CPhill

Notice that the first man has 32 choices, and the second man has 31 choices, and the 3rd man has 30 choices, etc.  By the time we get to the 26th man, he has 7 choices, thusly...

32  + 31 + 30+....+ 7 

1        2      3          26 

So, the number of choices  just equals the sum of the digits from 7 to 32....or, put another way...the sum of the first 32 digits  less the sum of the first 6 digits  =

32(33) / 2  -  6(7)/ 2  =

528            -   21   =

507

I think that's it.... [ I hope !! ]

Let's see...working with the LHS, we have

- sin(theta)/cot(theta)-sec(theta) = -cos(theta)

-sin(x) / [cos(x)/sin(x)] - 1/cos(x)

-sin^2(x) / cos (x) - 1/cos(x)

- [ 1 + sin^2(x) ] / cos(x) =  -cos(x)   ???.....nope....no way this is true....!!!

And

cos(theta)+tan(theta)xsin(theta)= sec(theta)

Working on the LHS, we have....

cos(x) + [ sin^2(x) / cos(x) ]

[cos^2(x) + sin^2(x) ] / cos(x) =

1 / cos(x)  =  sec(x).......  and this one is true....!!!!

1/z + x/y = 5/y - 3x / 2y     rearrange

3x / 2y  + x / y  = 5/y - 1/z    simplify

5x / 2y  = [5z - y ] / yz

5x = 2[5z - y ] / z

x  = (2/5) [ 5z - y ] / z = [ 10z - 2y ] /  [5z] = 2 - 2y/5z = 2 -  (2/5) (y/z)....but (y/z)  = 5/3..so we have

x =2 - (2/5)(5/3)  =  2 - 2/3  = 6/3 - 2/3 = 4/3

 And that's the truth....!!!!!

Rosala and I aren't lazy......we just live by this motto.....

Me too, rosala.....I don't want to have to work too hard.....!!!!

Here's another way to figure this without using much math....

100% is 20 times 5%

So....what is 20 times $14.80.....???

I seem to do this a little differently from everybody else on here

Forget the whole number part for a moment

Write the non-repeating part and the repeating part of the decimal = 45450

Subtract the non-repeating part from this....=  [45450 - 45  ] = 45405

Put this number over a fraction consisting of 9's and 0's.....the number of leading 9's  equaling the number of digits in the repeating part (3), and the number of trailing 0's equaling the number of digits in the non-repating part (2)..so we have

45405 / 99900 = 1009 / 2220

Add back the  whole number part.....so we have

1 + 1009/2220  = [2220 + 1009] / 2220 = 3229/2220   ...verify for yourself that this is the same as what we started with...!!!

My answer and Melody's are approximately the same -  depending on the level of rounding. She has taken a "ratio" approach, while I have have used a "trig" approach.......same dog, different fleas....

And heureka has provided a very nice LaTex answer....utilizing a combination of both things....!!!

A sector has an angle of 107.9 and an arc length of 5.92m. Find its radius

Let us convert 107.9 degrees to radians = 107.9 x pi / 180 = about 1.883 rads

And using  S = rΘ   where S= the arc length and Θ is in radians, we have

5.92m = r (1.883)     divide both sides by (1.883)

r = about 3.144m