Answer

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**Hint:**Here, we are going to solve the question directly, By converting the percentage of her grade which she needs to get in the entire exam, to the number of questions she needs to answer to get her grade in the entire exam. Then we need to find the number of questions which she answered correctly. After that we need to subtract the number of questions which she answered correctly, from the total number of questions which she needs to answer correctly to get her grade in the entire exam.

Then, we will have the number of questions which she needs to answer to get her required grade in the entire exam. Then we need to convert them to the percentage using the percentage formula.

**Formula Used:**

\[{\text{percentage = }}\dfrac{{{\text{value}}}}{{{\text{total value}}}}{{ \times 100}}\]

\[{\text{value = }}\dfrac{{{\text{percentage}}}}{{{\text{100}}}}{\times \text{ total value}}\]

**Complete step by step answer:**

It is given that the total number of questions \[{\text{ = 80}}\]

Now, the number of questions that she needs to be answered correctly to score \[{{75\% }}\] out of the total number of questions.

So we can use the formula and we write it as \[\dfrac{{75}}{{100}} \times 80 = 60\] questions.

\[\therefore \]The total number of questions she needs to answer correctly to score \[{{75\% }}\] is \[60\]questions

Arpita has answered \[{{65\% }}\] of the first \[{\text{40}}\] questions.

Again we use the formula,

Number of questions \[{\text{ = }}\] \[\dfrac{{65}}{{100}} \times 40 = 26\] questions

\[\therefore \]Number of questions she answered correctly on first 40 questions \[{\text{ = }}\] \[{\text{26}}\] questions

Number of questions she has to answer more to get \[{{75\% }}\] is \[{\text{60 - 26 = 34}}\] questions

Let the \[{\text{34}}\] questions which she has to answer more to get \[{{75\% }}\] be \[{{y\% }}\].

Now we use the formula we get,

\[{\text{value = }}\dfrac{{{\text{percentage}}}}{{{{100}}}}{\times \text{ total value}}\]

\[ \Rightarrow \]\[{\text{34 = }}\dfrac{{\text{y}}}{{{{100}}}}{{ \times 40}}\]

\[\Rightarrow {{y = 34 \times }}\dfrac{{{\text{100}}}}{{{\text{40}}}}{{ = 85\% }}\]

**She has to answer \[{{85\% }}\] of the remaining questions correctly to get \[{{75\% }}\] of the entire exam.**

**Note:**

There is an alternative method, for solving this problem,

Let the required percent of questions she has the answer to be \[{\text{x}}\].

So, we have to add that Arpita has answered \[{{65\% }}\] of the first \[{\text{40}}\] questions and \[{\text{40}}\] question she need to answer correctly for her grade in the entire exam to be \[{{75\% }}\] is equal to the out of \[{\text{80}}\] questions.

Now we can write it as,

\[\Rightarrow 65\% \] of \[40\]\[ + \]\[{{x\% }}\] of \[40\]\[{\text{ = }}\] \[{{75\% }}\] of \[{\text{80}}\].

We write it as the mathematical terms we get,

\[ \Rightarrow \] \[\dfrac{{{\text{65}}}}{{{\text{100}}}}{{ \times 40 + }}\dfrac{{\text{x}}}{{{\text{100}}}}{{ \times 40 = }}\dfrac{{{\text{75}}}}{{{\text{100}}}}{{ \times 80}}\]

Here we have to the value of x so we take that as LHS and remaining as RHS we get,

\[\Rightarrow \dfrac{{{\text{x}} \times {\text{40}}}}{{100}}{\text{ = }}\left( {\dfrac{{{\text{75}}}}{{{\text{100}}}}{{ \times 80 - }}\dfrac{{{\text{65}}}}{{{\text{100}}}}{{ \times 40}}} \right)\].

Taking the variable as LHS and remaining take as the reciprocal term we get,

\[\Rightarrow {\text{x = }}\left( {\dfrac{{{\text{75}}}}{{{\text{100}}}}{{ \times 80 - }}\dfrac{{{\text{65}}}}{{{\text{100}}}}{{ \times 40}}} \right) \times \dfrac{{100}}{{40}}\]

On simplifying the terms we get,

\[\Rightarrow {{x = ( 60 - 26 ) \times }}\dfrac{{\text{5}}}{{\text{2}}}\]

On subtracting the bracket term we get,

\[\Rightarrow {{x = 34 \times }}\dfrac{{\text{5}}}{{\text{2}}}\]

Let us divide the term we get,

\[\Rightarrow {\text{x = 17 }} \times {\text{ 5 }}\].

On multiplying we get,

\[\Rightarrow {\text{x = 85}}\]

**\[\therefore \] The Percentage of Number of questions that she needs to correctly answer to get \[75\% \] in entire exam is \[85\% \].**

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