Two questions:
(1) In a class every boy is friends with exactly 3 girls. Every girl is friends with exactly 2 boys. There are only 19 desks (each holding at most two students), and 31 of the students in the class study French. How many students are there?
(2) A smaller circle rolls along the inside curve of a larger circle. The diameter of the larger circle is 3 times that of the smaller circle. How many revolutions does the small circle have to make to complete the inside curve of the larger circle?
I think the answer is 3. If I cut the circles and lay them flat on the ground, the larger circle is 3 times longer than the smaller circle. So the smaller circle would have to revolve 3 times in order to equal the distance inside the larger circle. Is that correct?
yes on #2
What is the answer and solution to #1?
I can't quite get my head around this the way it is stated.
I see four restrictions:
(b -- boys, g -- girls)
b+g ≥ 31
b+g ≤ 38
g ≤ 3b
b ≥ 2g
If I graph this on a b-g axes graph, there is a region with about 80 ordered pairs that obey all 4 restrictions
e.g. b = 12 g = 22
b+g = 34 , which is less than 38 and greater than 31, check first two restrictions
third: is 22 < 3(12) ? yes
fourth: is 12 < 2(22) yes.
What don't I see ?
36?
Let's break down the two questions one by one:
(1) To determine the total number of students in the class, we need to use the information provided about the relationships between boys, girls, and desks.
Every boy is friends with exactly 3 girls, and every girl is friends with exactly 2 boys. Let's assume there are "x" boys in the class. Since each boy has 3 girl friends, the total number of girl friends in the class would be 3x. As each girl is friends with exactly 2 boys, the total number of girls would be (3x) / 2.
Now, let's add up the number of boys and girls in the class: x (boys) + (3x) / 2 (girls). Since every student in the class occupies a desk and there are a total of 19 desks, we can write the equation as:
x + (3x) / 2 = 19
To solve this equation, we can multiply both sides by 2 to eliminate the fraction:
2x + 3x = 38
5x = 38
x = 7.6
Since x represents the number of boys, which must be a whole number, we can conclude that there are 8 boys in the class.
To find the number of girls, we substitute the value of x into the equation:
(3x) / 2 = (3 * 8) / 2 = 12
Therefore, the number of girls in the class is 12.
To find the total number of students, we add the number of boys and girls:
Total number of students = Number of boys + Number of girls = 8 + 12 = 20
Therefore, there are 20 students in the class.
(2) Your assumption for the second question is incorrect. Rolling the circles along the ground does not give an accurate representation of the motion. Instead, let's consider the relationship between the circumferences of the larger and smaller circle.
The circumference of a circle is calculated using the formula C = πd, where C is the circumference and d is the diameter.
Let's assume the diameter of the smaller circle is "a". According to the problem, the diameter of the larger circle is three times that of the smaller circle, so it would be 3a.
To calculate the number of revolutions the smaller circle needs to make to complete the inside curve of the larger circle, we need to find the ratio of their circumferences:
Ratio of circumferences = Circumference of larger circle / Circumference of smaller circle
Circumference of larger circle = π * (3a)
Circumference of smaller circle = π * a
Simplifying the ratio, we have:
Ratio = (π * (3a)) / (π * a) = 3
Therefore, the smaller circle needs to make 3 complete revolutions to complete the inside curve of the larger circle.
Hence, the answer is 3.