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.    

Sorry, but you can not achieve a 50% winning season.    

_.

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.    

Sorry, but you can not achieve a 50% winning season.    

_.

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.      

_.

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)  

_.

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.    

_.

In a store window, there was a box of berries having a total weight of 200 kg.  The berries were 99% water, by weight.  After two days in the sun, the water content of the berries was only 90%, by weight.  What was the total weight of the berries after two days, in kg?  

You're starting with 200 kg of berries, and 99% of that is water.  

Since they're 99% water, that means the other 1% is solid stuff.  

1% of 200 kg means that there are 2 kg of solid stuff.   

Some of the water evaporates in the sun, but the solid stuff doesn't.  

After evaporation of some water, there's still the same 2 kg of solid stuff.  

Since the new water content is 90%, that means that now the solid stuff is 10%.  

So 2 kg (the solid stuff) is 10% of the total weight.  

That means the new total weight is 20 kg.  

_.

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.    

Sorry, but you can not achieve a 50% winning season.    

_.

In triangle PQR, M is the midpoint of PQ. Let X be the point on QR such that PX bisects angle QPR, and let the perpendicular bisector of PQ intersect AX at Y. If PQ = 36, PR = 22, QR = 26, and MY = 8, then find the area of triangle PQR    

I was trying to draw the figure, and then       

you threw me a curve when you said AX. 

Where did that come from?   

Then I saw that you have the values of all   

three sides.  You don't need anything else. 

Add all three sides, then take half of it.    

Call this "s".       

                                                                               36 + 22 + 26   

                                                                       s  =  ——————  =  42     

                                                                                      2    

Then use the following to get the area.         A  =  sqrt[(s)(s – a)(s – b)(s – c)]   

This is called Heron's Formula.  

                                                                      A  =  sqrt[(42)(42 – 36)(42 – 22)(42 – 26)]   

                                                                      A  =  sqrt(42 • 6 • 20 • 16)    

                                                                      A  =  sqrt(80,640)    

                                                                      A  =  283.97 square units    

That answer looks weird,but it has     

to be right ... it's Heron's Formula. 

_.

Will and Grace are canoeing on a lake.  Will rows at 50 meters per minute and Grace rows at 20 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 p.m. If they always row directly towards each other, and the lake is 3800 meters across from the west side of the lake to the east side, at what time will the two meet?  

Will's rate is 50 m/min.  

Grace's rate is 20 m/min.  

They're rowing toward each other,  

so their rate of closure is 70 m/min.  

The lake is 3800 meters across.  

The time it will take them to meet is given by  

                                                                                    Distance  

                                                                            T  =  –––––––  

                                                                                       Rate  

                                                                                    3800 m  

                                                                            T  =  –––––––  

                                                                                    70 m/min  

The time it takes them to meet                             T  =  54.29 min  

Note that (0.29 min • 60 sec/min)  =  17 seconds

The time on the clock is                                             2:54:17 pm 

_.

4^n=26 find n     

When I open the calculator on my iPhone   

then turn the phone on its side   

then select 2^nd functions    

one of the buttons is labeled log_y   

What that does, you can choose your own base    

and the phone will provide the log of any number  

I found that log₄(26) is  2.350219859 (rounded)  

check answer    

using the x^y function on my iPhone    

4^2.350219859  =  25.99999999    

Close enough for government work.  

_.