Bosco

All members of our painting team paint at the same rate.  If 20 members can paint a 2000 square foot wall in 40 minutes, then how long would it take the 20 members to paint a 6000 square foot wall, in minutes?   

It's the same number of painters, all painting at the    

same rate that they painted the 2000 sqft wall.  

The only thing that's different is that the new wall is    

3 times as big, so it will take 3 times as long to paint.   

                                3 • 40 minutes  =  120 minutes   

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If I give my brother 5 dollars, then we will have the same amount of money. If instead he gives me 25 dollars, then I'll have twice as much money as he will have. How much money does my brother currently have (in dollars)?   

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 giving each other money.   

The problem doesn't ask for My original amount but it's B + 10 = 95  

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.  

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Catherine rolls a standard 6-sided die six times. If the product of her rolls is  then how many different sequences of rolls could there have been? (The order of the rolls matters.)   

If you go to https://web2.0calc.com/questions/help-plz_51258

you can check out the earlier answer I gave, and GA's elaboration    

on the way to have done it using permutations.  There, I got 45.    

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The error is in answer #1 at   

(A + n )* 7 =  (B - n)     →  7A - 2B  = -5n   (1)   

(A - n) =  3 * ( B + n)  →  A - 3B  = 4n  (2)   

The first one should be  

(A + n)  =  7 • (B – n)    (Since Alice has 7 times as much as Bob, you   

                                      have to multiply Bob's by 7 to make them equal.)   

I do like Guest's reasoning in answer #1 though. 

So let's continue that, but with these other numbers.     

                                         (A + n)  =  7 • (B – n)  

                                          A + n    =  7B – 7n  

                                          A – 7B  =  – 8n            (1)  

                                         (A – n)  =  3 • (B + n)  

                                          A – n   =  3B + 3n 

                                          A – 3B  =  4n                (2)  

Multiply both sides  

of (2) by 2 and then   

add (1) and (2)  

                                            A – 7B  =  – 8n  

                                          2A – 6B   =     8n    

                                          3A – 13B  =  0  

Add 13B to both sides                  3A  = 13B   

Divide both sides by 13B          

                                                     3A        1      

                                                     –––  =  –––   

                                                     13B       1  

Multiply both sides by 13/3   

                                                 13       3A          13        1  

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

                                                   3       13B          3         1   

The digits cancel out   

of the left side, leaving                 

                                                  A         13   

                                                 –––  =  –––   

                                                  B           3   

_.

Find all values of x such that \sqrt{4x^2} + \sqrt{x^2} = 6 + 3x.   

I thought I had this figured out in an earlier posting of this same question,   

but a moderator, Alan, added an answer, as a correction of the mistake   

that I had made in my answer.  Here's a copy/paste of what Alan said:   

Note there is a difference between  x^2 = 4, for which x = ±2,  and  x = √4, for which x = +2 only.   

√(4x^2) = 2∣x∣    and    √(x^2) = ∣x∣   

so we have:   3∣x∣ = 6 + 3x    

_.

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 of that 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 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 see it for yourself.  :-)  

Would it be cheating, to use that website to avoid the drudgery of endless   

multiplications?  I do not consider it cheating to use any tool that has been  

made available.  It used to be slide rules, then calculators, now computers.   

_.

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 that there's 1% of 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.  

So, after the evaporation, there still remains 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.  

_.

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 

.

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  

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