In a certain company, the number of entry-level employees is 60\% of the total number of e… — SAT

In a certain company, the number of entry-level employees is $60\%$ of the total number of employees, and the remaining employees are managers. Over a year, the number of entry-level employees increases by $x\%$ , and the number of managers decreases by $y\%$ . As a result, the total number of employees in the company decreases by $4\%$ , and the ratio of the number of managers to the number of entry-level employees becomes $1$ to $3$ . What is the value of $x$ ?

Answer: 20

Answer

The value of

x

20

To find the value of

x

, represent the initial and final quantities of entry-level employees and managers algebraically. Using the

4\%

decrease in total employees and the final manager-to-entry-level ratio of

1

3

, set up a system of linear equations:

2y - 3x = 20

and

2y = 100 - x

. Solving this system for

x

yields

20

Step-by-Step Solution

Represent the initial employee counts algebraically.

Initial entry-level employees =

0.60T

, initial managers =

0.40T

, where

T

is the initial total.

Establishing initial quantities based on the given percentages.

Represent the final employee counts after percent changes.

Final entry-level =

0.60T(1 + 0.01x)

, final managers =

0.40T(1 - 0.01y)

Applying the percent increase and decrease to the respective initial quantities.

Use the total percent change to create a linear equation.

2y - 3x = 20

Setting the sum of the final counts equal to

0.96T

and simplifying.

Use the final ratio of managers to entry-level employees to create a second equation.

2y = 100 - x

Setting the ratio of final managers to final entry-level employees equal to

1/3

and simplifying.

Solve the system of equations for

x

x = 20

Substituting the second equation into the first to solve for the target variable.

Key Concept

Percents and Percent Change

Question

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Key Concept

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