of ShakeX, 31 liters of available syrup can produce up to 31/(3/5) = 51.67 liters of it. But we also need to check whether we have sufficient milk to produce this much or not.
To produce 51.67 liters of ShakeX, we need 51.67*(2/5) = 20.67 liters of milk. We have 23 liters which is more than required.
Had this not been the case? The teacher asked the students.
Some said to purchase more milk. Some said to do similar computation with milk and check for Syrup. Some said to reduce the quantity of ShakeX. There were different ideas expressed which can lead to various advanced concepts of Linear Programming and Economics.
However, for present, making 51.67 liters of ShakeX and 0 liter of ShakeO is feasible with the available resources. With this, the profit is;
Rs. 50*51.67 = Rs 2583. Also, about 2.33 liters of milk remains unutilized. The vendor can either purchase this much less milk or use it for some other purpose.
The other combination was using all the available milk and syrup but giving Rs.1720 profit only.
It was wow moment for most of the students. Utilizing all the resources may be a good objective but maximizing profit may require something different. Milk is available in lesser quantity but for maximized profit it is Syrup that is getting exhausted. So, buying more milk will only increase the quantity of unutilized milk, it will not increase the profit. However, if some extra syrup is purchased which can be used to make some more Shake, the profit can be increased.
The most important outcome of this lesson was not the numerical answer but the questions that followed it. A simple problem on ratios and equations created ripples that led students towards modelling, optimization, and decision-making. They discovered that using all available resources is not always the same as achieving the best outcome. Such moments reveal the true power of mathematics—not merely to solve problems, but to think more deeply about them.
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