What do the following two math problems have in common?
- 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?
- (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?
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?
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?
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?
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?
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?
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.