Chapter 2.7, Problem 108ES

To determine

a) The grades on the final to get A grade.

b) The grades on the final so as to lose B grade.

Solution:

a) With 100 marks in final exam, A is not possible.

b) To earn B, the score $y$ in the final exam should be $y\ge 68$.

Explanation:

Given:

The given is that the two examination grades are $86\text{ , 78 and }88$ and in order to earn an A in the course, the final average should be $\text{atleast }90$. Also to earn a B in the course the final average should be $\text{atleast 8}0$.

Concept and Formula Used:

The concept used here is the concept of linear inequality. The concept of inequality includes two types of inequalities:

1) At most inequality: this inequality is less than equal to $(\le )$ type inequality.

2) At least inequality: this inequality is more than equal to $(\ge )$ type inequality.

And the formula used here is the formula of an average.

   $\text{Formula of the average = }\frac{\text{sum of the numbers whose average is to be taken out}}{\text{total number of observations}}$

Calculation:

a) As to earn A, the final average should be $\text{atleast }90$, thus the inequality is of greater than type. As $100$ is the score in the final, thus the average obtained is given by, $$

   $\begin{array}{l}\text{Formula of the average = }\frac{\text{sum of the numbers whose average is to be taken out}}{\text{total number of observations}}\\ \Rightarrow \text{Average = }\frac{88+86+78+100}{4}\\ =\frac{352}{4}=88\end{array}$

As to earn A, the final average should be at least 90, but the average is 88 thus A is not possible.

b) As to earn B, the final average should be $\text{atleast 8}0$, thus the inequality is of greater than or equal to type.

Let $y$ be the score in final exam, so according to the given situation the inequality is

   $\begin{array}{l}\text{Formula of the average = }\frac{\text{sum of the numbers whose average is to be taken out}}{\text{total number of observations}}\\ \Rightarrow \frac{86+88+78+y}{4}\ge 80\\ \Rightarrow 4\left(\frac{86+88+78+y}{4}\right)\ge 4(80)\text{ (Multiplying both sides by 4)}\\ \Rightarrow \frac{4(86+88+78+y)}{4}\ge 320\\ \Rightarrow 86+88+78+y\ge 320\\ \Rightarrow 252+y\ge 320\\ \Rightarrow 252+y-252\ge 320-252\text{ (Subtracting 252 from both the sides)}\\ \Rightarrow y\ge 68\end{array}$

Thus to earn B, the score in the final exam should be greater than or equal to 68.

Conclusion:

a) With 100 marks in final exam, A is not possible.

b) To earn B, the score $y$ in the final exam should be $y\ge 68$.