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Answers by Michael F.

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Let the amount of added water be W gallons. After it is added we have 1/(1+W)=.95 You may take it from here.

If x=logbN and y=logb2 N we have bx  = (b2 )y=b2y or that y=x/2 or x=2y  Also logcx/logcy=logyx Now log38÷( log916×log410 )=log38/(2log316×log410)=log168/2log410=2log48/(2log410)=log108=3log102

The ancient Greeks thought this ratio gave a most pleasing appearance to the structure. It is also called the golden mean. The reason for this is that it is the mean proportional between 1 and 1+itself, that is the positive solution to the equation 1/g=g/(1+g).

The formula to use is sinA+sinB=2sin((A+B)/2)cos((A-B)/2) sin(x) + sin(3x) = 4 sin(x) cos^2(x) 2sin((x+3x)/2)cos((3x-x)/2)=2sin2xcosx=4sinxcosxcosx=4sinxcos2x

Let the first term be a, the common difference d. the ninth term is a+8d=0, which makes a=-8d The ratio of the 29th term to the 19th term is (a+28d)/(a+18d)=20d/10d=2

A+B+C=180 so that B+C=180-A A-B-C=A-(B+C)=A-(180-A)=2A-180 tan(x-y)=(tanx-tany)/(1+tanxtany) tan(2A-180)=(tan2A-tan(180))/(1_+tan2Atan(180))=tan2A/1-tan2A) since tan(180)=-1  

or you could divide the numbers and see whether the answer is <=>1, as first number 4 3/8=35/8,  second number 34/7 35/8÷34/7=35/8×7/34=245/272 <1 so 34/7 is larger.