Solving Word Problems with Wolfram|Alpha

October 4, 2012
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Peter Barendse
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What do the following two math problems have in common?

  1. If I have 12 apples, and Jane has 7, and then Jane gives 2 apples to me, how many more apples do I have than Jane?
  2. (12 + 2) – (7 – 2)

Answer: two things, actually. First, the two problems are essentially the same; the latter is just expressed in the language of symbolic mathematics. Second, Wolfram|Alpha can now solve both! That’s right, we’ve added math word problems to the kinds of questions you can ask Wolfram|Alpha!

Did you find the second problem easier? It’s no surprise that most people find “word problems” harder, or that the first step in solving them is to translate them into symbolic form. After all, this is why mathematical symbols were invented: to represent complex problems in a way that helps us organize our thinking in order to find their solutions.

Once expressed in symbols, solving a math problem is relatively easy. It can almost always be done by performing some purely mechanical process or “algorithm” (although it often helps to have a more intuitive understanding as well). Software programs like Mathematica know almost all these algorithms and can solve pretty much any symbolic problem you can think of.

But knowing how to solve symbolic math problems is like knowing how to use a saw to cut wood and a hammer to drive nails. You still can’t build a house unless you know where and when to use these tools. This is where Wolfram|Alpha can help.

Word problems are an important part of our mathematical education because real-life problems almost never present themselves to us in symbolic form. They are usually given in the form of a story, which we translate into symbolic math in order to solve. This is what Wolfram|Alpha does with elementary word problems: it not only gives you the answers, but it also helpfully translates the information in the problems into mathematical symbols, showing you the first (and most important) steps toward finding a solution.

Here’s what Wolfram|Alpha says if we enter the first problem: If I have 12 apples, and Jane has 7, and then Jane gives 2 apples to me, how many more apples do I have than Jane?

If I have 12 apples, and Jane has 7, and then Jane gives 2 apples to me, how many more apples do I have than Jane?

You can ask Wolfram|Alpha to solve any problem where people (or letters) have some common objects (like apples) and then gain, lose, or swap these objects. Wolfram|Alpha knows to ignore any irrelevant information and shows a calculation and illustration to help you understand how the numbers in the problem were used to find the solution.

Here’s an example of a word problem that tries to distract you with lots of irrelevant information and disorganized phrasing: Veronica owns 12 cats and 17 dogs. Maurice had 16 dogs and sells 3 to Veronica. How many cats are there in all if Maurice owns 1 cat?

Veronica owns 12 cats and 17 dogs. Maurice had 16 dogs and sells 3 to Veronica. How many cats are there in all if Maurice owns 1 cat?

Besides gaining, losing, and transferring objects among various entities, you can also solve word problems involving comparisons. The common core standards have many examples, including this one: Lucy has 3 fewer apples than Julie. Lucy has 2 apples. How many apples does Julie have?

Lucy has 3 fewer apples than Julie. Lucy has 2 apples. How many apples does Julie have?
Word problems can have several comparisons, which can involve multiplication/division as well as addition/subtraction: John has 3 cookies more than Bill. Bill has half as many cookies as John. How many cookies does John have?

John has 3 cookies more than Bill. Bill has half as many cookies as John. How many cookies does John have?

In addition to individual amounts and differences, Wolfram|Alpha can also tell you the total number of objects there are: A boy has 12 apples more than Jon. Jon has 2 apples less than George. If George has 3 apples, how many apples are there in all?

A boy has 12 apples more than Jon. Jon has 2 apples less than George. If George has 3 apples, how many apples are there in all?

Not all word problems involve people. Wolfram|Alpha also understands simple arithmetic problems like this one: How many is 3 apples more than 2 apples?

How many is 3 apples more than 2 apples?

There is an infinite variety of types of word problems, so of course, Wolfram|Alpha can’t cover all of them. However, we’ll be steadily improving the variety and difficulty of word problems that Wolfram|Alpha can solve, as well as enhancing the output. For example, we’ll soon be adding Step-by-step solutions to the systems of linear equations in the problems above.

We hope that students will use this feature to help them learn how to think through math word problems, which is one of the most challenging—and essential—parts of the elementary math curriculum.

23 Comments

What about rate problems, like “One train leaves the station at 3pm going 50 mph, and another train leaves at 4pm going 70mph. When do they meet?”

Posted by Rob Lockhart October 4, 2012 at 2:02 pm Reply

Hello Peter!

I am a Brazilian blogger, and constantly uses Wolfram Alpha ..

The features of Wolfram Alpha and its applications are becoming more refined.

I really liked your text and examples applied to this new functionality.

Hug!

Posted by Edigley Alexandre October 4, 2012 at 2:05 pm Reply

Cutting-edge Natural Language Processing with a constant drive towards practical applications. Thanks for sharing and inspiring all of us!

Posted by StatsTrade October 4, 2012 at 2:36 pm Reply

Will it also solve Einstein’s Puzzle?

Posted by Qwerty October 4, 2012 at 5:32 pm Reply

Wow this is great. Can’t wait to see this expanded.

Posted by Dillon Korman October 5, 2012 at 12:23 am Reply

So it is not long until WolframAlpha can do all the logic puzzles on the LSAT and enter law school. That would be cool.

Posted by Philip Maymin October 5, 2012 at 2:16 am Reply

Something I’d definitely recommend adding: age comparison problems (for instance: “John is half Fred’s age. Fred is ten years older than John. How old is Fred?”). I remember having a lot of those types of word problems when I took that level of math.

Posted by David October 5, 2012 at 6:52 pm Reply

This type of word problem is uneasy to solve because of confusions but WolframAlpha can solve it in a few moments.

Examples section is great for new users. I have started using WolframAlpha.

Posted by Madhav Tripathi October 17, 2012 at 3:27 am Reply

Are there plans to offer a rest API for Wolfram Alpha Pro?

Posted by Andy McSherry December 2, 2012 at 8:11 am Reply

Anastasia is with her father at the grocery store. She is wondering how much shopping is left. We have 70% of the items On our list. Her father said. Anastasia counts 14 items in the cart. How many grocery items are on the list.

Posted by Jacob provilla June 27, 2013 at 11:43 pm Reply

    let total items be x . 70% of x = 14 therefore , 0.7x = 14 , solve for x , get x = 20 .

    Posted by bhavya joshi July 17, 2013 at 7:44 am Reply

Wonderful! :)

Posted by Valerio Bozzolan August 5, 2013 at 2:08 am Reply

Hi! Really nice! But what about joint work? For example “First machine do the job in 11 hours, second machine in 5 hours. How long will they work together?”

Posted by Bob October 25, 2013 at 9:59 am Reply

our boys each order their own small pizza, William eat 2/3 of his pizza. Mike east 2/5 of his pizza. Julio eats 1/2 of his pizza. Who ate the least amount of pizza

Posted by Dalton December 17, 2013 at 8:35 pm Reply

We want to know the average sugar content per packet in a brand of instant oatmeal. We consider a box of 18 packets which have an average of 12.2g of sugar per packet. The oatmeal factory did a more comprehensive study of there product and found that they have 13.1g sugar per packet.

1. Subjects=
2. Observations=
3. Sample size=
4. Perimeter=
5. Statistic=

Posted by Bobby January 15, 2014 at 10:05 pm Reply

Peter,
Can this be done in Mathematica 9. I am using it to teach my kids maths and this would be very helpful.

Posted by Pradeep January 22, 2014 at 3:36 am Reply

Yes, Pradeep, this can be done in Mathematica 9, via the Wolfram|Alpha integration.

Just enter “==” into a blank cell and then type freeform input !

Posted by Peter Barendse January 24, 2014 at 2:17 pm Reply

What about this: I own a house in a school district that has a yearly budget of $100 million. How much of the budget will be funded by property taxes if the amount funded by state sales tax is $15 million and state income tax will be three times the amount funded by property taxes?

Posted by Jamee Smith March 3, 2014 at 8:05 pm Reply

The weight of a half full tank is 302 pounds. Full the tank weighs 564 pounds. Find the weight of th empty vessel.

Posted by N L March 13, 2014 at 9:20 pm Reply

Can someone help me solve this right away

a swimming pool is formed by the intersection of two circles whose radii are both 9m. find the area of the swimming pool formed?e can anyone solve this right

Posted by aaron March 19, 2014 at 4:54 am Reply

    I apologize that this is like a month late. It seemed like you needed the information really soon. Here’s the answer if you still want to know: 132.9 m^2

    Hopefully that will be helpful to you.

    Posted by William April 10, 2014 at 10:04 am Reply

A prescription was written for 270 tablets on the 14 of February. When does the prescription get run out if it is taken 4 times a day?

Posted by Mary March 20, 2014 at 8:07 pm Reply

Its great to be here. I hope to gain and be a mathematician in future

Posted by Aderogba March 23, 2014 at 5:02 am Reply
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