How old is Sandra? The correct answer explained

Hey, this is Presh Talwalkar. Chicago history trolled the internet by posting a puzzle that has nothing to do with Chicago history. This problem has over 3. 8 million views. How old is Sandra? 90% get it wrong. The puzzle reads, Sandra was asked her age. She answered, 80 minus 40 divided by 10 * 4. This little question sparked a lot of discussion. I read the top 50 replies to this post and categorized them into four different groups. The most common answer was 64, given by 44% of people. But notice, this is not even a majority of the top 50 comments. A significant number of people said the answer was 79, 16, or some other interpretation of this problem. So there was widespread disagreement. So how old is Sandra? Perhaps Sandra will be the only person who knows the answer. And maybe that was the point after all. Because as the old joke goes, you should never ask a woman her age. And of course there's a continuation that you should never ask a man his salary, and you should never ever ask a mathematician a viral problem. They will let you know that they hate these questions and they are sick of this problem. With the exception being Mind Your Decisions and Presh Talwalkar, because if you send this to me, I will be very happy to explain the correct answer according to the current usage of the order of operations. In order to evaluate 80 minus 40 divided by 10 * 4, we can use the order of operations commonly remembered by PEMDAS, which refers to parentheses, exponents, multiplication, division, and addition subtraction, or the corollary BODMAS, where parentheses has brackets and exponents has orders. There's also one more important rule, that if you have two operations of the same precedence, you want to evaluate them from left to right. So, in this expression, we do have a multiplication and a division. So, here's a division and here is a multiplication. Which one do we do first? According to this convention, we want to evaluate from left to right. So, we would evaluate 40 divided by 10 first. 40 divided by 10 is equal to four, so we evaluate that and we substitute in. We now have a multiplication and that takes precedence over the subtraction, so we evaluate 4 * 4, which gives 16. Substitute that in and we finally evaluate the subtraction, so we have 80 - 16 and that gives us the answer of 64. According to this convention, there is no other answer to this expression than 64. 80 - 40 / 10 * 4 is equal to 64 and that's the correct answer according to the modern interpretation of the order of operations. Now, I say the modern interpretation, but in fact, such a rule was written over 100 years ago. Here is a paper in the American Mathematical Monthly, which was first published in 1894 and this paper comes from 1917, relating to the order of operations in algebra. A snippet from this article says, "In case the signs of multiplication and division occur with no signs of addition, subtraction intervening, the operations are to be performed in order from left to right. Thus, 8 / 2 * 4 is equal to 8 / 2 evaluated first * 4, which equals 16 and not 8 divided by the group product 2 * 4, which would equal 1. A series of operations involving multiplication and division alone shall be performed in the order in which they occur from left to right. So, you would think this is the convention, it's written down, everyone understands it. But people have told me, "Wait, I learned a different rule in school. " And it's one of the true things I can say I learned by making these videos is that yes, in fact, some people did learn a different rule. Here is a book, Shop Mathematics from 1922. So, that would be contemporary to the 1917 article. What does this author say? "When the multiplication sign immediately follows the division sign, multiplication should be performed first. This rule is not always strictly followed. Therefore, it is better to use parentheses in order to avoid ambiguity. Thus, an expression like 10 / 2 * 5, according to many authorities, equals 1. While some claim that it equals 25. " So, I truly do believe that some people learn the rule that you need to evaluate multiplication before division.

So, you would need to multiply 10 * 4 first, that would be equal to 40. Then you would do 40 / 40, which gives 1, and 80 - 1 is equal to 79. And some people had to do it this way to get good grades in school. But other people learn the rule that you need to do multiplication and division from left to right. So, that would be 40 / 10, which gives 4. Then we have 4 * 4, which gives 16, and 80 - 16 is equal to 64. So, given the two possible interpretations, there are people who say that 40 / 10 * 4 has two equally valid answers. If you do the multiplication first, you get 1. But if you do the division first, you get 16. "It's just a poorly written and ambiguous expression," they would say. But, a student might be wondering, "What would happen if I press these buttons and input this into a calculator? Will it tell me that it's an invalid expression? Will it give me an answer? What do calculators actually evaluate this expression as? " I think it's an important lesson that students know most calculators and computer programs will go from left to right. When you input this expression into a calculator, it is going to use its internal set of rules and create a binary expression tree. A calculator that goes left to right is first going to evaluate 40 / 10. This will be the terminal root and the nodes, and then we are going to multiply by four. So, the binary expression tree is going to look like this. You will get an answer of 16. On the other hand, a calculator might do the multiplication first. So, we would have 10 * 4, and then we would do the division. So, we would have 40 divided by that result. This, of course, will give us the answer of one. This is a great time to illustrate binary expression trees to students so they learn something about programming and how calculators work. And, in fact, I have tested out this expression. It will be equal to 16 in Python, JavaScript, Java, C#, Excel, and most calculators will evaluate it this way. The day may come that you are reviewing someone else's code, they have written an expression like this, and you would be wise to know what the evaluation would be. Now, let us put Sandra's age question to calculators. When we put it on the Android calculator, we get an answer of 64. The same is true on an iPhone calculator. a Samsung calculator. It's also the same answer in Bing, and it's Wolfram Alpha. We are seeing a pattern. So, let's now return to the question of how old Sandra is. There was definite disagreement in this question, and more than 50% of people gave an answer besides 64. And, understandably, some people did learn a different rule in school. But, the only question to the people who gave these answers is why are pretty much all calculators giving the answer of 64? Were none of you involved in making calculators and computer programs? It would seem to me this is the strongest evidence that after 100 years mathematicians have finally come up with a consensus. We do the division and multiplication from left to right, and therefore 64 is the correct answer according to the modern interpretation of the order of operations. But, perhaps there's more to the puzzle. Perhaps Sandra is 79 years old. She learned the old rule, and she decided to troll the internet just by putting the expression according to the old rule. We will never actually know how old Sandra is, but we can evaluate this expression according to the modern interpretation of the order of operations. I want to conclude the video just by illustrating why these questions are important. Even if that question was ambiguous, students do fail unambiguous order of operations question. Here is a problem that was given to 12th grade students in America in 1992. 3 cubed plus 4 parentheses 8 minus 5 divided by 6. Students were given five choices. Some students even omitted an answer. Now, if you look at the distribution of choices, only 29% of students selected the correct answer. They were even allowed to use a calculator to evaluate this. That means 71% of students gave the wrong answer. There is genuine confusion about the order of operations even in an unambiguous question. A similar trend happened in Japan where 40% of engineers missed this question. What is 9 - 3 / 1/3 + 1 equal to? The order of operations is a topic that confuses many people. So the next time a problem goes viral, you know where to come for the correct

answer explained. Thanks for making us one of the best communities on YouTube. See you next episode of Mind Your Decisions where we solve the world's problems one video at a time.