In a company, the number of graduate employees and the undergraduate employees were in the ratio of 1 : 4. Ten more persons joined the company. Some of the employees acquired graduation degree and the number of undergraduates reduced by 6, thereby changin

Let the number of graduate employees be \( g \) and the number of undergraduate employees be \( u \). According to the problem, we have the initial ratio of graduates to undergraduates as: \[ \frac{g}{u} = \frac{1}{4} \] From this ratio, we can express \( u \) in terms of \( g \): \[ u = 4g \] Next, we are informed that 10 more persons joined the company, and some of the employees acquired graduation degrees, which resulted in a reduction of the number of undergraduates by 6. Let's denote the number of graduates who acquired degrees as \( x \). After these changes, the new number of graduates becomes \( g + x \) and the new number of undergraduates becomes \( u - 6 \). The new ratio of graduates to undergraduates is given as: \[ \frac{g + x}{u - 6} = \frac{1}{3} \] Cross-multiplying this ratio gives us: \[ 3(g + x) = u - 6 \] Substituting \( u = 4g \) into this equation, we have: \[ 3(g + x) = 4g - 6 \] Expanding and rearranging leads to: \[ 3g + 3x = 4g - 6 \] \[ 3x = 4g - 6 - 3g \] \[ 3x = g - 6 \] \[ x = \frac{g - 6}{3} \] Next, we need to ensure that \( x \) is a non-negative integer. Thus, \( g - 6 \) must be non-negative and divisible by 3, leading us to: \[ g \geq 6 \] \[ g - 6 \equiv 0 \mod 3 \] We can express \( g \) in terms of its minimum value. Let \( g = 6 + 3k \) for non-negative integer \( k \). Thus, we can substitute this back into our expression for \( u \): \[ u = 4g = 4(6 + 3k) = 24 + 12k \] Now, we compute the total number of employees initially: \[ \text{Total} = g + u = (6 + 3k) + (24 + 12k) = 30 + 15k \] The total number of employees will vary depending on \( k \). For \( k = 0 \): \[ \text{Total} = 30 + 15(0) = 30 \] For \( k = 1 \): \[ \text{Total} = 30 + 15(1) = 45 \] For \( k = 2 \): \[ \text{Total} = 30 + 15(2) = 60 \] Thus, the total number of employees in the company initially can take values like 30, 45, 60, etc., but since the count is logically constrained to the changes described, the initial solution of \( k = 0 \) (30 employees, when substituted into the conditions) should be verified for the consistency of the problem statement. Final check if \( g = 6 \) and \( u = 24 \): 1. Initial ratio: \[ \frac{g}{u} = \frac{6}{24} = \frac{1}{4} \] 2. After 10 join, and \( x = 0 \): \[ g + x = 6, \quad u - 6 = 24 - 6 = 18 \] New ratio: \[ \frac{6}{18} = \frac{1}{3} \] Both conditions satisfied correctly lead us to the final conclusion: Thus, the initial total number of employees in the company is: \[ \boxed{30} \]