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Paul has a mix of quarters and dimes in his piggybank. He has twice as many dimes than he has quarters. He counts all his coins and notices that he has   nine dollars in total. How many quarters does Paul have?     

D  =  2Q     

10D + 25Q  =  900     

10(2Q) + 25Q  =  900     

   20Q   + 25Q  =  900     

                45Q  =  900     

                    Q  =  20     

                     D  =  2Q     

                     D  =  2 • 20     

                     D  =  40     

check answer     

20 quarters is 5 dollars     

40 dimes     is 4 dollars     

total is             9 dollars     

_.     

  In right triangle BCD with D = 90, we have BC = 9 and BD = 5. Find sin B.     

  As I always tell everybody, sketch the triangle.     

  sin(B) = BD / CD     

  We know what BC and BD are, so one way is to use Pythagoras' to find CD.     

  CD²  =  BC² – BD²     

  CD²  =  9² – 5²  =  81 – 25  =  56     

  CD    =  sqrt(56)  =  7.4833              ––>>   sin(B)  =  5 / 7.4833     

                                                                      sin(B)  =  0.6682     

_.     

In triangle ABC , we have A = 90 and sin B = 4/17. Find cos C.     

sin(B) = AC / BC    Well, whaddya know.  Sketch the triangle 

                               and you find that cos(C) also = AC / BC     

                                therefore            cos(C) = 4 / 17     

_.     

With AB = AC = 1 and angle BAC = 60°, it is an equilateral triangle with side length 1. D lies on AB, E lies on the arc BC. The condition BD = DE is only met if E is identical to C and D is identical to A. F lies on DB, B lies on the arc DE. The condition DF = FG is only met if F is identical to B and G is identical to C.

Thus, the length DF = AB = 1.

  !

It is green for   40 out of  (40+5+35) seconds = 40 out of 80 seconds 

   the probability that  you see green  watching it for 75 seconds means

                            you will see green in that  75 seconds EVERYTIME  or  100 %   ....there is no 75 seconds where it is not going to have at least a portion of a green phase.

1,0     0,1     and   0,0    would produce a triangle with area = 1/2 

Well, 

2 ( 5+3) = 16     so   q   or  r  could be    5 or 3 

Break it into 4 pieces as shown:

 1^1 + 2^2 + 3^4 + 4^8    = 

  1   +   4   +  81  +  65536 =   65622          Units digit = 2 

He first circles '2'    then circles the powers of 2 :  cirlces 4   circles  8  

He then circles '3'   then circles the powers of 3 :  circles 9 

He then circles '4'    no powers of 4 are on the list so no circles   and :

He then circles 5    then 6  ,7  ,10 ,11, 12 , 13 , 14  and lastly   '15 '

Now, he starts crossing:

   He  crosses off 15    then 14 , 12, 10 , 8 , 6, 4, ,2

    He then crosses off 13    then crosses off 11 , 9 ,7,5,and lastly '3'   

Last circled = 15     last crossed = 3           3 X 15 = 45 

The maximum possible distance between their endpoints is the diameter of the circle, namely 2r.

 !

The value of the numerator of a particular fraction is one-half the value of its denominator. If the numerator is increased by 2 and the denominator is decreased by 2 , the value of the resulting fraction is 5/4 . What is the denominator of the original fraction?     

original fraction is   x / 2x     

therefore     

                                  x + 2           5     

                                ––––––   =   –––     

                                 2x – 2           4     

crossmultiply           (4)(x + 2)  =  (5)(2x – 2)     

                                     4x + 8  =  10x – 10     

                                        – 6x  =  – 18     

                                             x  =  3     

denominator of     

original fraction                   2x  =  6     

check answer     

(3 + 2) / (6 – 2)  =   5 / 4    True     

_.     

Circle O and circle P, with radii 3 and 8, respectively, are both tangent to line l at H. Enter all possible lengths of OP separated by commas.     

If they're on the same side of the line, OP = 5.     

If they're on opposite sides of the line, OP = 11.     

_.     

Last circled, 15     

Last crossed off, 3     15 x 3 = 45     

_.     

Siva starts at vertex A of square ABCD with side length 10. He walks towards vertex C, along the diagonal AC.  He then walks towards B, then towards D.  What is the length of his path?     

A to C (diag)     =  10 sqrt(2)  =  14.14     

C to B (side)     =  10              =  10.00     

B to D (diag)     =  10 sqrt(2)   =  14.14     

                                  total       =  38.28     

_.     

Find all solutions to the equation sqrt(2 - 3z) = 8.  Express your answer as a common fraction.     

                                           sqrt(2 – 3z)  =  8     

square both sides                      2 – 3z   =  64     

                                                       – 3z  =  62     

                                                            z  =  – 62 / 3     

_.     

Even perfect squares; 4, 16, 36, 64, 100, ...

Odd perfect cubes: 1, 27, 125...

Not enough information, but use that.