A group of friends decided to divide the $1,000 cost of a trip equally among themselves. When one of the friends decided not to go on the trip, those remaining still divided the $1,000 cost equally, but each friend's share of the cost increased by $50. How many friends were in the group originally?

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The description of the situation can be modeled by the difference of rates:

$$ \text{new rate} - \text{old rate} = \$50 $$

The rate is equal to the total cost divided by the number of friends. If \(f\) is the number of friends originally,

$$ \text{new rate}=\frac{\$1{,}000}{F-1} $$ $$ \text{old rate}=\frac{\$1{,}000}{F} $$

Putting it all together:

$$ \frac{1{,}000}{F-1} - \frac{1{,}000}{F}=50 $$ $$ \frac{1{,}000}{F-1}\cdot \frac{F}{F} - \frac{1{,}000}{F}\cdot \frac{F-1}{F-1}=50 $$ $$ \frac{1{,}000F-(1{,}000F-1{,}000)}{F(F-1)} =50$$ $$ \frac{1{,}000}{F^2-F}=50 $$ $$ \frac{20}{F^2-F}=1 $$ $$ F^2-F-20=0 $$ $$ (F-5)(F+4)=0 $$ $$ F=\boxed{5} $$

Once you've confirmed your rate expresions, we can just try different values of \(F\) and check if the difference is $50.

$$ \text{new rate}=\frac{\$1{,}000}{F-1} $$ $$ \text{old rate}=\frac{\$1{,}000}{F} $$

We can also check for whole integer answers since we need to obtain a difference of $50.

$$ \text{for }F=5$$ $$ 250-200=50   \checkmark$$