Subtracting a fraction by -1, simplifies to putting -n on the numerator

Question

Math Question: You share 60 dollars between a certain number of people.

Each person gets 1 dollar more than if you had shared the $60 between 2 more people.

How many people are you sharing the 60 dollars between?

I have the answer in the image below:

Question and Answer

I would like some explanation, firstly for what a represents (I'm thinking each person)?

And how $\dfrac {60}{n}-1$ simplifies to $\dfrac {60-n}{n}$

Also how it then becomes evaluated to $60n = (60-n)(n+2)$ (Perhaps cross multiplication?)

Cheers

Answer 1

Hint: If you have the fraction $\dfrac {1}{2}+ \dfrac {1}{3}$, what would you need to make the denominators the same?

Answer 2

A represents amount given to those people which are represented as n. This amount is a dollar less if you have the +2 for the people in the denominator. So we begin with two equations involving fractions. The picture then takes the left equation that's solved for a and uses substitution to put into the other equation on the right. Then, because we have a fraction equal to another fraction they use cross multiplication (not required but saved a step). Finally they get common denominators, foil and simplify numeratora, and move everything to one side which gives a quadratic equation. They use the diamond method to solve for n which gives two real solutions but throw away the negative answer because you can't have negative amount of people which leaves 10 as the answer.