1. In the figure shown, what is the value of x?x+<QSR+<UST=180 (straight line =180) and <R+<T=90 (as PRT is right angle)

(1) The length of line segment QR is equal to the length of line segment RS --> QRS is isosceles --> <PQS=<QSR=(180-R)/2 (as <PQS+<QSR+<R=180 --> 2<QSR+<R=180 --> <QSR=(180-R)/2). Not sufficient.

(2) The legnth of line segment ST is equal to the length of line segment TU --> UST is isosceles --> <SUT=<UST=(180-T)/2. Not sufficient.

(1)+(2) x+<QSR+<UST=180 --> x+(180-R)/2+(180-T)/2=180 --> x+(360-(R+T))/2=180 --> as R+T=90 --> x+(360-90)/2=180 --> x=45. Sufficient.

Answer: C.

2. An equilateral triangle ABC is inscribed in the circle. If the length of arc ABC is 24, what is the approximate diameter of the circle?Arc ABC is \(\frac{2}{3}\) of the circumference (as ABC is equilateral triangle) --> \(24=c*\frac{2}{3}\), hence circumference \(c=\frac{24*3}{2}=36=\pi{d}\) --> \(d\approx{11.5}\).

Answer: C.

3. Machine X and Y work at their respective constant rates. How many more hours does it take Machine Y, working alone, to fill a production order of a certain size than it takes Machine X, working alone?(1) Machine X and Y, working together, fill a production order of this size in two thirds the time that Machine X does.

(2) Machine Y, working alone, fills a production order of this size in twice the time that Machine X, working alone, does.

Let \(x\) and \(y\) be times needed for X and Y to to fill a production order of a certain size when working alone. Question: \(y-x=?\)

(1) \(\frac{1}{x}+\frac{1}{y} = \frac{1}{total \ time}\) (general formula) --> Total time needed for X and Y working together = \(\frac{xy}{x+y}\) (general formula) --> given \(\frac{xy}{x+y}=x*\frac{2}{3}\) --> \(2x=y\). Not sufficient.

(2) \(2x=y\), the same info. Not sufficient

(1)+(2) Nothing new. Not Sufficient

Answer: E.

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