A man earns $40 per hour as a consultant Additionally, he also earns $6 per hour as a content writer. He is only allowed to work 15 hours per week, but wants to make $450 per week. If n represent, the integer number of hours he works as a consultant, what is the least integer value of n?

Answer

To solve the problem step by step, we will define the variables and set up the equations based on the information given.

### Step 1: Define the Variables
Let:
- \( n \) = number of hours worked as a consultant
- \( y \) = number of hours worked as a content writer

### Step 2: Set Up the Equations
From the problem, we know:
1. The total hours worked per week is 15:
\[ n + y = 15 \]

2. The total earnings per week should be $450:
\[ 40n + 6y = 450 \]

### Step 3: Solve for One Variable
From the first equation, we can express \( y \) in terms of \( n \):
\[ y = 15 - n \]

### Step 4: Substitute into the Earnings Equation
Now substitute \( y \) in the second equation:
\[ 40n + 6(15 - n) = 450 \]

### Step 5: Simplify the Equation
Distributing the 6:
\[ 40n + 90 - 6n = 450 \]
Combine like terms:
\[ 34n + 90 = 450 \]

### Step 6: Isolate \( n \)
Subtract 90 from both sides:
\[ 34n = 450 - 90 \]
\[ 34n = 360 \]

### Step 7: Solve for \( n \)
Divide both sides by 34:
\[ n = \frac{360}{34} \approx 10.588 \]

### Step 8: Find the Least Integer Value of \( n \)
Since \( n \) must be an integer, we take the least integer greater than or equal to 10.588, which is:
\[ n = 11 \]

### Final Answer
The least integer value of \( n \) is **11**.

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