Bosco

How many of the same digits are found in the base 7 and base 8 representations of 609_{10}?     

609₁₀  =  1530₇  =  1141₈     

Looks like there's just one digit in both, namely 1 and there are 4 of them.     

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A florist sells some red, yellow and orange roses. She sells 60 red and yellow roses, 70 yellow and orange roses and 80 orange and red roses. Find the number of red roses, the number of yellow roses and the number of orange roses that the florist sells.     

R + Y  =  60     (eq 1)     

Y + O  =  70     (eq 2)     

O + R  =  80     (eq 3)     

subtract (2) from (1)      R + Y  =  60     

                                      Y + O  =  70     

                                      R – O  =  –10     (eq 4)     

add (4) and (3)              R – O  =  –10     

                                      O + R  =  80     

                                           2R  =  70     

                                             R  =  35     

substitute R=35 into (3)     O + 35  =  80     

                                                   O  =  45     

substitute O=45 into (2)     Y + 45  =  70     

                                                   Y  =  25     

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Modify one digit in the number 31743 can make it a multiple of 281.  What is the number after change     

31743 / 281  =  112.96+     Truncate the .96 ... all we want right now is the 112.     

112 x 281  =  Ruh Roh, this isn't going to work, so try 111.     

111 x 281  =  31753     Bingo.  Change the 4 to a 5.     

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Two boys and three girls are going to sit around a table with 5 different chairs. If the two boys want to sit next to Jill, in how many possible ways can they be seated?     

The only way both boys can sit next to Jill at the same time is one on each side of her.     

So, consider every configuration that contains B1 - J - B2 as a unit.  There are 2 more     

girls, so there are 3 "entities" call them.  How many ways to arrange 3 entities.     

This would be expressed as the permutation of three things taken three at a time.     

 3P3  =  6     Now have the two boys change sides of Jill and there are 6 more.     

Add them together and there are 12 total possible ways to seat the people.     

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Solve the system of equations 5x+6y=265x+6y=26 and -x+y=-14−x+y=−14 by combining the equations.    

Consider the     –x + y  =  –14 – x + y    

If you would add x – y to both sides, you get 0 = –14    

However, 0 = –14 cannot be,     

at least, not in our universe,     

so the premise is invalid.    

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Two runners start at vertices A and B simultaneously and run clockwise around the perimeter of square ABCD at the same speed. A drone flies so that it is always at the midpoint of the two runners.  What is the minimum distance between the two runners?  What is the maximum distance between the two runners?    

See this problem at https://web2.0calc.com/questions/geometry_69250    

There, I solved the problem using trending, and Chris solved it using calculus.    

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If x and y are integers such that 12^2 \cdot 18^3 = xy, find x+y.    

12² • 18³ = xy     ——>     xy = 144 • 5832 = 839,808     

x and y could be any of the many factors of 839,808     

so let's just use the two factors that we have already    

been given ... namely, 144 and 5832     

x + y = 144 + 5832  =  5976     

This is one sum.  There could be are others possible.     

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What is the range of the function g(x) = (6x + 7)(2x - 3)?    

This forms a parabola that opens upward, therefore the    

range is from the vertex's y-coordinate to positive infinity.    

y  =  (6x + 7)(2x – 3)  =  12x² – 4x – 21    

y'  =  24x – 4     

set y' = 0     —>>  x = 1/6     & sub back into original    

y  =  [(12) • (1/6)²]  –  (4)(1/6)  –  21    

y  =  1/3                 – 2/3           – 21   =  – 21.3333    

The range is from – 21.3333 to positive infinity    

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