SAT – Venn Diagrams

Q: Of the 20 students in the classroom, 14 are taking Calculus, 12 are taking Physics, and 4 are taking neither Calculus nor Physics. How many students are taking both Calculus and Physics?
 
Answer: 10
Use a Venn Diagram.  Assume for a moment that there is no overlap between Calculus and Physics.  Then, there would be 14 + 12 = 26 students taking either Calculus or Physics.  Add the 4 students taking neither, and that would be a total of 30 students.  The question states that there are only 20 students in the class.  So there must be some overlap after all.  How much?  30 – 20 = 10 students taking both Calculus and Physics.  Put 10 in the overlapping region.  To fill out the rest of the diagram: 14 – 10 = 4 students in Calculus but not Physics.  12 – 10 = 2 students in Physics but not Calculus.
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Geometry – Triangles

Q: In triangle ABC below, what is sin C ?
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Answer: 4/5
Explanation:  Use the Pythagorean Theorem or the 3-4-5 Pythagorean triple to find out that the hypotenuse is 10.
Then, sin C = (opposite)/(hypotenuse) = 8/10 = 4/5.
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