Systems of Equations Word Problem HELP PLEASE!!

Question

Okay so... I did the first two systems of equations word problems pretty easily but this one has me stumped. I have no idea what to do... help?

 

Ramiro earns $20 per hour during the week and $30 per hour for overtime on the weekends. One week Ramiro earned a total of $650. He worked 5 times as many hours during the week as he did on the weekend. Write and solve a system of equations to determine how many hours of overtime Ramiro worked on the weeked.

 

I'm guessing that one of them will end with =$650... and that's really all I know. It seems like just a bunch of totally unrelated words :(

Please explain!!!

Answer 1

System of equations is about turning words into a mathematical equation.

 

We need to make up some variables first of the things we don't know.  It's asking about overtime hours.   So let's assign that a variable: x= hours over time.  There is an other time we don't know, the number of hours worked during the week.  Let's give that a variable name to: y= hours during week.  Now we can start writing equations.

 

 

Let's start with:  Ramiro earns $20 per hour during the week and $30 per hour for overtime on the weekends. One week Ramiro earned a total of $650.

 

Let's identify some of the language.  Per means divide by so $20 per hour is the same as 20$/hour and $30 per hour is the same as 30$/hour.

 

How does one find money made?  If you multiply time spent, by income rate($/hour) then you have total cash.

 

Example if  Ramiro worked 5 hours during the week he would make: 5 hours* 20$/hours = 100$ 

 

We know the total amount of money he made.  Remember that we assigned variable for the actual hours he worked. To calculate the total money earned would be weekday rate*weekday hours+weekend rate*weekend hours

 

$650= 20*y+30* x; 

 

 

We have one equations with 2 unknowns.  We have to have at least the same number of equations as unknowns to get a concrete answer.

 

Let's look at the next bit of information: "He worked 5 times as many hours during the week as he did on the weekend"

This sentence is tricky because it can't be converted in the order it is written.  The main issue is where does the 5 belong?

 

(times will convert to a times in math terms "*")

 

Is the 5 multiplied by the x= hours over time or y= hours during week.

 

That is to say the options are 5x=y or 5y=x?

 

"As many"  in this case will mean that the second number mentioned get's the 5*  since it needs to be the smaller number. 

 

so the equation is y=5x;

 

Now you have 2 equations and 2 unknowns:

y=5x;

 

 

$650= 20*y+30* x;

The question is only for over time, so you need only solve for x.  I'll let you do the rest!

 

 

Answer 2

After reading the problem to understand it, assign variables.  If possible, assign a variable to the value the problem wants you to find.  Here, it is the amount of time Ramiro worked on the weekend.  Let that be x=time worked on weekend.

 

Now, translate:

  "worked 5 times as many hours during the week as he did on the weekend"  means   5x hours during the week

  Note:  the dollars per hour times the number of hours gives the dollars (hours cancel out)

  His total paycheck  is $650  =   dollars during the week    +    dollars during weekend

                                  $650 =             ($20/hr)(5x hr)              +          ($30/hr)(x hr)

    650 = 100x + 30x

    650 = 130x             (collect terms)

       5 hours = x                  (divide both sides by 130)

 

Checking (very important):

   Is  (20)(25) + (30)(5) = 650    ?

         500 + 150  =  650    ?

         650 = 650    ? yes

Answer 3

Let x=number of hrs worked during week (M-F) and y=number of hours worked on weekend.

 

20x+30y=650

 

But x=5y, so

 

20(5y)+30y=650

130y=650

y=5

x=5y=5(5)=25

 

So, Ramiro worked 25 hrs thru the week and 5 hours on the weekend.