Calculation:

Let the grade in the final test be x

Average of n numbers \(\displaystyle{a}_{{{1}}}+{a}_{{{2}}}+{a}_{{{3}}}+\ldots+{a}_{{{n}}}\) is given by \(\displaystyle{A}={\frac{{{a}_{{{1}}}+{a}_{{{2}}}+{a}_{{{3}}}+\ldots+{a}_{{{n}}}}}{{{n}}}}\)

Since the final carries double weight, it is considered as 2 numbers and the score is doubled,

i.e.,

\(\displaystyle{A}={\frac{{{87}+{59}+{73}+{2}{x}}}{{{5}}}}\)

Given average is 75

Therefore,

\(\displaystyle{5}={\frac{{{87}+{59}+{73}+{2}{x}}}{{{5}}}}\)

Multiplying both sides by 5

\(\displaystyle{75}\cdot{5}={\frac{{{87}+{59}+{73}+{2}{x}}}{{{5}}}}\cdot{5}\)

\(\displaystyle{375}={219}+{2}{x}\)

Subtracting 219 from both the sides,

\(\displaystyle{375}—{219}={219}+{2}{x}—{219}\)

\(\displaystyle{156}={2}{x}\)

\(\displaystyle{2}{x}={156}\)

Dividing both sides by 2,

\(\displaystyle{\frac{{{2}{x}}}{{{2}}}}={\frac{{{156}}}{{{2}}}}\)

\(\displaystyle{x}={78}\)

Therefore the score required in final test to get an average of 75 with double weight for final test is 78.

Let the grade in the final test be x

Average of n numbers \(\displaystyle{a}_{{{1}}}+{a}_{{{2}}}+{a}_{{{3}}}+\ldots+{a}_{{{n}}}\) is given by \(\displaystyle{A}={\frac{{{a}_{{{1}}}+{a}_{{{2}}}+{a}_{{{3}}}+\ldots+{a}_{{{n}}}}}{{{n}}}}\)

Since the final carries double weight, it is considered as 2 numbers and the score is doubled,

i.e.,

\(\displaystyle{A}={\frac{{{87}+{59}+{73}+{2}{x}}}{{{5}}}}\)

Given average is 75

Therefore,

\(\displaystyle{5}={\frac{{{87}+{59}+{73}+{2}{x}}}{{{5}}}}\)

Multiplying both sides by 5

\(\displaystyle{75}\cdot{5}={\frac{{{87}+{59}+{73}+{2}{x}}}{{{5}}}}\cdot{5}\)

\(\displaystyle{375}={219}+{2}{x}\)

Subtracting 219 from both the sides,

\(\displaystyle{375}—{219}={219}+{2}{x}—{219}\)

\(\displaystyle{156}={2}{x}\)

\(\displaystyle{2}{x}={156}\)

Dividing both sides by 2,

\(\displaystyle{\frac{{{2}{x}}}{{{2}}}}={\frac{{{156}}}{{{2}}}}\)

\(\displaystyle{x}={78}\)

Therefore the score required in final test to get an average of 75 with double weight for final test is 78.