Would you please explain the solution to Problem #31 in the calculator-allowed section of Practice Test #6?

This one is all about translating carefully between words and math. You can write two equations. Let’s use f for the original number of friends. If all f friends go on the trip, the cost per person is $20 less than if f-2 friends go on the trip. Cost per Person with f-2 Friends minus Cost per Person with f Friends is $20.

    \begin{align*}\dfrac{800}{f-2}-\dfrac{800}{f}&=20\end{align}

To solve, multiply by f(f-2):

    \begin{align*}f(f-2)\left(\dfrac{800}{f-2}\right)-f(f-2)\left(\dfrac{800}{f}\right)&=20f(f-2)\\800f-800(f-2)&=20f^2-40f\\800f-800f+1600&=20f^2-40f\\0&=20f^2-40f-1600\\0&=f^2-2f-80\\0&=(f-10)(f+8)\end{align}

That tells you that the number of friends originally in the group is either 10 or –8. Since the answer can’t be negative, it must be 10.

BONUS: The other way to go on this one is the way I actually did it the first time: just dividing 800 by a bunch of numbers in a row until I saw a difference of 20:

800/4 = 200
800/5 = 160
800/6 = 133.333
800/7 = 114.286
800/8 = 100
800/9 = 88.889
800/10 = 80

There you have it: 100 and 80 are 20 apart, and that’s the difference between splitting 800 over 10 people vs. over 8 people. That was fast!

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