AI Comparison – Mathematical Order of Operations for an Equation

This one has been a doozy to say the least! This was a big topic of discussion in the IT department at my client and it piqued my interest. Plus, I had been looking at doing one of these in honor of the school year starting. You would think that AI would be able to give you a simple answer to a simple math problem. To tell you the truth the two methods of interpretation were giving two answers that both had good arguments. I said both answers were correct depending on how you were taught! Everyone scoffed at me and told me I couldn’t do simple math because I couldn’t really tell them why off hand. I needed to go deeper!

So. What if this was some student in a classroom asking for help on this? I wanted to dive deeper into the topic and compare the AI

The Prompt

I posed the same initial question to Google Gemini Advanced, ChatGPT 4o, and Copilot Pro. Here is the original prompt question and the answers:

You are a mathematician. If you have an equation that says the following: 100 divided by 4(2+3), is 100 the numerator, why or why not, and what is the correct answer?

AI Responses

ChatGPT 4o

In the expression 100 / 4(2+3), 100 is indeed the numerator. The entire expression can be interpreted as:

100/4(2+3)​

To solve this expression:

1. Evaluate the expression inside the parentheses:2+3=5
2. Multiply the result by 4: 4×5=20
3. Divide 100 by 20:100 / 20=5

Thus, the correct answer is 5.

Explanation:

• In this expression, 100 is clearly the numerator because it is the number being divided by the entire denominator 4(2+3). <–THIS IS THE IMPORTANT INTERPRETATION
• The denominator is 4(2+3), which simplifies to 20.
• The division process gives the final result, 5.

This follows the standard order of operations (PEMDAS/BODMAS):

• Parentheses first.
• Then Exponents (not applicable here).
• Then Multiplication and Division (from left to right).

By applying these steps, we correctly solve the equation as described above.

I was like Okay, but the argument is that the other interpretation is reading strictly from left to right. Does that violate the Order of Operations?

My next question in the chain:

What if you are reading the equation from left to right i.e. 100 / 4(2+3) What is the answer then?

When reading the expression from left to right, you still need to follow the order of operations (PEMDAS/BODMAS), which dictates the sequence in which operations are performed. This means that multiplication and division should be handled before addition, but multiplication and division are treated equally and are performed from left to right.

The expression is:

100 / 4(2+3)

Step-by-Step Solution:

1. Evaluate the expression inside the parentheses:2+3=5
2. Substitute the value back into the expression:100/4(5)
3. Interpret the expression: When reading left to right, the expression is understood as: 100/4×5 Here, since division and multiplication are of equal precedence, you proceed from left to right. <–NOTE THE INTERPRETATION HERE AS WELL FROM MY PROMPT QUESTION.
4. First, divide 100 by 4: 100 / 4=25
5. Then multiply the result by 5: 25 × 5=125

So, if you interpret the expression as performing the operations strictly from left to right, the result would be 125.

Summary:

• Order of Operations Interpretation: 100 / 4(2+3) = 5
• Left-to-Right Interpretation: 100 / 4 × 5=125

The answer depends on how the equation is interpreted, but the most mathematically standard approach is to follow the order of operations, which gives the answer 5. The left-to-right interpretation, without considering the implicit multiplication priority, gives 125.

Copilot Pro

In the equation 100 / 4(2+3), 100 is indeed the numerator because it is the number being divided. The denominator is the entire expression ( 4(2+3) ).

To solve this equation, follow these steps:

1. First, simplify the expression inside the parentheses: ( 2+3 = 5 ).
2. Then, multiply this result by 4: ( 4 x 5 = 20 ).
3. Finally, divide 100 by the result: (100 / 20 = 5) .

So, the correct answer is 5.

My next question in the chain:

What if you are reading the equation from left to right i.e. 100 / 4(2+3) What is the answer then?

When reading the equation ( 100 / 4(2+3) ) from left to right, it’s important to follow the order of operations, also known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

1. First, simplify the expression inside the parentheses: ( 2+3 = 5 ).
2. Then, multiply this result by 4: ( 4 x 5 = 20 ).
3. Finally, divide 100 by the result: ( 100 / 20 = 5 ).

So, even when reading the equation from left to right, the correct answer is still 5.

Google Gemini Advanced

NOTE: I love Google Gemini the best for learning fundamental classroom subjects. I actually got two different responses from my question. The first time it interpreted it the one way and in the second chat, it interpreted it another way. Please continue and see what I did…

First String Responses

In the expression 100 ÷ 4(2+3), 100 is the numerator of the first term.

Here’s the breakdown:

The expression can be understood as:

• (100/4) * (2+3)

This is because multiplication and division have equal priority and are done from left to right.

Therefore:

• The first term is 100/4, where 100 is the numerator.
• The second term is (2+3).

The final calculation would be:

(100/4) * (2+3) = 25 * 5 = 125

My next question in the chain (Yes the whole thing is the prompt that is in green):

I was told this: Order of operations always takes precedence over left-to-right evaluation. The correct interpretation of the expression 100 / 4(2+3)

100 / 4(2+3) involves treating the multiplication in the denominator as a priority:

First, complete the calculation inside the parentheses.
Then, multiply the result by 4.
Finally, perform the division.
This approach leads to the correct answer of 5.

So, left-to-right reading does not override the order of operations; it only applies within the context of operations that are of equal precedence (e.g., multiplication and division).
Why are you telling me the exact opposite?

Answer:

There seems to be a misunderstanding in what you’ve been told. While the order of operations (PEMDAS/BODMAS) is indeed crucial, it’s essential to apply it correctly. The information you received seems to imply that implicit multiplication (like 4(2+3)) always takes precedence over any other operation, which isn’t accurate.

The correct interpretation, following the order of operations and left-to-right rule, is as follows:

1. Parentheses: (2 + 3) is evaluated first, resulting in 5.
2. Multiplication and Division (from left to right): Now we have 100 / 4 * 5
   • We encounter division first (from left to right), so we perform 100 / 4 = 25
   • Then we perform the multiplication: 25 * 5 = 125

Therefore, the correct answer is 125, not 5.

Key Points:

• Order of Operations: PEMDAS/BODMAS is the guiding principle.
• Equal Precedence: Multiplication and division have equal precedence.
• Left-to-Right: When operations have equal precedence, they are performed from left to right.
• Implicit Multiplication: While implicit multiplication can sometimes be treated with slightly higher precedence, it doesn’t override the fundamental left-to-right rule within the same precedence level.

The misunderstanding likely stems from the misconception that implicit multiplication always forms a grouped term, which isn’t the case unless explicitly indicated by parentheses. To get an answer of 5, the expression would need to be written with explicit grouping using parentheses: 100 / (4 * (2 + 3))
In conclusion, the left-to-right rule is an integral part of the order of operations and should not be disregarded.

NOTE: This totally blew my mind, that Gemini stuck to it’s guns and did the exact opposite of ChatGPT and Copilot Pro, but I understood. ChatGPT and Copilot Pro treated implicit multiplication for the denominator (which is how I was taught in school), where Google says the rule of left-to-right reading gives the other answer, Copilot and ChatGPT just said it was standard practice. I could NOT accept this, so I opened a new chat and put in the same prompt. Here is the second go around.

Second String Responses

Absolutely, I’ve been sharpening my skills in handling order of operations problems, and I’m happy to help. Let’s evaluate the expression: 100 / 4(2+3)​

Steps to solve: 1. Add the numbers inside the parentheses: 100 / 4(5)

2. Multiply the numbers: 100 / 20

3. Simplify the fraction: 5

Answer: 5

My next question in the chain (Yes the whole thing is the prompt that is in green):

In a previous chat you told me this: No, the answer to 100 / 4(2+3) is still 125, not 5. And yes, 100 is the only numerator in this expression. The confusion might arise from the way implicit multiplication (like 4(2+3)) is sometimes perceived. While some might argue it has slightly higher precedence, it doesn’t fundamentally change the left-to-right rule for operations of equal precedence.

Here’s the breakdown:
Parentheses: (2 + 3) is evaluated first, resulting in 5.
Left to Right: We then have 100 / 4 * 5
We encounter division first (from left to right), so we perform 100 / 4 = 25
Then we perform the multiplication: 25 * 5 = 125
Therefore, the final answer is 125.

Key Points:
Order of Operations: PEMDAS/BODMAS is the guiding principle
Equal Precedence: Multiplication and division have equal precedence
Left-to-Right: When operations have equal precedence, they are performed from left to right
Implicit Multiplication: It doesn’t override the fundamental left-to-right rule
To get an answer of 5, the expression would need to be written with explicit grouping using parentheses:
100 / (4 * (2 + 3))

Why am I getting conflicting answers?

Answer:

You’re getting conflicting answers because there is a genuine ambiguity in how some people interpret implicit multiplication, particularly when it involves expressions in parentheses.

• The Traditional PEMDAS/BODMAS Interpretation: This is the one I originally provided, and it leads to the answer 5. It treats multiplication and division as having equal precedence and evaluates them strictly from left to right.
• The “Implicit Multiplication Binds Tighter” Interpretation: This is the one leading to the answer 125. It suggests that implicit multiplication (like the 4(5) in your problem) should be done before explicit division, even though they’re technically of equal precedence.

Why the Disagreement?

• Lack of Universal Standard: There isn’t a single, universally accepted rule for how to handle implicit multiplication in relation to division. Different textbooks, calculators, and even mathematicians might have slightly different interpretations.
• Historical Precedence: In some older texts, implicit multiplication was indeed given slightly higher precedence. This might still influence how some people approach these types of problems.
• Practical Considerations: In many real-world contexts (especially in physics and engineering), formulas are written in a way that assumes implicit multiplication binds tighter. This is because it often leads to more concise and readable expressions.

So, which answer is “correct”?

Unfortunately, there’s no single definitive answer. Both interpretations have their merits and drawbacks.

• If you’re taking a standardized test or following a specific curriculum: It’s crucial to know which interpretation they’re using.
• In other contexts: It’s best to use parentheses to make your intentions clear and avoid ambiguity.

My Apologies: I apologize for the initial confusion. I should have been more explicit about the potential for different interpretations in this specific scenario.

Key Takeaway: The most important thing is to be aware of this potential ambiguity and to communicate your mathematical expressions clearly to avoid misunderstandings.

The Challenge

I challenge you to use AI platforms like Google Gemini to explore mathematical principles further. Experiment with prompts that address specific ambiguity and its challenges, such as the equation that was expressed here. By actively engaging with AI, you can unlock a wealth of insights and creative solutions to learn any skill, even mathematics.
Vetting your data is important and I also challenge educators to challenge and validate the data these AI platforms are using for THEIR learning. I still believe that AI will not take over the human element and this is a great example of that.
Whatever you do, please have a good attitude towards AI and the process, you don’t know who it might affect in the future!

Final Conclusion

Why are these responses correct? Why are they incorrect? What is your experience with this problem? Were you able to understand why the other AI Platforms gave different answers and how that relates to how Generative AI works? Please leave your comments below and connect with me on Linked-In

About Lance Lingerfelt

Lance Lingerfelt is an M365 Specialist and Evangelist with over 20 years of experience in the Information Technology field. Having worked in enterprise environments to small businesses, he is able to adapt and provide the best IT Training and Consultation possible. With a focus on AI, the M365 Stack, and Healthcare, he continues to give back to the community with training, public speaking events, and this blog.