In the diagram below, a diameter of each of the two smaller circles is a radius of the larger circle. If the two smaller circles have a combined area of square unit, then what is the area of the shaded region, in square units?
[figure]
Solution
Let the radius of the large circle be . Then, the radius of the smaller circles are . The areas of the circles are directly proportional to the square of the radii, so the ratio of the area of the small circle to the large one is ( is of .) This means the combined area of the 2 smaller circles is half of the larger circle, and therefore the shaded region is equal to the combined area of the 2 smaller circles, which is .
Abby, Bridget, and four of their classmates will be seated in two rows of three for a class picture. If the seating is randomly assigned, what is the probability that Abby and Bridget are both in the same row?
Solution
Total ways to seat 6 people = 6! = 720. Ways Abby and Bridget are in the same row: choose which row (2 ways) × arrange Abby and Bridget in 2 of 3 spots × arrange others = 2 × P(3,2) × 4! / (arrangements) = 1/3.
What is the correct ordering of the three numbers frac{5}{19} , frac{7}{21} , and frac{9}{23} , in increasing order?
Solution
The value of frac{7}{21} is frac{1}{3} . Now we give all the fractions a common denominator.
frac{5}{19} implies frac{345}{1311}
frac{1}{3} implies frac{437}{1311}
frac{9}{23} implies frac{513}{1311}
Ordering the fractions from least to greatest, we find that they are in the order listed. Therefore, our final answer is boxed{textbf{(B)} frac{5}{19}<frac{7}{21}<frac{9}{23}} .
A number of students from Fibonacci Middle School are taking part in a community service project. The ratio of -graders to -graders is , and the the ratio of -graders to -graders is . What is the smallest number of students that could be participating in the project?