Question

error analysis a company has two branches that are growing rapidly. branch 1 opened with 26 clients and they get 3 new clients each week. branch 2 opened with 39 clients and they get 12 new clients each week. a manager of the company wants to compare the number of clients each branch has, n, with respect to the number of weeks since the company opened, w. the manager incorrectly uses the system of equations below to compare the number of clients for each branch. what error might the manager have made? a. the manager multiplied by w instead of adding w. b. the manager switched the constant and coefficient of w in both equations. c. the manager combined the rates in one equation and the number of clients each branch started with in the other equation. d. the manager used the rate for branch 1 with the number of clients branch 2 started with and used the rate for branch 2 with the number of clients branch 1 started with.

Explanation:

Step1: Define correct equations

For Branch 1: $n_1 = 26 + 3w$
For Branch 2: $n_2 = 39 + 12w$

Step2: Analyze each error option

• Option A: Multiplying by $w$ would give $n=26w+3w$ or similar, which is not a mix-up of rate/starting value.
• Option B: Switching constant and coefficient would give $n_1=3+26w$, $n_2=12+39w$, which is not the described cross mix.
• Option C: Combining rates/starting values in one equation would be a different structure, not cross-assigning.
• Option D: This matches the error of using Branch 1's rate (3) with Branch 2's starting clients (39), and Branch 2's rate (12) with Branch 1's starting clients (26), creating incorrect equations like $n_1=39+3w$, $n_2=26+12w$.

Answer:

D. The manager used the rate for Branch 1 with the number of clients Branch 2 started with and used the rate for Branch 2 with the number of clients Branch 1 started with.