Bosco

Alice and Bob each have a certain amount of money. If Alice receives $n$ dollars from Bob, then she will have $7$ times as much money as Bob. If, on the other hand, she gives $n$ dollars to Bob, then she will have $3$ 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  =  7 • (B –n)    

                                                                    A + n  =  7B – 7n    

                                                                    A – 7B  =  –8n              (1)    

                                                                    A – n  =  3 • (B + n)    

                                                                    A – n  =  3B + 3n       

                                                                    A – 3B  =  4n                (2)             

Repeat (1) here                                             A – 7B  =  –8n            (3)    

Multiply both sides of (2) by 2                      2A – 6B  =    8n            (4)    

Add (3) and (4)                                            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      

                                                                        A           13            

                                                                      ––––  =  ––––    

                                                                        B             3            

So the ratio of Alice's $ to Bob's $ is               A : B  =  13 : 3       

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Fundamental error in the attempted solution at   

"If Alice receives n dollars from Bob, then she will have A+n=7B"   

This should be  

If Alice receives n dollars from Bob, then she will have A + n = 7(B – n)  

etc.   

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Is it okay to just see two squares, side by side?  If so, the total area is 36 + 36 = 72 square units

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If k^n=nk^n-1 is true. Then can 2 = 1? As 2 = 2^1 = 1*2^0 = 1 * 1 = 1?    

kⁿ  =  nk^n–1 is not true. 

kⁿ  =  kk^n–1 is true.   

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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 95%, 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 95%, that means that now the solid stuff is 5%.  

So 2 kg (the solid stuff) is 5% of the total weight.  

That means the new total weight is 2 / 0.05 = 40 kg.  

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Thank you.  I tried to give the answer a heart, but it wouldn't add one.  

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"The formula for the sum of an infinite series is a/(1-r), where a is the first term in the series and r is the common ratio i.e. the number that each term is multiplied by to get the next term in the sequence. To find r, divide any term in the series by the prior term."  

I didn't know that, I just copied it off the internet.  Let's see if I can use it right.    

Considering  10 + 10r + 10r^2 + 10r^3 + 10r^4 + ... = 8    

                      The first term is 10 and it looks like r is r. 

So       

                                        10      

                                      –––––  =  8   

                                       1 –  r    

                                            10  =  8 – 8r   

                                             8r  =  8 – 10   

                                               r  =  –0.25   

Can it really be that easy?  Would somebody who knows how      

to solve this kind of problem please check these figures?                      

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Hi Chris.

I'm so glad to see that you're back.  We've all missed you, everybody has been asking about you.  

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What is the smallest four digit multiple of 9 with four distinct digits?    

I did a little bit of logic and a little bit of brute force.    

We want the lowest four digit number.  Well, it has to start with a 1.  So we've got that.

What second digit would be the lowest.  Obviously, 0.  So we've got 10 _ _.   

For the third digit, it can't be 0 or 1 so let's go with 2.  Now we have 102 _.

The fourth digit can't be 0 or 1 or 2.  It has to be 3 or larger. 

That's the end of the logic.  Now for the brute force. 

Tried 1023, not evenly divisible by 9.       

Tried 1024, not evenly divisible by 9.   

Tried 1025, not evenly divisible by 9.  

Tried 1026, Bingo.  It's 114.  So that's your number right there:  1026.  

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