Bosco

Levans writes a positive fraction in which the numerator and denominator are integers, and the numerator is 2 greater than the denominator. He then writes several more fractions. To make each new fraction, he increases both the numerator and the denominator of the previous fraction by 1. He then multiplies all his fractions together. He has 3 fractions, and their product equals 10. What is the value of the first fraction he wrote? 

      x+2           x+3            x+4            10     

   ––––––  •  ––––––  •  ––––––  =  ––––––     

        x             x+1            x+2              1            cancel the x+2's     

       1             x+3            x+4             10     

   ––––––  •  ––––––  •  ––––––  =  ––––––     

        x             x+1             1                 1            cross multiply     

                               x² + 7x + 12  =  10x² + 10x     

                            –9x² – 3x + 12  =  0                multiply by –1     

                              9x² + 3x – 12  =  0                divide by 3     

                              3x² + x – 4  =  0     

                         (3x + 4)(x – 1)  =  0     

                                              x  =  –4/3     discard – the problem specifies an integer     

                                              x  =  1     

   Therefore, the first fraction is (1+2) / 1  =  3 / 1     

   _.     

   Two circles intersect at two points, $P$ and $Q$.     

   The equations of the two circles are $x^2 + (y - 1)^2 = 1$ and $(x - 1)^2 + y^2 = 1$.     

   Find the length PQ.     

                x² + (y – 1)²  = 1 

       and   (x – 1)² + y²  = 1     

   Use Desmos to plot both circles and note     

   that they intersect at the origin and at (1,1)     

   That makes PQ = sqrt(2)     

   _.     

   Jon teaches a fourth grade class at an elementary school where class sizes are always at least $20$ students and at most $60$.     

   One day Jon decides that he wants to arrange the students in their desks in a rectangular grid with no gaps.     

   Unfortunately for Jon he discovers that doing so could only result in one straight line of desks.     

   What is the largest number of students that Jon could have in his class?     

   This one is a breeze.  The number is 59 because 59 is the largest (20 < x < 60) that's a prime number.     

   A prime number has no factors other than 1 and itself.  So, there can be only a single row of 59 desks.     

   _.     

   What is the square root of 64     

   The sqrt(64) = + 8 . . . everybody knows it's 8, but     

   sometimes they forget that negative 8 is also a root.     

   _.     

   How do I solve a y= system using the combination method? For example, y = 6x +4 and y = 4x - 2?     

   The combination method, more commonly called the elimination method, is a way to solve a      

   system of equations by adding or subtracting the equations to eliminate one of the variables.     

   Using the provided example:              y  =  6x + 4     

                                                               y  =  4x – 2     

                                  subtract                0  =  2x + 6     

                                                          – 2x  =  6     

                                                             2x  =  – 6     

                                                               x  =  – 3     

                OR set the y's equal      6x + 4  =  4x – 2     

                                                      2x         =      – 6     

                                                               x  =  – 3     

   _.     

   All members of our painting team paint at the same rate.     

   If 20 members can paint a 2000 square foot wall in 40 minutes, then how long would it take     

   the 20 members to paint a 6000 square foot wall, in minutes?     

   20 members   ==>>    paint 2000 ft² / 40 min  =  50 ft² per min     

   20 members   ==>>    paint 6000 ft² /  x          =  50 ft² per min     

                                               6000 ft²                =  50 ft² per min • x     

                                          6000 ft² / 50 ft²/min  =  x     

                                                                        x  =  120 min     

   Another way to do it is to consider the 6000 ft² wall as equivalent to three 2000 ft² walls     

   So, because one wall takes 40 minutes, three walls would take (3 • 40 min) = 120 min     

   _.     

   Find the maximum area of a right triangle with hypotenuse 1m.     

   The area of a quadrilater for a given perimeter is maximized when the quadrilateral is a square.     

   So let's assume the same holds true for a triangle, by considering the triangle as half a square.     

   If the triangle is half a square, that means both legs are the same length.     

   by Pythagoras' Theorem                                          a² + b²  =  c²     

   the legs are equal, which means a = b therefore           2a²  =  1²     

                                                                                            a    =  sqrt(1/2)     and b is the same     

   Area of a triangle   =  (1/2) • base • height     

                                 =  (1/2) • sqrt(1/2) • sqrt(1/2)     

                                 =  (1/2) • (1/2)     

                                 =  1/4 meter²     

   _.