Question 1162699
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A setup for solving the problem using formal algebra....<br>
Let x be the number who went on the expedition.  Then x+15 is the number that were originally going to go.<br>
The cost per person originally was 3200/(x+15); the cost now is 3200/x.<br>
We are told the difference between those two costs is $48:<br>
{{{3200/x-3200/(x+15) = 48}}}<br>
Multiply by the common denominator x(x+15) to clear fractions:<br>
{{{3200(x+15)-3200(x) = 48(x)(x+15)}}}
{{{48000 = 48(x^2+15x)}}}
{{{1000 = x^2+15x}}}
{{{x^2+15x-1000 = 0}}}<br>
To solve that by factoring, you need to find two integers with a difference of 15 whose product is 1000.  If your mental math is good, you should find 25 and 40 work.  So<br>
{{{(x+40)(x-25) = 0}}}<br>
Ignore the negative solution, since it makes no sense in the problem.<br>
{{{x = 25}}}<br>
ANSWER: 25 people went on the trip.<br>
CHECK:
3200/25 = 128
3200/40 = 80
128-80 = 48<br>
You should know how to set up and solve the problem like that, using formal algebra.<br>
But if a formal algebraic solution is  not required, you can solve the problem much faster if your mental math is good.<br>
In the original problem, you need to find two different pairs of two positive integers that give a product of 3200; you are looking to have the differences between the numbers in the two pairs equal to 15 and 48.<br>
A bit of playing around with the numbers will find
40*80 = 3200
25*128 = 3200<br>
The difference between 40 and 25 is 15; the difference between 80 and 128 is 48.  So those numbers are the ones we want.<br>
So originally there were 40 members paying $80 each, and in the end there were 25 members paying $128 each.<br>