systems of equations word problem

Question

From 2001 to 2005 the average number of hours that a user spent on the internet each week increased by 110%. this percent increase amounted to 11 hours. Find the average number of hours that a user spent on the Internet each week in 2001 and 2005. 

Answer 1

Let's start by defining our terms.

 

Average hours of 2001= H01

 

Average hours of 2005 = H05

 

We know that the users spent an average of 11 mores hours online  2005 than they did in 2001

 

Expressed as an equation, this reads as follows:

 

H01 + 11 = H05

 

Simultaneously, the problem tells us this represents an 110 percent increase.

 

A percentage is just a ratio multiplied by 100.

 

So we can express this statement as follows:

 

(H05/H01) * 100 = 110

 

Which simplifies to:

 

(H05/H01) = 1.10

 

 

Now we have 2 equations

 

Equation 1: (H05/H01) = 1.10  and Equation 2: H01 + 11 = H05

 

I like to solve systems of equations by substitution, so that's the method I'll use here.

 

H01 + 11 = H05

 

H05/H01 = 1.1

 

(H01 + 11)/H01 = 1.1

 

H01 + 11 = 1.1 * H01

 

(1.1 * H01) - H01 = 11

 

.1H01 - 11

 

H01 = 110

 

H05 = H01 + 11 = 110 + 11 = 121

 

So we spent 110 hours on the web in 2001 and 121 hours on the web in 2005.

 

Does this all make sense?

 

Importantly, we can check this math using the other equation:

 

(H05/H01) * 100 = 110

 

Does (121/110) * 100 = 110?

 

It does!

 

Let me know if any more help is needed!

 

 

 

 

Answer 2

We have only two data points:   (2001,x1) and (2005,x2).  All we can do is assume that the growth in-between in linear.

 

Growth or Decline in percent has a base (or starting) value and the final value is expressed as a percent of that base value.

 

For this problem, (x2/x1) = 110%  or we could write  x2=(110%)x1.  We may express this as 110% or 110/100 or 1.10 as needed by the problem.

 

"What is the average number of hours that a user spent on the internet each week in 2001?"

    Well the raw value of the increase (x2-x1) is 11 hours.

        x2-x1=11

         x2 = x1+11

  and

      x2/x1 = 1.10

       x2 = 1.10*x1

set these equal to find x1:

      x2 = x1+11 = 1.10*x1

                   11  = 1.10x1 - x1         [subtract x1 from both sides]

                   11 = 0.10x1                [collect terms]

                 110 = x1                     [divide both sides by 0.10]

    In 2001, the average number of hours is 110.

 

And, that means, x2=x1+11=121 hours in 2005.

 

Checking (very important):

   Is  from 110 to 121 a growth of 10%   ?

        110+11=121   ?yes

   Is the 10% growth=11   ?yes