Bosco

Real numbers a and b satisfy    

a + ab = 250    

a - ab = -240    

Enter all possible values of a, separated by commas.   

i doubt that you'll find an easier problem   

than this in your entire algebra textbook. :-)    

                                                                                a + ab  =    250    

                                                                                a – ab  =  –240   

Just add the two together   

                                                                              2a           =     10    

I just kinda kept the a's lined up   

with each other.  After you divide  

both sides by 2, the final answer is                                   a  =  5  

The problem doesn't ask for the      

value of b, but it is, of course,                                           b  =  49    

_.

Alice and Bob each have a certain amount of money. If Alice receives n dollars from Bob, then she will have 4 times as much money as Bob. If, on the other hand, she gives n dollars to Bob, then she will have 8 times as much money as Bob. If neither gives the other any money, what is the ratio of the amount of money Alice has to the amount Bob has?      

The first case:   

                                                         (A + n)  =  (4) • (B – n)  

                                                          A + n  =  4B – 4n  

                                                          A – 4B  =  –5n             (1)  

The second case:   

                                                         (A – n)  =  (8) • (B + n)  

                                                          A – n  =  8B + 8n  

                                                          A – 8B  =  9n               (2)  

Multiply both sides of (1) by 9           9A – 36B  =  –45n  

Multiply both sides of (2) by 5           5A – 40B  =    45n  

Add these two together                    14A – 76B  =  0  

Add 76B to both sides    

                                                         14A            =  76B  

Divide both sides by 76B      

                                                        14A           1        

                                                        ––––  =   ––––  

                                                         76B          1              

Multiply both sides by 76 / 14  

                                                         76         14A        76          1   

                                                        ––––  •  ––––  =  ––––  •  ––––      

                                                         14         76B        14           1      

The 76 and the 14 cancel out  

on the left side, leaving  

                                                          A           76  

                                                        ––––  =  ––––  

                                                          B           14  

Reduce the fraction  

                                                          A           38 

                                                        ––––  =  ––––  

                                                          B            7 

_.

Find the number of ways of arranging the numbers 1,2 ,3, 4, 5, 6, 7, 8, 9   

in a row so that the product of any two adjacent numbers is even.  

For two numbers to have an even product, at least one of them must be even.  

That is, an odd number can never be adjacent to another odd number.  

So, odds and evens have to alternate, such as the example shown in the problem.  

Furthermore, since there are more odd numbers, the first number has to be odd.  

There are five odd numbers, and five positions into which they can be situated.  

So, there are 5 • 4 • 3 • 2 • 1 = 120 ways the odd numbers can be arranged.  

And, there are four even numbers, and four positions where they can be located.  

So, there are 4 • 3 • 2 • 1 = 24 ways that the even numbers can be arranged.  

Any even number arrangement can be combined with any odd number arrangement.  

Therefore, the number of possible mixes is 120 • 24 = 2880.   

_.

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

p - 2q = 3  

q - 2r = -2 + q  

p + r = 9 + p  

Let's number those equations, for identification only:  

(1)  p – 2q  =  3  

(2)  q – 2r  =  –2 + q  

(3)  p + r  =  9 + p  

Per equation (2), r = 1  

Per equation (3), r = 9  

You can't have r equalling two different things.  There is no solution.  

_.

Find the number of ways of arranging the numbers 1,2 ,3, 4, 5, 6, 7, 8, 9 in a row so that the product of any two adjacent numbers is even.  

Find the number of ways of arranging the numbers 1,2 ,3, 4, 5, 6, 7, 8, 9   

in a row so that the product of any two adjacent numbers is even.  

For two numbers to have an even product, at least one of them must be even.  

That is, an odd number can never be adjacent to another odd number.  

So, odds and evens have to alternate, such as the example shown in the problem.  

Furthermore, since there are more odd numbers, the first number has to be odd.  

There are five odd numbers, and five positions into which they can be situated.  

So, there are 5 • 4 • 3 • 2 • 1 = 120 ways the odd numbers can be arranged.  

And, there are four even numbers, and four positions where they can be located.  

So, there are 4 • 3 • 2 • 1 = 24 ways that the even numbers can be arranged.  

Any even number arrangement can be combined with any odd number arrangement.  

Therefore, the number of possible mixes is 120 • 24 = 2880.   

_.

My square patio is tiled with square tiles, all the same size. All the tiles are gray, except the tiles along the two diagonals, which are all yellow. (The corners are yellow, the center is yellow, and all the tiles along the diagonal in between are yellow.) If there are 5 yellow tiles, how many gray tiles are there?   

Here's the key info:   

                                   The corners are yellow...  

                                    the center is yellow...  

                                    there are 5 yellow tiles...  

There are four corners and one center.  

That's five yellow tiles right there.  

That's all of them.  There are no more yellow tiles.  

So the patio has to be a 3 x 3 square.  Draw it if you doubt me.  

The 3 x 3 patio has nine tiles in total.  

Five tiles are yellow, therefore the other four tiles are gray.  

_.

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

p - 2q = 3  

q - 2r = -2 + q  

p + r = 9 + p  

Let's number those equations, for identification only:  

(1)  p – 2q  =  3  

(2)  q – 2r  =  –2 + q  

(3)  p + r  =  9 + p  

Per equation (2), r = 1  

Per equation (3), r = 9  

r can't be two different things.  The system has no solution.  

_.

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

p - 2q = 3  

q - 2r = -2 + q  

p + r = 9 + p  

Let's number those equations, for identification:  

(1)  p – 2q  =  3  

(2)  q – 2r  =  –2 + q  

(3)  p + r  =  9 + p  

Per equation (2), r = 1  

Per equation (3), r = 9  

You can't have it both ways.  The system has no solution.  

_.

what is the size of an exterior angle of a regular polygon with 180 sides.   

The sum of the interior angles   

of a regular polygon is   

                                                   total  =  (n – 2)(180^o)   

                                                   where n is the number of sides  

The total for 180 sides is           

                                                   total  =  (180 – 2)(180^o)  

                                                   total  =  178 • 180^o  =  32,040^o   

That's the total.  Each angle

would be the total divided by  

the number of sides.     

                                                                                            32,040^o  

                                                   a single interior angle  =  ––––––––    

                                                                                               180   

                                                   a single interior angle  =  178^o     

I see now that I should have anticipated that answer and done it in my head. 

An exterior angle is 180^o minus the interior angle. Visualize extending a side   

beyond the angle.  A straight line contains 180^o   

                                                    exterior angle  =  180^o – 178^o  =  2^o   

_.

I have to paint one side of a wall.  The wall is 12 meters tall and 120 meters long.  Each gallon of paint covers 150 square feet. If a foot is approximately 0.3048 meters, then what is the smallest whole number of gallons I can buy and have enough paint to cover the whole wall?  

You're going to have to convert those meters to square feet,  

so I think it's better to do it from the start.  Even though we'll  

have fractions to contend with, a calculator makes that easy.  

                                                                       12 m  

1 foot is 0.3048 meter, so the height is       –––––––––  =  39.3701 ft   

                                                                   0.3048 m/ft  

                                                                       120 m  

1 foot is 0.3048 meter, so the length is       –––––––––  =  393.7007 ft   

                                                                   0.3048 m/ft  

Area is height times length     (39.3701 ft)(393.7007 ft)  =  15,500.0359 sq ft  

1 gallon will cover 150 sq ft  

so divide that into the area               15,500.0359 sq ft  

                                                        –––––––––––––––  =  103.3335 gallons  

                                                           150 sq ft/gallon  

You can't buy just part of a gallon of paint,  

so we have to round that up to whole gallons                       104 gallons  

_.