Geography tests

On three 150-point geography tests, you earned grades of 88%, 94%, and 90%. The final test is worth 250 points. What percent do you need on the final to earn 93% of the total points on all tests?

Correct result:

p =  97.2 %

Solution:

$\ \\ e_{1}=150 \cdot \ \dfrac{ 88 }{ 100 }=132 \ \\ e_{2}=150 \cdot \ \dfrac{ 94 }{ 100 }=141 \ \\ e_{3}=150 \cdot \ \dfrac{ 90 }{ 100 }=135 \ \\ \ \\ \dfrac{ p }{ 100 } \cdot \ 250+e_{1}+e_{2}+e_{3}=\dfrac{ 93 }{ 100 } \cdot \ (3 \cdot \ 150+250) \ \\ \ \\ b=\dfrac{ 93 }{ 100 } \cdot \ (3 \cdot \ 150+250) - e_{1} -e_{2} - e_{3}=\dfrac{ 93 }{ 100 } \cdot \ (3 \cdot \ 150+250) - 132 -141 - 135=243 \ \\ \ \\ p=100 \cdot \ \dfrac{ b }{ 250 }=100 \cdot \ \dfrac{ 243 }{ 250 }=\dfrac{ 486 }{ 5 }=97.2 \%$

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