Writing Equations for Word Problems

Date: 09/01/98 at 18:25:24
From: Tigger
Subject: Writing Equations for Word Problems

In Algebra 1 we need to write equations for these word problems and 
then solve them:

1. An oil painting is 16 years older than a watercolor by the same 
   artist. The oil painting is also three times older than the 
   watercolor. How old is each? Choices for the watercolor's age: 
   4, 8, or 12.

2. The gym is 21 years newer than the auditorium. The gym is also one-
   fourth as old as the auditorium. How old is each building? Choices 
   for the auditorium: 26, 27, or 28.

My answer for number 1 is that the watercolor painting is 8 years old, 
and the oil painting is 24 years old.

My answer for number 2 is that the gym is 7 years old and the 
auditotrium is 28 years old.

Am I right?

Date: 09/02/98 at 16:14:59
From: Doctor Kate
Subject: Re: Writing Equations for Word Problems

Dear Tigger:

Yes, you are indeed right. Congratulations! Here's a hint:

You can often tell you are right about a problem without asking 
someone. If you have come up with only one answer, you should make 
sure that your problem should have only one answer. Sometimes problems 
can have more than one answer. 

However, here you can see that the paintings and buildings have only 
one age each, not several. So for the question about paintings, all you 
have to do is check that your answer fits. We know that the oil 
painting is 16 years older than the watercolour painting. Is 24 years 
old 16 years older than 8 years old? Yes, it is. Now the other thing we 
know is that the oil painting is three times as old as the watercolour 
painting. Is 24 years old three times as old as 8 years old? Yes, it 
is. So you are correct.

Now you try it with your second question, and let me know if I haven't 
been clear.

- Doctor Kate, The Math Forum
  http://mathforum.org/dr.math/   

Date: 09/05/98 at 22:50:22
From: Anonymous
Subject: Re: Arkansas

Dr. Math,
 
Yes, it worked. Thank you. Could you explain how you got the answers?

Tigger

Date: 09/16/98 at 16:29:43
From: Doctor Kate
Subject: Re: Arkansas

All right, Tigger. Let's look at the first problem.

1. You would like to find equations that show the information in the 
problem. What information do we have?  I see these bits of information:

A) we have two paintings: an oil painting and a watercolour painting
B) the oil is 16 years older than the watercolour
C) the oil is three times older than the watercolour

All the information here is about the ages of these two paintings, 
so let's make some variables for the ages. Let's call them x and y.  
I'll write it like this:

   Let x represent the age of the oil painting.
   Let y represent the age of the watercolour painting.

I've picked these because that's what we want to know: How old is each 
painting? It's usually a good idea to make variables for what you want 
to find out (and we want to find out the ages).

Now I'm going to construct equations using the other information 
(B and C). We know that the oil painting is 16 years older than the 
watercolour. That is to say, the age of the oil painting is 16 greater 
than that of the watercolour. Or, in symbols:  x = 16 + y. This is 
like writing a very short form of the sentence: "The age of the oil 
painting is equal to 16 more than the age of the watercolour painting." 
Equations are a lot like fancy ways of writing normal sentences.

Now you try to make an equation out of the third piece of information:  
the oil is three times older than the watercolour. Then keep reading.

Here's what I got: x = 3y. That's just like this sentence: "The age of 
the oil painting is equal to three times the age of the watercolour 
painting."

To review, here are our equations and their meanings:

   x = 16 + y    The age of the oil is equal to 16 more than the age 
                 of watercolour.
   x = 3y        The age of the oil is equal to three times the age of
                 watercolour.       

Do you understand where my equations have come from?  Let me know if 
you don't. If you do, now it's just math! We have:

   x = 16 + y
   x = 3y

So we can write 3y = 16 + y (since both sides equal x).

And then we can solve, like this:

   3y = 16 + y
   3y - y = 16 + y - y
   2y = 16
   y = 16/2 = 8

So y (the age of the watercolour) is 8.
And x = 3y = 3*8 = 24 (the age of the oil).

Then you can check this answer as I showed in my last e-mail. You 
should try the second problem to see if you get the right answers.

- Doctor Kate, The Math Forum
  http://mathforum.org/dr.math/