- #1
maxkor
- 84
- 0
In summary, the conversation is about finding the value of a fraction, specifically $\frac{a}{b}$, in a given problem. The helper suggests using Pythagoras' theorem and provides some hints for solving the problem. The asker then confirms their answer as correct.
- #2
I like Serena
Homework Helper
MHB
- 16,336
- 258
maxkor said:
Hi maxkor! (Smile)
We ask that our users show their progress (work thus far or thoughts on how to begin) when posting questions. This way our helpers can see where you are stuck or may be going astray and will be able to post the best help possible without potentially making a suggestion which you have already tried, which would waste your time and that of the helper.
Can you post what you have done so far?
- #3
maxkor
- 84
- 0
View attachment 4533
Let 1/2b radius of the big circle, let r radius of the smaller circle
Let $c=1/2a=1/2b−r,
b=2c+2r,
a=2c.$
So $\frac{a}{b}=\frac{2c}{2c+2r}=\frac{c}{c+r}$
Small circles respectively tangential to the large circles so
$z=c+2r,t=a−r=2c−r$
Is this right?
Let 1/2b radius of the big circle, let r radius of the smaller circle
Let $c=1/2a=1/2b−r,
b=2c+2r,
a=2c.$
So $\frac{a}{b}=\frac{2c}{2c+2r}=\frac{c}{c+r}$
Small circles respectively tangential to the large circles so
$z=c+2r,t=a−r=2c−r$
Is this right?
Attachments
- #4
Opalg
Gold Member
MHB
- 2,778
- 13
Use Pythagoras in the triangles $CXY$, $DXY$ (where $Y$ is the centre of one of the footballs) to find two expressions for $XY^2$ in terms of $a$, $b$ and $r$. Putting those expressions equal to each other will give you an equation connecting $a$, $b$ and $r$.
You already know that $r = \frac12(b-a)$ (from your equation $c = \frac12a = \frac12b-r$). Substitute that value of $r$ into your equation, and it will give you the connection between $a$ and $b$.
Attachments
Last edited:
- #5
maxkor
- 84
- 0
Is $\frac{a}{b}=\frac{\sqrt{2}}{2}$ correct answer?
- #6
Opalg
Gold Member
MHB
- 2,778
- 13
Yes! (Happy)maxkor said:
FAQ: Find $\frac{a}{b}$ in the Circle of Balls
What is the Circle of Balls problem?
The Circle of Balls problem is a mathematical puzzle where there are several balls arranged in a circle, with one ball missing. The goal is to find the value of a fraction represented by the number of balls on one side of the missing ball divided by the total number of balls in the circle.
How do you solve the Circle of Balls problem?
To solve the Circle of Balls problem, you need to first count the number of balls on one side of the missing ball and the total number of balls in the circle. Then, express this as a fraction and simplify it to find the value of the missing ball.
Are there any strategies or tricks to solve the Circle of Balls problem?
Yes, there are a few strategies that can make solving the Circle of Balls problem easier. One strategy is to divide the circle into equal sections and count the number of balls in each section. Another strategy is to use the fact that the sum of opposite sides of a circle is always equal to the total number of balls.
Is there a limit to the number of balls in the Circle of Balls problem?
No, there is no limit to the number of balls in the Circle of Balls problem. The number of balls can vary and the problem can still be solved using the same method.
What are the real-world applications of the Circle of Balls problem?
The Circle of Balls problem can be used to develop critical thinking and problem-solving skills. It can also be applied in various fields such as statistics, game theory, and computer science. Additionally, it can be used as a fun and challenging puzzle for children and adults alike.