Bosco

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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In triangle ABC, the measure of angle A is x degrees, the measure of angle B is 2x degrees and the measure of angle C is 6x degrees. What is the value of x? Express your answer as a decimal to the nearest tenth.    

A = x     B = 2x     C = 6x    

A + B + C  =  180^o    

x + 2x + 6x  =  180^o    

              9x  =  180^o     —>>     x = 20.0^o    

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Rationalize the denominator of 5/(3 - sqrt(3)), and write your answer in the form (A + B*sqrt(3))/C where the fraction is in lowest terms. What is A + B + C?    

5 / (3 – sqrt(3))    

Multiply both top and bottom by (3 + sqrt(3))     

to get the difference of squares in the denom.    

[ 5 / (3 – sqrt(3)) ] • [ (3 + sqrt(3)) / (3 + sqrt(3)) ]    

result  5 • (3 + sqrt(3)) / (9 – 3)    

               15 + 5sqrt(3) / 6    

A = 15     B = 5     C = 6    

A+B+C = 15 + 5 + 6  =  26    

_.    

A city park commission received a donation of playground equipment from a parents' organization. The area of the playground needs to be 256 square yards for the children to use it safely. The playground will be rectangular.

In a different plan, the sides can be of any length as long as the rectangular area remains 256 square yards. What dimensions of the rectangular area provide the least perimeter of fencing?    

You get the most area for the amount of perimeter if the figure is a square.    

sqrt(256) = 16     so the dimensions would be 16 x 16    

_.    

A class of 30 students recently took a test. If 20 students scored 80, 8 students scored 90, and 2 students scored 95, then what was the class average (the mean) on this test?    

                                                       20 x 80  = 1600    

                                                         8 x 90  =   720    

                                                         2 x 95  =   190    

add them up                                                    2510    

                                                    2510 / 30  =  83.67    

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Two liters of a 30%  acid solution are mixed with three liters of a 70% acid solution. What percent of the mixture is acid?    

2 liters @ 30% acid  =  0.6 liter acid & 1.4 liter water    

3 liters @ 70% acid  =  2.1 liter acid & 0.9 liter water    

adding them up            2.7 liter acid & 2.3 liter water    

This is a total 5.0 liter solution, 2.7 liters of which is acid    

2.7 / 5.0  =  0.54  ——>>  The new solution is 54% acid    

Confirm plausibility:  2.7 is a little more than half of 5.0    

_.    

How do you write the following equation in standard form? Determine a, b, and c.    

8x^2+x=-78x2+x=−7    

     8x² + x  =  –78x² + x  =  –7    

     86x² + 7  =  0     That's your standard form.    

     a = 86     b = 0     c = 7    

_.    

Let assume Rohan age is X. If we 1/4 his age and add 8 then we find this is equal to 14. Write this expression in equation?    

The equation is                          (x / 4) + 8 = 14    

Work it on out as follows             x / 4  =  14 – 8    

                                                    x / 4  =  6    

                                                         x  =  24    

_.    

A rectangle in the coordinate plane is shown below.  A line with slope 2 splits this rectangle into two regions with equal area.  What is the y-intercept of this line?  The rectangle has coordinates (-1,-1), (-1,1), (1,1), and (1,-1).    

Those coordinates draw a square, centered on the origin.    

Any straight line through the origin will bisect the square.   

Conversely, any line that bisects the square will pass through the origin.    

Therefore, the y-intercept is  (0, 0)    

_.    

A line and a circle intersect at A and B as shown below. Find the distance between A and B.    
The circle is x^2 + y^2 = 1, and the line is y = x.    

This is an easy one, I don't even have to draw it. 

The circle is centered on the origin with a radius of 1. 

The line crosses through the origin and intersects the circle.  

The distance between A & B is the diameter of the circle.  =  2 units      

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