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.

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