Stretching a Vacation

Question

Help, I don't know how to solve this problem. Thank you!

 

A student saved $1,200 for a trip to Europe. By cutting $20 from her daily expense she was able to stay three extra days. How long had she originally plan to be gone?

Answer 1

The key to word (story) problems is to translate the words and phrases into precise, concise math expressions --- variables and operations.

If possible, assign a variable to the value being found ("How long had she originally plan to be gone?").  Let that be x.

Also, let’s assign C as the planned cost per day.

  "saved $1,200 for a trip"  means that 
     the planned cost per day times the planned number of days = $1200.
         That is,   C * x = $1200

  "By cutting $20 from her daily expense she was able to stay three extra days"   means  
        Actual cost per day times the actual number of days = $1200
        (C - $20) * (x+3) = $1200

O.K. now the easy math:

Using F-O-I-L on the second equation, we get:
         C*x + 3C -20x -60 = 1200
          1200 + 3C – 20x -60 = 1200       (replace C*x from first equation)
             3C – 20x – 60 = 0                (subtract 1200 from both sides)
          3(1200/x) – 20x – 60 = 0           (replace C using first equation)
           3600  - 20x^2 – 60x = 0           (multiply all terms by x)
           x^2 + 3x -180 = 0              (divide all terms by -20, and rearrange)

Now, either by factoring or by using the quadratic equation,
     We find that the roots are   12  and   -15     (but the days can’t be negative)

The number of days originally planned was 12.

Checking (very important):
   With $1200 saved from 12 days, the cost/day was planned to be $100.
   But, with a $20 reduction per day, the new cost is $80.
   Is   $100*12 = ($100-$20)*15   ?
        $1200  =  $80*15    ?       yes!

 

(PLZ note that I sometimes use '*' for multiply, and sometimes leave it out.)