Bosco

Is the Queen able to move any number of squares,     

as is the case on a chessboard?   

If so, your winning strategy would be to move first,  

then just move her all the way to the other corner.  

_.

A nonagon has angles that measure 140°, 140°, 130°, 140°, 140°, 145°, 155°, 140°, and a. What is a?    

You would calculate the sum of all the angles,   

then subtract the total of the angles you know.   

The sum of the internal angles of a polygon is  S = (n – 2)(180^o)    

where n is the number of sides of the polygon.  

                                                             S  =  (9 – 2)(180^o)   

                                                             S  =  7 • 180^o   

                                                             S  =  1260^o   

The total of the known angles is                    1130^o   

if neither of us made a mistake.  

You'd better check my addition.   

Subtract                                                 a  =  130^o     

_.

Find the coefficient of y^4 in the expansion of (2y-5+3y^2)^6.   

That's way too much multiplication to do by hand.  

Fortunately, a useful tool has been provided for us, at   

https://www.symbolab.com/solver/expand-calculator

Per the "expand" tool on that page, the coefficient of y⁴ is 375.  

But don't take my word for it, go there and check it for yourself.  :-)  

Is it considered cheating, to use a tool to avoid the drudgery of endless

multiplications?  I don't consider it cheating to use any tool that has been  

made available.  The abacus, the slide rule, the calculator, the computer.      

_.

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 just 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  

It can't be both at the same time, so the system has no solution.  

^.

We will have the same amount of money if I give my brother $5. Instead, if he offers me $25, I'll have twice as much cash as he does.  

Well........ if he only offers $25 nothing changes.  

He would have to actually hand over the $25.  

But I think I know what you mean.  

Let M stand for My original amount  

Let B stand for Brother's original amount   

                                                             (M – 5)  =  (B + 5)   

                                                             (M + 25)  =  (2)(B – 25)  

                                                              M – 5  =  B + 5   gives us  M = B + 10    (1)  

                                                              M + 25  =  2B – 50                                 (2)  

Substitute the value of M from (1)   

into the equation (2)                             B + 10 + 25  =  2B – 50   

Combine like terms                                          85  =  B    

Brother has 85 dollars before they start swapping moneys.   

My amount is 95 dollars   

Check answer  

If I give my brother 5, then I'll have 90 and so will he, so we have the same.  

If my brother gives me 25, then I'll have 120 and my brother will have 60.  

_.

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?  

You mean 45 minutes earlier.  Obviously, if you drive faster, you get there faster.  

This problem makes use of

the following relationship:                   Distance = Velocity x Time 

                                                           D  =  V • T  

case 1                                                 D  =  (40) • (T)  

case 2                                                 D  =  (50) • (T – 45)  

Since the Distance, D, is the  

same for both cases, let's set       

the "V•T"s equal to each other.             (50)(T – 45)  =  (40)(T)  

                                                               50T – 2250  =  40T  

Subtract 40T from both sides                  10T – 2250  =  0  

Add 2250 to both sides                                       10T  =  2250  

Divide both sides by 10                                           T  =  225   (this is in minutes)  

Divide minutes by 60 to get hours                           T  =  225 min / 60 min/hr  =  3.75 hours  

Plug this T back into original equation                     D  =  (40 mi/hr) • (3.75 hr)  =  150 miles  

_.

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.   

_.

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? 

We're going to have to convert metric units to imperial units,  

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 a third of a gallon of paint,  

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

^.

Real numbers a and b satisfy  

a + ab = 250  

a - ab = -240  

Enter all possible values of a, separated by commas.  

You probably will never find a simpler

problem in your algebra textbook, lol.  

                                                    a + ab  =    250   

                                                    a – ab  =  –240  

Just add the two together.  

                                                  2a          =      10    

Divide both sides by 2.  

                                                            a  =  5    

The problem didn't ask for b.  

_.

One ordered pair (a,b) satisfies the two equations ab^4 = 48 and ab = 72. What is the value of b in this ordered pair?    

To find b, consider                                                  ab⁴  =  48  

We will divide both sides by ab.  

Since ab=72, we will divide the left side  

by "ab" and the right side by its equal 72.  

                                                                              ab⁴         48  

                                                                             ——   =   ——  

                                                                              ab           72  

Note that ab⁴ = (ab) * (b³)  

Cancel ab out of the left side.  

Reduce 48/72 on the right side.  

                                                                               b³           2  

                                                                             ——   =   ——  

                                                                                1            3  

                                                                                 b   =   cube root of (2 / 3)  

_.