A friend has a \( 84 \% \) average before the final exam for a course. That score includes everything but the final, which counts for \( 15 \% \) of the course grade. What is the best course grade your friend can earn? What is the minimum score your friend would need on the final to earn a \( 75 \% \) for the course? Give answers accurate to at least one decimal place.
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To find the best course grade your friend can earn, we first need to consider that their current average is \( 84\% \), which accounts for \( 85\% \) of the total course weight. The final exam counts for \( 15\% \). The best score they can get on the final is 100%. Calculating the maximum course grade: \[ \text{Course grade} = 0.85 \times 84 + 0.15 \times 100 \] \[ = 71.4 + 15 = 86.4 \] The best course grade your friend can earn is \( 86.4\% \). Next, to find out the minimum score needed on the final to earn at least a \( 75\% \) for the course, we set up the equation: \[ 0.85 \times 84 + 0.15 \times x = 75 \] Where \( x \) is the score on the final. Solving for \( x \): \[ 71.4 + 0.15x = 75 \] \[ 0.15x = 75 - 71.4 \] \[ 0.15x = 3.6 \] \[ x = \frac{3.6}{0.15} = 24 \] Thus, your friend needs at least \( 24\% \) on the final exam to earn a \( 75\% \) course grade.