Bosco

Laverne starts counting out loud by 5's.  She starts with 2.  As Laverne counts, Shirley sums the numbers Laverne says.  When the sum finally exceeds 100, Shirley runs screaming from the room.  What number does Laverne say that sends Shirley screaming and running?   

Laverne's           Shirley's   

Counting          Summation   

     2                           2   

     7                           9   

   12                         21   

.  17                         38   

   22                         60   

   27                         87   

   32                       119    

_.

I drove to the beach at a rate of 40 miles per hour. If I had driven at a rate of 50 miles per hour instead, then I would have arrived 45 minutes later. How many miles did I drive?   

When you go faster you'll get there sooner, so the problem

must actually mean "would have arrived 45 minutes earlier."   

Distance  =  Rate • Time    

There are two cases:    (1)  D  =  (40)(t)   

                                     (2)  D  =  (50)(t –  45 / 60)   

Regarding 45 / 60  -  Since the speed is given in miles per hour, the 45 minutes   

must be converted to hours to keep units consistent.  We could have converted    

the speed to miles per minute, but that would have unnecessarily added tedium.  

Since the distance is the same in both cases, set them equal and solve for time.   

                                            (50)(t – 45 / 60)  =  (40)(t)   

                                                     50t – 37.5  =  40t    

                                                                10t  =  37.5   

                                                                    t  =  3.75   

Substitute t back into case (1)                    D  =  (40)(3.75)   

                                                                   D  =  150    

check answer     

Substitute t back into case (2)                    D  =  (50)(3.75 – 0.75)   

                                                                   D  =  (50(3.00)    

                                                                   D  =  150    

_.

Find the ordered triple (p,q,r) that satisfies the following system:   

p - 2q = 3   

q - 2r = -2 + q   

p + r = 9 + p    

from q – 2r = –2 + q    we find that r = 1   

from p + r = 9 + p        we find that r = 9    

Since r can't equal both 1 and 9    

the system has no solution.   

_.

At a cafeteria, Mary orders two pieces of toast and a bagel, which comes out to $3.15. Gary orders a bagel and a muffin, which comes out to $3.50. Larry orders a piece of toast, two bagels, and three muffins, which comes out to $8.15. How many cents does one bagel cost?  

                                           Mary          2T + B = 315 ............. —>  T = (315 – B) / 2       

                                           Gary          B + M = 350 ................—>  M = (350 – B)      

                                           Larry          T + 2B + 3M = 815  

                                           Substitute for Toast & Muffin into Larry's order  

                                           [(315 – B) / 2]  +  2B  +  [3 • (350 – B)]  =  815   

                                           [(315 – B) / 2]  +  2B  +  (1050 – 3B)  =  815   

Multiply both sides by 2       (315 – B)  +  4B  +  (2100 – 6B)  =  1630    

Combine like terms              –3B + 2415  =  1630     

                                                        –3B  =  – 785     

                                                            B  =  261 ²/₃  ¢    

_.

The product of a set of distinct positive integers is $630$.   

If one of the numbers is 42, what are the other two numbers?    

630 divided by 42 is 15.  15 has only two factors.   

Therefore the three integers are 42 and 5 and 3.   

edited to add:  15 has only two factors other than 1 and itself.   

_.

A positive integer is called terrific if it has exactly $12$ positive divisors.  A positive integer is called fine if it has exactly 2 positive divisors.  Which numbers are fine?     

I can't figure out the question about "good" numbers,  

in fact, I'm not even certain I understood the question.  

But I think I can understand this one.  

Positive integers that happen to be Prime Numbers     

have exactly two positive divisors.  

_.

Starting with the A moving one letter at a time vertically, horizontally, or diagonally, how many different paths spell ARCH?   

        A   

     RRR   

  CCCCC   

HHHHHHHH   

From the A you have 3 choices to the R.   

From the R you have 3 choices to the C.   

From the C you have 3 choices to the H.   

Total possible paths to spell ARCH is 3 • 3 • 3  =  27   

_.

Starting with the A moving one letter at a time vertically, horizontally, or diagonally, how many different paths spell ARCH?   

        A   

     RRR   

  CCCCC   

HHHHHHHH   

From the A you have 3 choices to the R.   

From the R you have 3 choices to the C.   

From the C you have 3 choices to the H.   

Total possible paths to spell ARCH is 3 • 3 • 3  =  27   

_.

Catherine rolls a standard $6$-sided die three times.  If the product of her rolls is $600,$ then how many different sequences of rolls could there have been?  (The order of the rolls matters.)   

If Catherine rolls the largest side of the die all three times,   

then the product of her three rolls would be 6 • 6 • 6 = 216.  

There's no way to get a product of 600 with only three rolls.  

_.