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The LCM is obtained by taking the highest powers of all primes:
Question: A microbiologist is preparing a nutrient solution and needs to mix two components in a ratio of 5:3. If she uses 40 liters of the first component, how many liters of the second component are needed?
Solution: The ratio of the first component to the second is 5:3. Let $ x $ be the amount of the second component. Then:
Question: An engineering student designs a rectangular garden with a fixed area of 120 square meters. What is the smallest possible perimeter of the garden?
Solution: Let the length be $ l $ and the width be $ w $. The area is given by:
To minimize $ P(l) $, we find the critical point by differentiating:
Lets compute the total probability by summing over the valid cases.
For a fixed region (say region A), the number of foxes choosing it follows a binomial distribution:
But since only one region can have 6 or more (as $ 6+6 > 8 $), and if 6 choose region A, the other 2 must split between the other two, none exceeding 2, so no overlap.
Simplify: divide numerator and denominator by 3: