Bosco

Five times a number is divided by $2$ more than that number.  

If the result is $6,$ then what was the original number?  

                                          5n  

                                       –––––  =  6  

                                        n + 2  

                                          5n     =  6n + 12  

                                            n     =  –12  

check answer  

                       (5)(–12) / (–12) + (2)  =  ___  

                                    (–60) / (–10)  =  6  

_.

tangent = opposite over adjacent  

tan(80^o)  =   pole / 22  

5.671  =  pole / 22  

pole  =  (5.671)(22)  =  124.74 rounded to nearest foot  =  125  

_.

At a cafeteria, Mary orders two pieces of toast and a bagel, which comes out to 3.15.  Gary orders a bagel and four muffins, which comes out to 7.85.  Larry orders a piece of toast, two bagels, and three muffins, which comes out to 9.75.  How many cents does one bagel cost?  

wiseowl, what's up with the about a million postings?  Is this your homework or what?  

T is for toast  

B is for bagel  

M is for muffin  

                                         2T + B  =  315    (I'm using cents because the answer will be in cents.)  

                                         B + 4M  =  785  

                                         T + 2B + 3M  =  975  

from 1^st equation              2T = (315 – B)   so   T = (315 – B) / 2   

from 2^nd equation             4M = (785 – B)   so   M = (785 – B) / 4  

Now we have T and M in terms of B so substitute it into the 3^rd equation.  

                                          (315 – B)           2B          3 • (785 – B)  

                                          ––––––––   +   ––––   +   ––––––––––  =  975  

                                                 2                  1                    4  

Let's get rid of those denominators by multiplying both sides times 4.  

                                           2 • (315 – B)  +  4 • 2B  +  1 • 3 • (785 – B)  =  4 • 975   

Multiply it out                       (630 – 2B)  +  8B  +  (2355 – 3B)  =  3900  

Combine like terms              3B  =  915  

                                              B  =  305  ......  One bagel costs 305 ¢  

Plugging into original equations, a toast costs 5¢ and a muffin costs 120¢ 

_.

Will and Grace are canoeing on a lake.  Will rows at $50$ meters per minute and Grace rows at $30$ meters per minute. Will starts rowing at $2$ p.m. from the west end of the lake, and Grace starts rowing from the east end of the lake at $2{:}45$ p.m. If they always row directly towards each other, and the lake is $2800$ meters across from the west side of the lake to the east side, at what time will the two meet?   

Will has a 45 minute head start, so he's already traveled (45 min)(50 m/min) = 2250 m  

before Grace even begins to row.

So, the actual distance of closure for Will and Grace is (2800 m) – (2250 m) = 550 m  

rate times time equals distance  

Since they meet, they will both row the same amount of minutes.  Call it t.  

Therefore, the setup is                (50)(t) + (30)(t)  =  550  

                                                                       80t  =  550  

                                                                           t  =  550/80  =  6.875  

The 6 is minutes and the .875 is the fraction of a minute  

(60 sec/min)(0.875 min) = 52.5 sec ... let's round that to 52 sec   

So the clock time they meet is 6 min 52 sec after Grace starts.

Grace started at 2:45, so add 6 min 52 sec and the clock will read 2:51:52 pm when they meet  

_.

Catherine rolls a standard $6$-sided die six times.  If the product of her rolls is $100,$ then how many different sequences of rolls could there have been?  (The order of the rolls matters.)   

The prime factorization of 100 is 2 • 2 • 5 • 5  

There are other factors of 100, but you can't use any of them because  

they're larger than 6 and none of the faces on the die is larger than 6.  

For a product of 100 with 6 rolls, the numbers have to be 2, 2, 5, 5, 1, 1  

in some order, and the problem stipulates the order of the rolls matters. 

For the 1^st roll, there are 6 possibilities   

For the 2^nd roll, there are 5 possibilities  

For the 3^rd roll, there are 4 possibilities  

and so on.  

So the total number of possible sequences is 6 • 5 • 4 • 3 • 2 • 1  =  720  

OR the numbers could be 4, 5, 5, 1, 1, 1  

in which case it's still 6! to roll those so it's 720 ways to get 100 that way, too. 

Mind that this is not a probability, this is just the number of ways to get 100.  

_.

                   –5  and  +4  

_.

I think this problem requires the Pythagorean Theorem.  

Consider the triangle as pictured. 

x  =  upright leg (i.e., the portion of the pole still standing)  

40  =  bottom leg  

50 – x  =  hypotenuse  

x² + 40²  =  (50 – x)²  

x² + 1600  =  2500 – 100x + x²  

100x  =  900  

        x  =  9  (the short piece of the pole)  

50 – 9  =  41  (the long piece of the pole)  

41 – 9  =  32  (the difference between the lengths of the two pieces)  

_.

In a row of five squares, each square is to be colored either red, yellow, or blue, so that no two consecutive squares have the same color. How many ways are there to color the five squares, if there must be at least three yellow squares?  

Squares 1, 3, and 5 have to be the yellow ones. 

That leaves squares 2 and 4.  They can be

                                                                         red red  

                                                                         blue blue  

                                                                         red blue  

                                                                         blue red  

for a total of four ways to color the squares to satisfy the conditions of the problem.  

_.