How many students were in the group initially?

Question

A group of students decide to divide the $400 profit from their book sale among themselves. When they realize that there are three additional students with whom they must split the profits, they once again divide the $400 equally, and each students share of the profits decreases by $30. How many students were in the group initially?

 

Please explain how to solve problems like this, as well so that I know how to approach/ do similar ones in the future.

Answer 1

Let n = the number of students to begin with, so that n+3 = the number afterwards.

Now translate the English into math.

400/(n + 3) = the amount each student will receive after the 3 new students are added

400/n is the amount each would have received originally.

Therefore the equation is'

 

400/(n + 3) = (400/n) - 30 = (400 - 30n)/n

Cross multiply:

400n = 400n + 1200 - 30n² - 90n

Collect terms and divide through by 30:

n² + 3n - 40 = 0

(n - 5)(n + 8) = 0

The root n = - 8 is extraneous and n = 5.

 

Initially each student would have received 400/5 = 80; after the 30 reduction that would be 50.

And 400/8 = 50...so the answer does, in fact ssolve the problem.

 

Answer 2

Start with what you know and break down the problem. So you start and know that N₀ (some initial number of students) are dividing 400 amongst themselves, so write that down. 400/N₀=x (x is each student's share). When 3 students are added (N₀+3), the shares of the same amount of money are decreased by 30 (x-30). So here you have 400/N₀=x and 400/(N₀+3)=(x-30). Translating words into symbols makes a paragraph much easier to handle, and take it one phrase at a time