Let’s solve the problem step by step.
1. Total number of questions answered by Elena in Mathematics
Elena scored 80% in Mathematics, with a total of 40 questions. To find out how many questions she answered correctly: \[ \text{Correct answers} = 80% \text{ of } 40 = 0.80 \times 40 = 32 \] If she answered 32 questions correctly, she likely attempted all the questions in Mathematics as it is common for objective tests. So, we assume she answered all 40 questions.
Total questions answered in Mathematics = 40
2. Percentage score in Science
Elena was able to answer two-thirds of the science questions correctly. The total number of science questions is 30. To find the number of questions she answered correctly: \[ \text{Correct answers in Science} = \frac{2}{3} \text{ of } 30 = 20 \] To find her percentage score in Science: \[ \text{Percentage} = \left(\frac{\text{Correct answers}}{\text{Total questions}}\right) \times 100 = \left(\frac{20}{30}\right) \times 100 = \frac{2000}{30} \approx 66.67% \] Rounding to the nearest whole number gives her a score of 67%.
3. Percentage of questions answered incorrectly in English Language
Elena scored 45 out of 60 in English Language. To find out how many questions she got wrong: \[ \text{Wrong answers in English} = 60 - 45 = 15 \] To find the percentage of questions she got wrong: \[ \text{Percentage wrong} = \left(\frac{\text{Wrong answers}}{\text{Total questions}}\right) \times 100 = \left(\frac{15}{60}\right) \times 100 = 25% \]
4. Number of Science questions scored wrong
From question 2, we know that she correctly answered 20 science questions out of 30. Thus, the number of wrong answers in Science is: \[ \text{Wrong answers in Science} = 30 - 20 = 10 \]
Solving for \( n \)
Given the equation: \[ 4n - 3 = \frac{1}{256} \] First, we add 3 to both sides: \[ 4n = 3 + \frac{1}{256} \] We can express 3 as a fraction: \[ 3 = \frac{768}{256} \] So, \[ 4n = \frac{768 + 1}{256} = \frac{769}{256} \] Next, divide both sides by 4: \[ n = \frac{769}{4 \times 256} = \frac{769}{1024} \]
Thus, the value of \( n \) is: \[ \boxed{\frac{769}{1024}} \]