Bosco

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?  

In order to avoid having two yellows next to each other,  

the three yellows must occupy squares 1, 3, and 5. 

That leaves squares 2 and 4 to account for, and two   

colors do it with. There are only four ways. 

Red Red     Blue Blue     Red Blue     Blue Red   

So, the answer is four ways.  

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Given an isosceles trapezoid with bases of 8 and 18 and an area of 120 square units, what is the number of units in the length of one of the non-parallel sides?  

Area of a trapezoid is the height times half the sum of the top base plus the bottom base. 

                                                 A  =  (h) • (b_top + b_bottom) / 2  

                                             120  =  (h) • 26 / 2  

from this we get                         h  =  120 / 13  =  9.2308   

The height dropped from the top left corner intersects the bottom base at 5 from the left.  

The non-parallel side of the trapezoid is the hypotenuse of the right triangle thus formed.  

Use Pythagoras Theorem          c²  =  (9.2308)² + (5)²  =  110.207  

                                               side  =  sqrt(110.207)  =  10.5   

When I was doing this on the calculator, I didn't round the numbers.  

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A right triangle has a hypotenuse of 10 m and a perimeter of 24 m. In square meters, what is the area of the triangle?  

Since the hypotenuse is an integer and the perimeter is an integer, the other two legs must be integers too. 

Just by recognition, a Pythagoras Triple of 6-8-10 will work.  That makes the area (1/2)(6)(8) = 24 meter²  

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Billy purchased 100 shirts at a cost of $15 each and planned to resell them. The first week he sold some of them for $25 each. The second week he sold the rest for $20 each. If his total profit was $600, how many shirts did he sell the first week?  

            n  =  number of shirts sold the 1^st week at $10 profit  

(100 – n)  =  number of shirts sold the 2^nd week at $5 profit  

            [ 10 • (n) ]  +  [ 5 • (100 – n) ]  =  600    

                        10n + 500 – 5n  =  600  

                                            5n  =  100  

                                              n  =  20  shirts sold the first week  

check answer  

                                        20 shirts • 10 profit  =  200  

                                        80 shirts •   5 profit  =  400  

                                                      total profit  =  600  

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The area of a trapezoid is $80$. The length of one base is $16$ units greater than the other base, and one base of the trapezoid is $12$. Find the length of the median of the trapezoid.  

The length of the median is equal to the average of the two bases.  

So, with one base 12, and the other base 16 longer, we have  

                  12 + (12 + 16)        40  

median  =  ––––––––––––  =  ––––  =  20  

                             2                   2  

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Let a1, a2, a3, ..., a10, a11, a12 be an arithmetic sequence. If a1 + a2 = 4 and a2 + a3 = 5, then find a1.  

                                                         a₁ + a₂ = 4   &   a₂ + a₃ = 5  

Since it's an arithmetic sequence, the difference between adjacent terms is the same.  

Call that difference, i.e., the increment, let's call it "n".  Just keep this in mind for a while.  

                                                         a₂ + a₃ = 5    

                                                         a₁ + a₂ = 4    

Subtract one from the other  

                                                         a₃ – a₁ = 1  

Since a₃ is (a₁ + n + n)  

                                                         (a₁ + n + n) – a₁ = 1  

                                                          2n = 1  therefore n = 0.5  

Referring back to the top   

                                                          a₁ + a₂ = 4  

a₂ = (a₁ + 0.5) so plug it in  

                                                           a₁ + (a₁ + 0.5) = 4  

                                                           2a₁ + 0.5 = 4  

                                                           2a₁ = 4 – 0.5 = 3.5    

                                                            a₁ = 3.5 / 2  

                                                            a₁ = 1.75   

check answer  

                                                            a₁ = 1.75  

                                                            a₂ = 2.25        a₁ + a₂ = 4  

                                                            a₃ = 2.75        a₂ + a₃ = 5 

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Let x be a real number such that 625x = 9.  Then 125x = ____  

                                                    625x   =     9  

                                                    625x           9  

                                                   –––––  =  –––––   

                                                    625           625  

                                                                       9  

                                                           x   =  –––––   

                                                                     625  

                                                                       9  

                                                  125 • x   =  ––––– • 125  

                                                                     625  

                                                                     9 • 125         1,125  

                                                     125 x   =  –––––––  =  ––––––     

                                                                       625              625  

                                                                       9  

                                                     125 x   =  ––––   (in decimal form 1.8)        

                                                                       5  

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