To determine the minimum number of hours the student must work at the third job, let's break the problem into steps:

Step 1: Income from the first job

The student can work up to 20 hours per week at the school job at a rate of $10 per hour. If he works all 20 hours, his total income from this job is:

$\text{Income from first job} = 20 \times 10 = 200 \, \text{dollars.}$

Step 2: Income from the second job

The student works 8 hours on Saturdays at a rate of $12 per hour. His total income from this job is:

$\text{Income from second job} = 8 \times 12 = 96 \, \text{dollars.}$

Step 3: Total income needed

His weekly expenses are $400. The total income needed to cover his expenses is:

$\text{Income from all jobs} \geq 400.$

Step 4: Income from the third job

The total income from the first and second jobs is:

$\text{Total from first two jobs} = 200 + 96 = 296 \, \text{dollars.}$

The remaining income required to meet the $400 expense is:

$\text{Remaining income needed} = 400 - 296 = 104 \, \text{dollars.}$

The third job pays $9 per hour. Let $h$ represent the number of hours the student must work at the third job. The equation for the third job income is:

$9h \geq 104.$

Step 5: Solve for $h$

Divide both sides by 9:

$h \geq \frac{104}{9} \approx 11.56.$

Since the student cannot work a fraction of an hour, round up to the nearest whole number:

$h = 12 \, \text{hours.}$

Final Answer:

The student must work a minimum of 12 hours at the third job.

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Would you like a detailed explanation of any step or further assistance?

Related Questions:

1. How would the solution change if his expenses increased to $450 per week?
2. What is the total income he earns if he works 12 hours at the third job?
3. If the third job paid $10 per hour, how would that affect the number of hours needed?
4. What would happen if he worked fewer than 20 hours at the first job?
5. How much extra income does he earn if he works 15 hours at the third job?

Tip:

When solving for minimum requirements, always round up fractional hours to ensure constraints are met.