Bosco

What is the greatest prime factor of the sum of the arithmetic sequence 1 + 2 + 3? 

1 + 2 + 3  =  6  

the factors of 6 are 2 x 3  

greatest  =  3  

both 2 and 3 are prime, the larger is 3  

_.

                   Base_upper + Base_lower   

Median  =  ––––––––––––––––––––   

                                 2    

                   12 + (12 + 16)          12 + 28   

Median  =  ––––––––––––––  =  ––––––––   

                               2                         2   

Median  =  20   

The length of the median of a trapezoid is    

the average of the lengths of the two bases.    

The area has nothing to do with this problem.    

_.

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  ¢    

They usually construct these problems to come out with an even number.   

I've gone back over the steps a hundred times and cannot find a mistake.    

_.

Find the number of ordered pairs (a,b) of integers such that   
\frac{a + 2}{a + 1} = \frac{b}{8}.  

I don't know how to read that shorthand but I'm going to assume it means   

                                                     (a + 2)          b    

                                                    –––––––  =  –––    

                                                     (a + 1)           8    

I don't know how to solve this, so, as my geometry teacher used to say    

"Go as far as you can, then see how far you can go."  

Let's try some numbers and see if we can find a pattern    

When a is 0, we get 2 / 1  so in order to preserve equality b = 16    

                 1             3 / 2                                                          12    

                 2             4 / 3  won't work, no integer is 1/3 of 8    

                 3             5 / 4                                                          10    

                 4             6 / 5  won't work, no integer is 1/5 of 8         

                 5             7 / 6  won't work    

                 6             8 / 7  won't work    

                 7             9 / 8                                                            9    

I think I can see intuitively that no a > 7 will result in an integer b.    

Let's go the other direction from zero and see what happens.  

When a is –1, we get   1 / 0  zero in denominator is a no-no      

                 –2, we get   0 / –1  so                                              b = 0    

                 –3, we get –1 / –2  so                                                    4    

                 –4, we get –2 / –3  won't work    

                 –5, we get –3 / –4  so                                                    6      

                 –6, we get –4 / –5  won't work  

                 –7, we get –5 / –6  won't work            

                 –8, we get –6 / –7  won't work      

                 –9, we get –7 / –8  so                                                    7      

It looks like the equality stops working when (a+1)          

is less than negative 8 and greater than positive 8.     

I can't see a general pattern – other than, in the ones that work the denominator

is counting up in base 2 but I don't know if that means anything.  If if this were my

homework, I'd go out on that limb and say that the answer is  8 ordered pairs.   

_.

Fill in the blanks to make a quadratic whose roots are $1$ and $-1.$
x^2 + ___ x + ___  

Since the roots are +1 and –1 then the factors are (x – 1) and (x + 1) respectively.    

Multiplied out, the result is x² – 1    

Therefore the first blank holds a          0    

and the second blank holds a            –1    

_.

Laverne starts counting out loud by 5's. She starts with 1. As Laverne counts, Shirley sums the numbers Laverne says. When the sum finally exceeds 40, Shirley runs screaming from the room. What number does Laverne say that sends Shirley screaming and running? 

Laverne's           Shirley's  

Counting          Summation  

     1                           1  

     6                           7  

   11                         18   

.  16                         34  

   21                         55    

_.

Chuck deposits $2000 into a bank account that compounds weekly at an annual interest rate of 15% Assuming there are no other transactions, what will the balance be after 10 years, in dollars?  

Since the yearly interest rate for Chuck's account is 15% (same as 0.15) and there  

are 52 weeks in a year, then the amount earned the first week is (2000)(0.15 / 52)  

I'm going to change (0.15 / 52) to its decimal equal.  I'll round that value to 0.00288   

only for the convenience in writing it in this post.  In my calculator I will not round it,  

but maintain the entire 15-digit decimal in the calculations.  

Since the bank is adding (0.00288) of the current balance to the account each week,  

the total in the account (principal plus interest) after the first week is (2000)(1.00288).  

Then each week, the balance is (1.00288) times what it was the week before.  

Let's make a little tabulation, to demonstrate clearly the progression of the balance.   

Balance at end of 1^st week      (2000)(1.00288)  

Balance at end of 2^nd week     (2000)(1.00288)(1.00288)  

Balance at end of 3^rd week      (2000)(1.00288)(1.00288)(1.00288)   

Balance at end of 4^th week      (2000)(1.00288)(1.00288)(1.00288)(1.00288)   

Balance at end of n^th week      (2000)(1.00288)ⁿ   

Balance at end of 520^th week   (2000)(1.00288)⁵²⁰ =  8944.04 dollars   

_.

       2 • 33 • 4   

      ––––––––  =  24    

            11   

You can find this solution in your head.  

Just look at it, note that 33 / 11 is 3 and  

you have 2 • 3 • 4.  Piece of cake.   

_.