Bosco

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?   

Let the smallest integer be represented by x.   

Since the integers are consecutive, each following   

integer is one larger than the integer before it.  

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

                                                             7x + 21  =  21x + 126   

Subtract 7x from both sides   

                                                                     21  =  14x + 126               

Subtract 126 from both sides                     

                                                                  –105  =  14x   

Divide both sides by 14   

                                                                        x  =  –7.5   

check answer    

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

                                                                                                           –31.5  =  –31.5  

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The equation  y=-16t^2 +80t describes the height (in feet) of a projectile launched from the ground at 80 feet per second. At what  t will the projectile reach 36 feet in height for the first time? Express your answer as a decimal rounded to the nearest tenth.    

y  =  –16t² + 80t describes the height, so let's just set y to the desired height and see what happens.   

                                                                  36  =  –16t² + 80t      

Subtract 36 from both sides.                       0  =  –16t² + 80t – 36   

Multiply both sides by –1.                         16t² – 80t + 36  =  0  

Divide both sides by 4, to get     

more manageable coefficients.                   4t² – 20t + 9  =  0   

Factor.                                                        (2t – 1)(2t – 9)  =  0    

                                                                    2t  =  1      ----->  t  =  0.5 sec   

                                                            and  2t  =  9      ----->  t  =  4.5 sec   

The problem asks for the time  

to get to 36 feet on the way up,       

so it's the smaller answer.   

check answer    

                                                          y  =  –16t² + 80t    

                                                          y  =  –(16 • 0.25) + (80 • 0.5)    

                                                          y  =  –4 + 40    

                                                          y  =  36       

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@anitbiotics   >>> sigh, what did I do for my message to deserve moderation... <<<    

Mine have been getting flagged for moderation, too.  Like you, I also wondered what I had done to get in the doghouse, so I messaged a Moderator and asked.  It turns out that many posts are getting flagged for moderation.  I don't know how the posts are selected.  

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Keiko tosses one penny and Ephraim tosses two pennies. What is the probability that Ephraim gets less heads than Keiko gets?  Express your answer as a common fraction.    

For Ephraim to get fewer heads than Keiko, then Keiko has to get heads and Ephram no heads.  

If Keiko does not get heads, Ephram can't get fewer, because it would have to be fewer than zero.  

The probability that Keiko's toss comes up heads is 1 / 2.   

Ephraim's toss can come up one of four ways:  they are HH, TT, HT, or TH   

Only one of the results contains zero tails, so the probability of that is 1 / 4.   

The probability of both things happening simultaneously is  (1 / 2) • (1 / 4)  =  1 / 8       

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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 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.  

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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 .   

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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As noted by my friend in Answer #1, this problem makes use of an unusual assumption.  However, it can be solved, although the value of "n" will require an amount of imagination.  Hey, the square root of minus one is an imaginary number, but that doesn't stop people from using it.  I'm solving the problem for you, so you can see how to solve it, whether or not the answer satisfies your sense of propriety. 

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?    

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

                                         A + n    =  4B – 4n    

                                         A – 4B  =  –5n                   (1)    

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

                                          A – n    =  8B + 8n   

                                          A – 8B  =  9n                    (2)   

Multiply both sides  

of (1) by 9                        9A – 36B  =  –45n              (3)    

Multiply both sides  

of (2) by 5                        5A – 40B  =    45n              (4)      

Add (3) and (4)               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    

                                                      A          76                         38

                                                    –––   =   –––   reduces to   –––   

                                                      B          14                          7   

It works if you accept the  

concept of negative money.  

That is, n must equal –2 dollars.    

Say Alice has 38 and Bob has 7        

                                                     (38 + n)  =  4 • (7 – n)     

                                                      38 + n   =  28 – 4n    

                                                             5n  =  –10     

                                                               n  =  –2       

and               

                                                     (38 – n)  =  8 • (7 + n)    

                                                      38 – n   =  56 + 8n      

                                                          –9n   =  18    

                                                               n  =  –2       

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Equilateral triangle APQ is inscribed inside square ABCD. Find the area of square ABCD.    

cosine of angle BAP = BA/PA    

cosine of angle DAQ = DA/QA     

BA = DA                             both being sides of a square     

PA = QA                             both being sides of an equilateral triangle    

therefore angle BAP = angle DAQ   

angle PAQ = 60^o                 being an angle of an equilateral triangle    

therefore angle BAP = (1/2)(90^o – 60^o) = 15^o       

cosine BAP = BA/PA   

cosine 15^o  = 0.9659                     from the tables        

therefore    BA/PA = 0.9659   

                         BA = (PA)(0.9659)    

                         BA = (4)(0.9659)

                         BA = 3.8637   

                         Area of the square = (3.8637)²  =  14.928 square units    

I did some rounding on each decimal fraction that I wrote down above,       

but as I was doing the calculations I left the entire fraction.       

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(3^3-2^4)*6    

I assume you want this multiplied out    

                                                                     (3³ – 2⁴) • 6    

Perform the exponent operations                 (27 – 16) • 6    

                                                                             (11) • 6    

                                                                                 66   

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