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Compute the number of baseball ticket sold for adults and children

The admission fee at a baseball is $4.50 for children and $8.50 for adults. On a certain night 2,500 enter the baseball game and $20,650 is collected. How many children and how many adults attended the baseball game?

Using the calculator shown below, teachers and students can simulate different scenarios by changing the right hand side value of equation 1, equation 2 or both. Learning how to analyze a problem and write the given condition 1 or constraint 1 or equation 1 that must be satisfied is the first step that every student of algebra must master.

In Algebra textbook constraint number 1 or condition 1 is written as equation 1. Total count of adult and children that bought baseball ticket on a certain night is 2,500. Using algebra lesson learned, a student must master to represent the problem he/she is trying to solve by a letter variable. You can assign letter variable " a " = adult ticket count and letter variable " c " = children ticket count. Using textbook algebra variable you can assign letter variable " x "= children ticket count and letter variable " y " = adult ticket count. Now you are ready to write the first constraint given or condition that must be satisfied. Application in real work, in business contract or engineering contract there is always constraint given or condition given that must be satisfied. So a student well trained in Algebra to identify all the given constraint will have a good preparation in workplace environment.

How many children ticket was bought? you don't know. So you assign it as letter variable " x " = number of children ticket bought . How many adult ticket was bought? you don't know. So you assign it as letter variable " y " = number of adult ticket bought. But you know that the total number of tickets for children and adult is 2,500.

Written mathematically you will write: x + y = 2,500 this is your given constraint or condition that must be satisfied when you do the validation check. In our automated calculation you must write 1 x and 1 y otherwise our algorithm will assume by default the value "null" or zero which is not correct. Also memory recall lesson learned about multiplication identity 1 x = x and 1 y = y.

Learning how to write the given condition 2 or constraint 2 that must be satisfied.

How much is the collected sales from children ticket ? you don't know, but you know it must be $4.50 multiply by children ticket sold. Written mathematically as 4.50 x. How much is the collected sales from adult ticket? you don't know, but you know it must $8.50 multiply by adult ticket sold. Written mathematically as 8.50 y. You also know that if you add the children and adult ticket sold the total money is $20,650.

Written mathematically as 4.50 x + 8.50 y = 20, 650 this is condition 2 or constraint 2 that must be satisfied. Do not use comma " , " symbol in your number entry because our algorithm will interpret it as Not a Number (NaN).

X + Y = Equation 1 or Condition 1

X + Y = Equation 2 or Condition 2

number of children ticket, c or X =

number of adults ticket, a or Y =

Solution Using Generative AI using LLM like ChatGPT

Below is the solution generated by ChatGPT

Let's use algebra to solve this problem:

Let's call the number of children who attended the baseball game "c" and the number of adults who attended "a".

We can set up two equations based on the given information:

The total number of people who attended the game is 2,500: c + a = 2,500

The total amount collected in admission fees is $20,650: 4.5c + 8.5a = 20,650

Now we can solve for one of the variables in terms of the other in the first equation:

c = 2,500 - a

We can substitute this expression for "c" into the second equation:

4.5(2,500 - a) + 8.5a = 20,650

Distribute the 4.5:

11,250 - 4.5a + 8.5a = 20,650

Combine like terms:

4a = 9,400

a = 2,350

So there were 2,350 adults who attended the game. We can substitute this value back into the first equation to find the number of children:

c + 2,350 = 2,500

c = 150

Therefore, there were 150 children who attended the game.

Solution Using Generative AI using LLM like CoPilot

As you can see, you can assign any letter variable like "c" or "x" and "a" or "y" the answer is the same.

This is one example of INVBAT.COM - A.I useful immediately to students, teachers, employees, and parent.

IN-V-BAT-AI uses explainable Artificial Intelligence (AI) to automate repetitive solved problem like this example so we can use our brain power to solve harder new problem then automate again once it is solved.

How this automatic calculation happen? Learning how to compute determinant of 2x2 matrix and applying Cramer's rule makes this automatic calculation possible. That is advance mathematics but you can visit this link to learn more. Visit Determinant of 2 x 2 Matrix for more explanation on how this automatic scenario analysis becomes possible

How can you prove that your answer is correct and valid? Simply substitute the value of x and y to equation 1 and equation 2. If you get the answer from equation 1 are both equal then probably the answer is correct. To make sure it is really correct you evaluate equation 2 and if the answer are both equal. Then you have evidence that the answer for x and y is valid and correct for both equations because you have satisfied the two given conditions or constraint otherwise your answer is not valid.

Now student and teacher will no longer need to check the test exam paper because it is done automatically. This will save teacher a lot of time in checking paper.

Learning Algebra is very important skill every student must master. Why? Because algebra give student the training and stamina to press on , to formulate, to grapple with complex problem. It trains student to reflect on their thinking during problem solving process, develop the habits of persistent. It trains them to gather proof and evidence for critical analysis. In algebra student are given the training to analyze a problem situation, determine the question(s) to be answered, organize given information, determine how to represent the problem.

Today with machine learning and artificial intelligence the hard part of doing actual computation is now easy as you experience from using this software tool. But the concept of Algebra remains the same. Algebra connect the quantitative relationship of physical things like number of tickets that you want to solve. Algebra deals with representing the physical objects as letter variable like let letter X represent the number of tickets sold to children and let the letter Y represent the number of tickets sold to adults. Now the teacher job is to help the student to realize that every letter or variable they see in algebraic equation represent a physical object. Trying to convince their student that Algebra is a cool tool because when they are solving algebraic equation they are like playing with physical material in the laboratory but not spending any money for material cost because it is done using paper and pencil or simulation in computer software. Algebra is used in many computer simulation modelling.

When they actually work with actual material things their skills in using Algebra will save them time and money.

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How can IN-V-BAT-AI be used in classrooms ?

The IN-V-BAT-AI solution can be a valuable tool in classrooms, enhancing both teaching and learning experience. Here are some ways it can be utilized:

⋆ Personalized Learning : By storing and retrieving knowledge in the cloud, students can access tailored resources and revisit concepts they struggle with, ensuring a more individualized learning journey.

⋆ Memory Support : The tool helps students recall information even when stress or distractions hinder their memory, making it easier to retain and apply knowledge during homework assignments or projects.

⋆ Bridging Learning Gaps : It addresses learning loss by providing consistent access to educational materials, ensuring that students who miss lessons can catch up effectively.

⋆ Teacher Assistance : Educators can use the tool to provide targeted interventions to support learning.

⋆ Stress Reduction : By alleviating the pressure of memorization, students can focus on understanding and applying concepts, fostering a deeper engagement with the material.

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