"What Dragonlord Placidusax Revealed About Ancient Dragon Thrones – Don’t Miss This Epic Race!
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Solution: Let total days be $ x $. Ratio $ \frac{3}{8} = \frac{15}{x} $. Cross-multiply: $ 3x = 120 $. Divide: $ x = 40 $. \boxed{40}
Question: The average of $ 4z - 3 $, $ 2z + 5 $, and $ z + 1 $ is 6. Solve for $ z $.
Solution: Average equation: $ \frac{(4z - 3) + (2z + 5) + (z + 1)}{3} = 6 $. Simplify numerator: $ 7z + 3 = 18 $. Subtract 3: $ 7z = 15 $. Divide: $ z = \frac{15}{7} $. \boxed{\dfrac{15}{7}}
Question: A science journalist notes that 60% of their articles are about climate science. If they wrote 45 articles on climate science, how many total articles have they written?
Solution: Let total articles be $ x $. $ 0.6x = 45 $. Divide: $ x = \frac{45}{0.6} = 75 $. \boxed{75}Question: If $\frac{a + b}{a - b} + \frac{a - b}{a + b} = 3$, find the value of $\frac{a^2 + b^2}{a^2 - b^2}$.
\frac{a + b}{a - b} + \frac{a - b}{a + b} = 3
Let’s combine the terms on the left-hand side:
\frac{(a + b)^2 + (a - b)^2}{(a - b)(a + b)} = 3
Expand the numerator:
Adding them: