Word Problem GMAT, GRE, SAT

The workforce of a certain company comprised exactly 10,500 employees after a four-year period during which it increased every year. During this four-year period, the ratio of the number of workers from one year to the next was always an integer. The ratio of the number of workers after the fourth year to the number of workers after the second year is 6 to 1. The ratio of the number of workers after the third year to the number of workers after the first year is 14 to 1. The ratio of the number of workers after the third year to the number of workers before the four-year period began is 70 to 1. How many employees did the company have after the first year?

(A) 50
(B) 70
(C) 250
(D) 350
(E) 750





10,500/6 = 1,750 which is the number of workers after the second year 

We need a number of workers after the third year. 

Let it = X. We have some clues to get to X: 
- 10,500/X is an integer greater than 1. 
- X/1,750 is an integer greater than 1. 

We note that 10,500/1750 = 6. So we need two integer factors of 6 and neither factor can be 1. This means that the factors are 2 and 3. 

Therefore we have two possibilities: either X is 3,500 or X is 5,250. 

But we know that the ratio of the number of workers after the third year to the number of workers after the first year is 14:1. 

3,500/14 = 250 
5,250/14 = 375 

So the number of workers after the first year is either 250 or 375. 

But this has to be an integer ratio with the number of workers after the second year, which is 1,750. Of the two possibilities, only 250 satisfies the condition, because 1,750/250 = 7 and 1,750/375 = 4 2/3. 

Therefore the number of workers after the first year is 250.

C