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Turning Word Problems to Equations


Date: 03/08/2001 at 17:13:12
From: Agatha
Subject: Word Problems

How do you turn English sentences into math equations?


Date: 03/08/2001 at 18:50:26
From: Doctor Achilles
Subject: Re: Word Problems

Hi Agatha,

Thanks for writing to Dr. Math.

Word problems can be difficult to translate into math equations. If 
you have a specific example or a few specific examples that you need 
help with, I'll be able to help more.

I'll give you some rules that should work for most word problems. 
Don't be overwhelmed by how many there are - they are all pretty 
straightforward. I'll give you some examples as I go through them, 
then some more afterward.

Let's try translating these example English sentences:

   (1) Three times a number is six less than three-tenths of another 
       number.

   (2) Seven into a number equals two divided by another number.

   (3) The sum of two numbers equals the product of those numbers.

   (4) The quotient of two numbers equals the result of subtracting 
       the first number from the second.

The first thing to do on any word problem is to assign a variable to 
every number in the problem. 

  For (1), let's call the first number w and the second number x. 
  For (2), let's call the first number y and the second number z.
  For (3), let's call the first number n and the second number m. 
  For (4), let's call the first number p and the second number q.  

We could have picked any letters we wanted, I just chose those at 
random. We now have:

   (1) Three times w is six less than three-tenths of x.

   (2) Seven into y equals two divided by z.

   (3) The sum of n and m equals the product of n and m.

   (4) The quotient of p and q equals the result of subtracting p 
       from q.

The next thing to do is translate English names for numbers into the 
numbers themselves:

   (1) 3 times w is 6 less than 3/10 of x.

   (2) 7 into y equals 2 divided by z.

   (3) The sum of n and m equals the product of n and m.

   (4) The quotient of p and q equals the result of subtracting p 
       from q.

Here are a few basic translation rules:

RULE 1: "Is" or "equals" can be translated to = . If we do that we 
get:

   (1) 3 times w = 6 less than 3/10 of x

   (2) 7 into y = 2 divided by z

   (3) The sum of n and m = the product of n and m

   (4) The quotient of p and q = the result of subtracting p from q

RULE 2: "Times" or "multiplied by" can be translated  to * 
(multiplication). "Of" can also be translated as * when the "of" 
comes after a fraction or percent. That changes (1) to:

   (1) 3*w = 6 less than (3/10)*x

Note: 3*w is the same as 3w (the * is often omitted). Sometimes the 
words "times" or "of" are also omitted. So if we had:

   (1') Three w = 6 less than three-tenths x

That would translate to:

   (1') 3w = 6 less than (3/10)x

Which is equivalent to (1) above.

Another note on rule 2: the English word "twice" can be translated to 
2*.

RULE 3: "Less than" and "subtracted from" can be tricky to translate. 
Let me start with a simple example of an English sentence that uses 
"less than":

     What is 2 less than 6?

Well, 4 is 2 less than 6. So how do we get there? We take 6 and 
subtract 2. So "a less than b" can be translated b - a. Similarly, 
"subtracting p from q" means q - p. If we apply that to (1), we get:

   (1) 3w = (3/10)x - 6

And if we apply this to (4) and we get:

   (4) The quotient of p and q = q - p

A note on rule 3: "More than" works just like "less than" except it 
uses addition. So "5 more than n" would be translated n + 5. Also, 
"adding a to b" works just like "subtracting p from q," but with 
addition, so it would be translated to b + a.

RULE 4: "Divided by" gets translated directly to / (division). So 
(2) can be re-written as:

   (2) 7 into y = 2/z

RULE 5: "Into" is a little tricky. Let's start with a simple example 
of an English sentence that uses "into":

     How many times does 5 go into 15?

Well, 5 goes into 15 three times. So how do we get there? We take 15 
and divide it by 5. So "a into b" can be translated b/a. If we apply 
that to (2), we get:

   (2)  y/7 = 2/z

RULE 6: "The sum of" can be translated as +. "The sum of a and b" is 
a + b. If we apply this to (3), we get:

   (3) n + m = the product of n and m

NOTE: If you are given something like "n and m equals ..." (in other 
words, if you aren't told to do the sum or the difference, or the 
product or the quotient), then I would usually assume that you are to 
add. So:

   (3') n and m = the product of n and m

is equivalent to (3) above.

RULE 7: "The product of" works just like "sum of" except you use * 
(multiplication) instead of addition. So (3) is:

   (3) n + m = n*m

RULE 8: "The quotient of" is a / (division) operation. "The quotient 
of p and q" is translated to p/q. Note: this is DIFFERENT from "the 
quotient of q and p," which is q/p. Apply this to (4):

   (4) p/q = q - p

RULE 9: "The difference between" is a - (subtraction) operation. 
"The difference between a and b" is a - b. Note: this is DIFFERENT 
from "the difference between b and a," which is b - a. (I didn't give 
you any examples that use rule 9.)


How do all these rules get us from a complicated word problem to an 
equation?

Check out one example of these rules in action in our archives:

  Turning a Sentence into a Variable Expression
  http://mathforum.org/dr.math/problems/timkey.9.7.96.html   

Let me finish with a more complicated example that uses some of these 
rules. This comes from another archive page:

  Marble Puzzle
  http://mathforum.org/dr.math/problems/liu3.10.96.html   

   "Jason and Bob have 193 marbles altogether. Bob has 47 marbles less 
   than Jason. If Jason gives Bob 15 marbles, how many more marbles 
   does Jason have more than Bob?"

Step 1: Make the problem into a list of sentences:

   (5) Jason and Bob have 193 marbles altogether.

   (6) Bob has 47 marbles less (fewer) than Jason.

   (7) If Jason gives Bob 15 marbles, how many more marbles does Jason 
       have than Bob?

Step 2: Rewrite each sentence so the meaning stays the same, but so 
that they use the same phrases as our rules above. Then translate them 
into equations. Let's call the number of marbles Jason has right now 
j. Let's call the number of marbles Bob has right now b. Then we 
have:

   (5) The number of marbles Jason has added to the number of marbles 
       Bob has is 193.

   (5) j + b = 193


   (6) The number of marbles Bob has is 47 fewer than the number of 
       marbles that Jason has.

   (6) b = j - 47

In order to translate (7) using the rules above, we're going to have 
to introduce some new values. If Jason gives Bob 15 marbles, then the 
number of marbles Jason has will change. Let's call the new number k. 
The number of marbles Bob has will also change. Let's call that new 
number c.

What do we know about k and c? We know that k is 15 less than j, and c 
is 15 more than b. Let's make that into a couple of equations:

   (8) k = j - 15

   (9) c = b + 15

Then we can translate (7) into:

   (7) How many more is k than c?

In other words,

   (7) What is the difference between k and c?

   (7) What is k - c?

Step 3: For questions, pick a new variable, let's use a . Change the 
phrase "what is" in the question into a sentence that says "a equals"; 
(7) now reads:

   (7) a = k - c

The goal of the question is to find the value of a. Here are our 
equations:

   (5) j + b = 193

   (6) b = j - 47

   (7) a = k - c

   (8) k = j - 15

   (9) c = b + 15

Step 3: Solve the equations. 

We can solve for the value of j by substituting j - 47 for b in (5). 

   (5)        j + b = 193

       j + (j - 47) = 193

              j + j = 193 + 47

                 2j = 240

                  j = 120

Take that value of j and plug it into (6) to find b.

   (6) b = j - 47

       b = 120 - 47

       b = 73

Now you can find k and c using equations (8) and (9).

   (8) k = j - 15

       k = 120 - 15

       k = 105

   (9) c = b + 15

       c = 73 + 15

       c = 88

Once you have k and c, you can find a, which is what you were asked to 
find.

   (7) a = k - c

       a = 105 - 88

       a = 17

Take your time with this stuff. I hope all this helps. If you have any 
questions about this or if there is a specific word problem you need 
translating, please write back.

- Doctors Achilles and TWE, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
Elementary Word Problems
Middle School Algebra
Middle School Word Problems

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