choutowne

Let G be Grogg's current age, let D be Grogg's dad's current age.

1. "Seven years ago, Grogg's dad was 9 times as old as Grogg":
   D - 7 = 9(G - 7). We can rearrange and rewrite to get:
   D = 9G - 56.
 

2. "Four years ago, Grogg's dad was 6 times as old as Grogg":
   D - 4 = 6(G - 4), We can rearrange and rewrite to get:
   D = 6G - 20.

Now we have the two equations:
1. D = 9G - 56
2. D = 6G - 20

Next, we can set the right-hand sides of Equation 1 and Equation 2 equal to each other, since we know their left hands sides are equal:
9G - 56 = 6G - 20
9G - 6G - 56 = -20,
3G = 36, 

So we have: G = 12
 

Now that we have Grogg's age, we can find his dad's age. Using the first equation:
D = 9G - 56 = 9(12) - 56
D = 52
 

Thus, Grogg's dad is currently 52 years old. 

You can check this answer with equation 2, if you'd like. 

Thank you very much!! 

Your detailed explanation made it much easier to understand the answer.

And sorry I didn't clarify, A must be a single-digit positive integer, so the only value possible in that set is 6, which is the answer :)

Thank you so much!

Your explanation helped a lot.

Nevermind, I figured it out with symmetry! If anyone's wondering, it's \(\boxed{96}\)

Nevermind, thanks, I got the answer \(\boxed{\frac{16}{7}}\)! Thanks for your help!

Thanks for the answer, it helped me a lot!

5sqrt(21)/41 wasn't correct, but your response helped me figure out what to do next.

I followed your solution and got mostly the same results except that I got 40/41 as cos(PAQ) instead of 34/41.

I plugged that into the relationship between the sine and cosine of PAQ and got 9/41 as the final answer, which turned out to be correct. 

\( sin(PAQ) = \boxed{9/41}\)

You just copied CPhill's answer to my previous question...

Thank you so much! I forgot to mention but I believe the answer should be in the form of a common fraction. However, I'm not sure how I should convert it. (It's a program where one inputs the answer and doesn't take sin/cos etc. in the expression)  

Sorry, I'm a little lost. There are multiple points in your answer which confuse me. 
Why is XY = XZ = radius (I'm assuming R is radius) because X lies on the perpendicular bisectors of all three sides of the triangle? Points Y and Z lie on a circle centered on O, so OY and OZ would be the radius, unless you're referring to another circle? 

What is the inscribed circle? 

How did you get 48/(2 * sqrt(3)) = 12sqrt(3)?

Sorry, and thanks.