Age and Money

Date: Tue, 13 Dec 1994 18:38:36 AST
Comments: NB*net - New Brunswick's Regional Network 
   1-800-561-4459
From: Richard Seguin
Subject: Grade 9 (Richard Seguin)

Could you help me solve this. I am a grade 9 student. 

1> Frank is eight years older than his sister. In three years he will be 
twice as old as she is. How old are they now?

2> Karen is twice as old as Lori. Three years from now the sum of 
their ages will be 42. How old is Karen?

3> Dave has six times as much money as Fred, and bill has three times 
as much money as Fred. Together they have 550.00. How much does 
each have?

4> To find the length of a certain rectangle you must triple the width and
add 5m. If the perimeter of the rectangle is 74m, find the dimensions. 

I hope you can help me solve these. I had no luck!  

                                             Richard 

Date: Wed, 14 Dec 1994 09:53:08 -0500 (EST)
From: Dr. Sydney
Subject: Re: Grade 9 (Richard Seguin)

Dear Richard, 

        Thanks for writing Dr. Math.  These word problems can get a 
little confusing, but usually things work out if you assign all of your 
unknowns a letter name and write the information down in an equation.  
Let's try the first one...

1) The first thing to figure out is what you are trying to figure out.  
In the first problem, we want to know Frank's age and his sister's age.  
The next thing to do is assign letters to unknowns.  Let's call Frank's 
current age f, and let's call Frank's sister's current age s.  Now let's 
translate what the sentences in the problem say to mathematical 
equations.

a) "Frank is 8 years older than his sister."  This means if we add 8 to
Frank's sister's age we would get Frank's age, right?  So, we have:

                                s + 8 = f

b) "In three years he will be twice as old as she is."  In three years,
Frank will be three years older than he is now, so he'll be f+3 years old.
Similarly, his sister will be s+3 years old.  At that time, he will be twice
as old as she is, so if we were to multiply Frank's sister's age in three
years by 2, we would get Frank's age in three years.  So, we have:

                        2 (s + 3) = f + 3

So, now we have a systems of equations -- 2 equations, 2 unknowns. 
Simplify these and solve:

                        2s + 6 = f + 3

                So,     f = 2s + 3

Substituting this in the first equation, we get:

                        s + 8 = 2s + 3

Simplify to get:

                        s = 5

                If s = 5, then what must f be?  Plug into either equation to
get: f = 13.  So, Frank's sister's age is 5 and Frank's age is 13.  You 
can always check your answer by seeing if your answer makes sense 
in the problem.  Check that Frank is 8 years older than his sister: 
13 is 8 more than 5, so this does make sense.  Now check that in three 
years he will be twice as old as she is.  In three years they will be 16 
and 8 years old.  16 is 2 times 8, so this works.  So, we did everything 
right.  Did that make sense to you? 

I bet now you can do some more of these problems.  Mainly they 
involve translating the sentences into math equations.  If you have any 
more problems or are confused by anything I said, please feel free to 
write back.  

--Sydney