Bosco

In the single-elimination NCAA Tournament, 64 teams compete in 6 bracketed rounds. How many games are there in total?     

64 teams would play 32 games ——>  32 teams go home, 32 teams play again     

32 teams would play 16 games ——>  16 teams go home, 16 teams play again     

16 teams would play   8 games ——>    8 teams go home,   8 teams play again     

  8 teams would play   4 games ——>    4 teams go home,   4 teams play again     

  4 teams would play   2 games ——>    2 teams go home,   2 teams play again      

  2 teams would play   1 game   ——>    1 team goes home, 1 team takes the trophy home     

                                   –––––––     

add up all games       63 games      

_.     

In a box there are red and blue balls. If you select a handful of them with eyes closed, you have to grab at least 5 of them to make sure at least one of them is red and you have to grab at least 10 of them to make sure both colors appear among the balls selected. How many balls are there in the box?     

"you have to grab at least 5 of them to make sure at least one of them is red"     

This tells me that there are four blue balls.     

"you have to grab at least 10 of them to make sure both colors appear among the balls selected"     

This tells me that there are nine red balls.     

4 blue + 9 red  =  13 balls total     

_.     

The distances from a point on the circumcircle of an equilateral triangle to the two nearest vertices are 3 and 6.     

Find the distance from the point to the furthest vertex.     

Draw it and the answer is obvious.  There's a third of the circle's circumference from each vertex to the next.     

From the point to the two nearest vertices is 3 to one and 6 to the other.  So a third of the circumference is 9.     

You start from the point and go the direction that's 6 to a vertex, and then it's 9 to the next vertex.  6 + 9 = 15     

_.     

If c > a > 0 and a - b + c = 0, find the larger root of ax^2 + bx + c = 0.     

The first one I thought of is          x² + 3x + 2 = 0     

                                                   (x + 1)(x + 2) = 0     

                                                                      x = – 1     

                                                                      x = – 2     

The larger root:    – 1 is larger than – 2.     

The criteria c > a > 0 and a – b + c = 0 also work for  x² + 4x + 3   (x + 1)(x + 3)     

                                                                             and  x² + 5x + 4   (x + 1)(x + 4)     

                                                                             and  x² + 6x + 5   (x + 1)(x + 5)     

                                                                             and  x² + 7x + 6   (x + 1)(x + 6)     

                                                                             and you can keep going as long as you want to.     

_.     

Triangle DEF is an equilateral triangle with a side length of 3.     

A point P is chosen at random within triangle DEF.     

What is the probability that the length DP is at most 1?      

Draw an equilateral triangle DEF, side 3.     

Draw a circle, center at D, 1 inch radius.     

DP will be < 1 if it falls in or on the sector of the circle     

that's inside the triangle.     

Area of entire circle  =  (3.1416) • 1²  =  3.1416     

Area of sector inside triangle  =  (60^o / 360^o) • 3.1416  =  0.5236     

Area of triangle (per formula)  =  [ sqrt(3) / 4 ] • 3²  =  3.8971     

Probability  =  0.5236 / 3.8971  =  0.1344  ——>  as a percent  =  13.44%     

_.     

Twice the sum of a number and seven is three times the number.     

   2 • (x + 7)  =  3x     

       2x + 14  =  3x     

                –x  =  –14     

                  x  =  14     

_.     

Ms. Montgomery invested $800 at 8% simple interest at the beginning of each year for a period of ten years. Find the total accumulated value of all the investments at the end of the 10-year period.     

Beginning year balance           End of year balance     

  1st yr               800.00            1.08 •   800.00 =    864.00     

  2nd yr              864.00            1.08 •   864.00 =    933.12     

  3rd yr               933.12            1.08 •   933.12 =  1007.77     

  4th yr             1007.77            1.08 • 1007.77 =  1088.39     

  5th yr             1088.39            1.08 • 1088.39 =  1175.46     

  6th yr             1175.46            1.08 • 1175.46 =  1269.50     

  7th yr             1269.50            1.08 • 1269.50 =  1371.06     

  8th yr             1371.06            1.08 • 1371.06 =  1480.74     

  9th yr             1480.74            1.08 • 1480.74 =  1599.20     

10th yr             1599.20            1.08 • 1599.20 =  1727.14     

Another way to calculate the final balance would be 800 • 1.08¹⁰  =  1727.14     

_.     

state the value of c so it is a perfect square trinomial. the equation is 4x^2-24x+c.     

Figure out the factors.  Since it's a perfect square, both factors will be the same.     

The first term will be sqrt(4x²).  So first we have           (2x      )(2x      )     

The coefficient of the x¹ term is negative     

so the sign in the middle will be negative                      (2x –   )(2x –   )     

To determine the constant, you have to figure      

2x times what, plus 2x times what, equals 24x      

With a little thought, you realize it has to be 6.              (2x – 6)(2x – 6)     

Multiplied out, it is 4x² – 24x + 36, so c = 36     

_.     

 The Gonzalez family has a combined monthly income of $1850. Their food budget is $ 380 per month.     

 What percent of their monthly income is budgeted for food? (Round your answer to the nearest whole percent.)     

 Percent is 100 times the part of the whole divided by the whole.     

 %  =  100 • 380 / 1850     

       =  100 • 0.2054     

       =  20.54    ——>>  rounded as per problem is 21%     

_.