Problem Analysis:

The student received a grade of 7 for the first course and decides to complete more courses, each with a grade of 10. The courses are of equal weight.

The questions are:

1. (a) Create an expression to calculate the average grade after $x$ additional courses graded 10.
2. (b) Solve for how many courses $x$ are needed for the rounded average grade to equal 10.

Solution:

(a) Forming the Expression for the Average

Let:

• $x$ be the number of additional courses graded 10.
• Total courses = $1 + x$ (1 initial + $x$ new courses).
• Total sum of grades = $7 + 10x$ (grade 7 for the first + 10 for each new course).

The formula for the average grade is: $\text{Average} = \frac{\text{Total sum of grades}}{\text{Total courses}}$ Substitute the values: $\text{Average} = \frac{7 + 10x}{1 + x}$

(b) Solving for $x$ to Make the Average Round to 10

To find when the rounded average equals 10, the actual average must satisfy: $\frac{7 + 10x}{1 + x} \geq 9.5$ This is because an average of 9.5 or higher rounds up to 10.

1. Start with the inequality: $\frac{7 + 10x}{1 + x} \geq 9.5$

2. Eliminate the denominator (since $1 + x > 0$ for $x \geq 0$): $7 + 10x \geq 9.5(1 + x)$

3. Distribute 9.5 on the right-hand side: $7 + 10x \geq 9.5 + 9.5x$

4. Simplify: $10x - 9.5x \geq 9.5 - 7$ $0.5x \geq 2.5$

5. Solve for $x$: $x \geq \frac{2.5}{0.5} = 5$

Final Answer:

• (a) The expression for the average grade is: $\frac{7 + 10x}{1 + x}$

• (b) The student needs to complete at least 5 additional courses with a grade of 10 for the rounded average to equal 10.

Do you need further clarification or steps for this solution?

Related Questions:

1. How would the solution change if the first grade were different, e.g., 6 or 8?
2. How do you calculate the exact average when $x = 5$?
3. How does rounding impact the required number of courses?
4. Can you generalize this formula for any initial grade and any new grade?
5. How many additional courses are needed if the average must be exactly 9.8 instead of rounding to 10?

Tip: Always check the boundary conditions when rounding numbers in inequalities.