Exam Inequality Fraction
1. Problem 16: A student scores 10 out of 15 on Part 1 and 4 out of 10 on Part 2. To pass the exam, the student needs at least 54% of the total marks.
2. Calculate total marks obtained by the student:
$$\text{Total score} = 10 + 4 = 14$$
3. Calculate total maximum marks:
$$\text{Total max marks} = 15 + 10 = 25$$
4. Calculate the percentage scored by the student:
$$\text{Percentage} = \frac{14}{25} \times 100 = 56\%$$
5. Since the student scored 56%, which is more than 54%, the student passes the examination.
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6. Problem 17: Solve the inequality
$$2x - 1 \leq 11$$
7. Add 1 to both sides:
$$2x \leq 12$$
8. Divide both sides by 2:
$$x \leq 6$$
9. Solution: All values of $x$ less than or equal to 6 satisfy the inequality.
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10. Problem 18: Solve the equation
$$1 \frac{1}{5} + \frac{22}{d} = 1 \frac{3}{4}$$
11. Convert mixed numbers to improper fractions:
$$1 \frac{1}{5} = \frac{6}{5}, \quad 1 \frac{3}{4} = \frac{7}{4}$$
12. Substitute into equation:
$$\frac{6}{5} + \frac{22}{d} = \frac{7}{4}$$
13. Subtract $\frac{6}{5}$ from both sides:
$$\frac{22}{d} = \frac{7}{4} - \frac{6}{5}$$
14. Find common denominator and subtract:
$$\frac{7}{4} - \frac{6}{5} = \frac{35}{20} - \frac{24}{20} = \frac{11}{20}$$
15. So,
$$\frac{22}{d} = \frac{11}{20}$$
16. Cross multiply:
$$22 \times 20 = 11 \times d$$
$$440 = 11d$$
17. Divide both sides by 11:
$$d = 40$$
18. Final answer for $d$ is 40.