FN wrote:
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 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 very trickily-worded question.
I'll use the notation \(Y_n \to Y_k = R\) to signify the ratio increase from year \(n\) to year \(k\), where \(R\) is an integer \(> 1\) (the number of workers increased every year and is an integral ratio).
Determine the number of workers on year 2:
\(Y_2 = \frac{Y_4}{Y_2 \to Y_4} = \frac{10500}{6} = 1750\)
Since \(Y_2 \to Y_4 = 6\) is a 2-year gap, there must be two intermediate ratios, the product of which is 6.
Therefore each ratio is a combination of the prime factors of 6. We do not currently know this combination.
\(Y_2 \to Y_3 \,\bigg\vert\, Y_3 \to Y_4 = \{2,3\}\).
Since \(Y_1 \to Y_3 = 14\) is a 2-year gap, there must be two intermediate ratios, the product of which is 14.
\(Y_1 \to Y_2 \,\bigg\vert\, Y_2 \to Y_3 = \{2,7\}\).
From the intersection of the above two statements, we can see that: \(Y_2 \to Y_3 = 2\). From this, we can conclude:
\(Y_1 \to Y_2 = 7\\\\
Y_3 \to Y_4 = 3\)
Since we know \(Y_2\) from above, we can obtain \(Y_1\) as follows:
\(Y1 = \frac{Y_2}{Y_1 \to Y_2} = \frac{1750}{7} = 250\)
\(Y_0 \to Y_3 = 70\\\\
Y_1 \to Y_3 = 14\\\\
\therefore Y_0 \to Y_1 = 5\)
\(Y_0 = 50\)
\(Y_0 \to Y_1 = 5 \,\vert\, Y_1 = 250\\\\
Y_1 \to Y_2 = 7 \,\vert\, Y_2 = 1750\\\\
Y_2 \to Y_3 = 2 \,\vert\, Y_3 = 3500\\\\
Y_3 \to Y_4 = 3 \,\vert\, Y_4 = 10500\)