Bosco

All members of our painting team paint at the same rate. If $10$ members can paint a $1000$ square foot wall in $10$ minutes, then how long would it take for $5$ members to paint a $1000$ square foot wall, in minutes?    

This is a special case of a rate x time problem,    

so it's possible to take a short cut in this case.    

With half as many painters, and with all else equal,   

the job would take twice the time.  So, 20 minutes    

_.    

   Evaluate $a^3 - \dfrac{1}{a^3}$ if $a - \dfrac{1}{a} = 0$.   

                               1                            1   

       What is  a³ –  –––    when    a  –  ––– = 0    

                               a³                           a   

        For a – (1/a) to equal 0  the only way is when a = +1     

         If so, then a³ – (1/a³) = 0 as well   

_.     

You have a total supply of $1000$ pieces of candy, and an empty vat.  You also have a machine that can add exactly $5$ pieces of candy per scoop to the vat, and another machine that can remove exactly $3$ pieces of candy with a different scoop from the vat.  When these two machines are done, there is only one piece of candy left in the vat.  What is the smallest possible number of times the the first machine added candy to the vat?    

Let the 1st machine cycle two times to add         10 pieces    

Let the 2nd machine cycle three times to remove   9 pieces    

                                                                                1 piece remains    

_.

Real numbers a and b satisfy    
a + ab = 250    
a - ab = -240    
Enter all possible values of a, separated by commas.    

given the two equations    

                                                     a + ab  =    250    

                                                     a – ab  =  –240    

simply add the two equations    

                                                    2a         =      10    ==>>    a = 5          

_.   

Find the discriminant of the quadratic 5x^2 - 2x + 8 - 2x^2 + 6x + 2.  

Collect and combine like terms    

& arrange them as ax² + bx + c               5x² – 2x + 8 – 2x² + 6x + 2     

                                                                3x² + 4x + 10        

The discriminant is b² – 4ac                    D  =  (4)² – (4)(3)(10)    

                                                                 D  =  16 – 120    

                                                                 D  =  – 104    

_.

The number $100$ has four perfect square divisors, namely $1,$ $4,$ $25,$ and $100.$   
What is the smallest positive integer that has exactly $2$ perfect square divisors?   

I reckon that would be four ... it has two perfect square divisors, namely one and four.   

_.   

          Fill in the blanks with three distinct positive integers.   
          1/___ + 1/___ + 1/___ = \frac{13}{12}   

           1 / 1  +  1 / 18  +  1 / 36  =  39 / 36  =  13 / 12     

Note that the above is the fractions 36/36 + 2/36 + 1/36 = 39/36   

_.   

Find the discriminant of the quadratic 5x^2 - 2x + 8 - 2x^2 + 6x + 2 - 2x^2 + 4x - 1.    

First, collect and combine like terms   

and arrange them as ax² + bx + c           (5x² – 2x² – 2x²) + (–2x + 6x + 4x) + (+8 + 2 – 1)    

                                                                 x² + 8x + 9    

The discriminant is b² – 4ac                     D  =  8² – (4)(1)(9)    

                                                                 D  =  64 – 36    

                                                                 D  =  28    

_.