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Question: A palynologist collects pollen samples from three layers of sediment. The first layer contains $ \frac{3}{8} $ kg of pollen, the second $ \frac{5}{12} $ kg, and the third $ \frac{7}{16} $ kg. What is the total mass of pollen collected, expressed as a simplified fraction of a kilogram?
Solution: We compute $ \frac{3}{8} + \frac{5}{12} + \frac{7}{16} $. The least common denominator of 8, 12, and 16 is 48. Convert each fraction:
\frac{3}{8} = \frac{18}{48}, \quad \frac{5}{12} = \frac{20}{48}, \quad \frac{7}{16} = \frac{21}{48}
\frac{18 + 20 + 21}{48} = \frac{59}{48}
Since $ \frac{59}{48} > 1 $, the total is $ \frac{59}{48} $ kg, which is an improper fraction. It cannot be simplified further.
$\boxed{\frac{59}{48}}$
Question: A climate researcher in New York City analyzes energy usage data from three boroughs. Over one week, the total CO₂ emissions were proportional to the ratios $4:7:9$. If the second borough emitted 147 metric tons, how many metric tons did the third borough emit?
Solution: The ratio of emissions is $4:7:9$. Let the common multiplier be $x$. Then the second borough emitted $7x = 147$ metric tons. Solving for $x$:
x = \frac{147}{7} = 21