Question 1146604
<br>
Let x be the actual number of days of the vacation.  Then<br>
1200/x = daily amount spent
1200/(x+2) = daily amount spent if the vacation were 2 days longer<br>
The amount spent per day would have been $50 less if the vacation had been 2 days longer:<br>
{{{1200/(x+2) = 1200/x-50}}}<br>
Divide by the common factor of 50 to make the numbers smaller that you have to work with:<br>
{{{24/(x+2) = 24/x-1}}}<br>
Multiply by the least common denominator, (x)(x+2):<br>
{{{24x = 24(x+2)-x(x+2)}}}
{{{24x = 24x+48-x^2-2x}}}
{{{x^2+2x-48 = 0}}}
{{{(x+8)(x-6) = 0}}}
{{{x = -8}}} or {{{x = 6}}}<br>
Clearly the negative value makes no sense in the problem, so<br>
ANSWER: the vacation lasted x=6 days<br>
CHECK:
1200/6 = 200
1200/8 = 150
200-150 = 50