Bosco

Our badminton team has finished 75% of its season.  So far, we have won 20% of the games we played.  What percent of the remainder of our games must we win in order to finish the season with the same number of wins as losses?    

You have won 20% of 75% of the total number of games    

in the season.  That means you've won 15% of the total    

number of games in the season.  

To achieve a 50% winning season, you will need to win   

another 35% of the total number of games in the season.  

However, there are only 25% of the total number of games   

in the season left to play.  There aren't enough games left.    

Simply stated, you can not achieve a 50% winning season.    

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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  

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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 referencing 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 both at the same time.  The system has no solution.  

.

That's the way I was going to do it, when I saw you had already posted this. 

1 in the middle leaves 20.  Run them out to each corner and it's 5 to each corner.   

That makes the square 11 tiles wide, thus the square comprises 121 tiles total.  

121 total, minus the 21 yellow tiles, leaves 100 gray tiles.   

_.

There are  members in the Trumpington marching band, and when they line up in rows of 26, there are 4 band members left over. If  is an integer and there are fewer than 1000 band members, what is the maximum number of people that could be in the Trumpington marching band?       

I don't know how to work modulus problems,

I don't think they taught it when I was in school.   

In fact, I never even heard of it until I saw it in this website. 

So I'm going to kind of wing it.     

There are fewer than 1000 members,  

so start with 999.   

Divide 999 by 26 and see what the answer is.   

                                                                                         999 / 26  =  38.423   

Take the whole number part of that quotient, 

which is 38, and multiply it by 26. 

                                                                                           38 • 26  =  988   

So 988 is the number of band members that will  

fill up 38 rows of 26 members per row with none    

left over.  We need 4 left over, so just add it on.    

                                                                                          988 + 4  =  992   

And there you have it.  992 band members is the most,   

short of 1000, that will fit into rows of 26 members per row,     

and have 4 musicians left over.  Probably the piano players;   

those Steinways are hard to trundle around on a football field.

_.

Pearl writes down seven consecutive integers, and adds them up. The sum of the integers is equal to 21 times the largest of the seven integers. What is the smallest integer that Pearl wrote down?   

Call the smallest integer x.  Then each following integer  

will be 1 larger than the integer that preceded it.  

x + (x + 1) + (x + 2) + (x + 3) + (x + 4) + (x + 5) + (x + 6)  =  21 • (x + 6)  

                                                                          7x + 21  =  21x + 126  

Combine like terms                                                –14x  =  105   

Divide both sides by –14                                             x  =  –7.5  

The problem fails already, since 7.5 is not  

an integer, whether it's positive or negative.            NO SOLUTION  

But let's press ahead and see if this would have worked.   

Check answer by substituting the value into the problem.   

(–7.5) + (–7.5 + 1) + (–7.5 + 2) + (–7.5 + 3) + (–7.5 + 4) + (–7.5 + 5) + (–7.5 + 6)  =  (21)(–7.5 + 6) 

(–7.5) + (–6.5) + (–5.5) + (–4.5) + (–3.5) + (–2.5) + (–1.5)  =  (21)(1.5) 

                                                                                  –31.5  =  –31.5  

So, –7.5 would have been a good answer IF the problem

had not specified that the numbers have to be integers.  

_.

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 sooner.  

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          225 minutes  =  3.75 hours  

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

_.

The formula for the sum of the internal angles of a regular polygon is

sum = (n – 2)(180) where n is the number of sides.   

For a pentagram, n is 5         so sum = (3)(180) = 540   

That's the sum, so an individual internal angle is a fifth of that, i.e., 108.  

That makes FED equal 180 – 108 = 72   

The sum of the angles of a triangle is 180, so FDE = 180 – (90 + 72) = 18  

Angle FDE is 18^o   

had to edit because I misspelled polygon, grrr.  

_.

Five times a number is divided by 7 more than the number. If the result is 0, then what was the original number?  

                                                        5n    

                                                    ––––––  =  0      

                                                      n + 7  

The only number that works is       n  = 0     

_.

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.   

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