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A list of ten distinct, positive integers has a median of 5.5.  What is the smallest possible average of the ten positive integers?  Explain your ansswer in complete sentences.     

The median is the value that there are an equal number of values smaller and larger than it.     

It would look like this:  n  n  n  n  5  6  n  n  n  n     

The smallest distinct positive numbers are  1  2  3  4  on the left     (0 is not positive)     

                                                             and  7  8  9  10  on the right.     

The average of the above listed numbers is  55 / 10  =  5.5     

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A standard six-sided die is rolled twice. What is the probability that the product of the rolls is 3 or 12?     

Ways to get   3      1, 3     3, 1                            =  2 ways     

Ways to get 12      2, 6     3, 4     4, 3     6, 2      =  4 ways     

                                                              total      =  6 ways     

Total number of possible rolls is 6 • 6                =  36 ways     

Probability of desired product is  6 / 36              =  1 / 6     

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Two motorcyclists start at the same point and travel in opposite directions. One travels 10 mphfaster than the other. In 2 hours they are 224 miles apart. How fast is each traveling?     

MC1 rate  =   x mph    

MC2 rate  =  (x + 10) mph     

        time  =  2 hr     

2 • (x)  +  2 • (x + 10)  =  224     

2x + 2x + 20  =  224    

        4x + 20  =  224     

                 4x  =  204     

                  x  =  51 mph     MC1     

       51 + 10  =  61 mph     MC2     

check answer     

MC1 ...  2 • 51  =  102 miles one direction     

MC2 ...  2 • 61  =  124 miles the other direction     

MC1 + MC2      =  224 miles apart     

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how much is 55L in a gallon?     

Do you mean convert 55 liters to gallons?     

(55 liters) • (0.264172 gallon/liter)  =  14.5295 gallons     

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Sam wants to color the three sides of an equilateral triangle.  He has two different colors to choose from.  In how many different ways can Sam color the sides of the triangle?     

Call the colors R and B     

     R R R     B B B     

     R R B     B B R     

     R B R     B R B     

     B R R     R B B     

I make it 8 ways ... can you think of another combination?     

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A circle with a radius of 14 feet is cut into 8 equal pieces. How many square feet are 3 of the pieces? Use 22/7 for π.     

A.  1,155 square feet     

B.  231 square feet     

C.  22 square feet     

D.  11 square feet     

                    A  =  π r²     

                    A  =  (22/7)(14)(14)     

                    A  =  (22)(2)(14)     

                    A  =  616     

                    616 / 8  =  77     

                    3 • 77  =  231     Answer is B.     

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Kari has a round trampoline with a diameter of 4 meters. What is the trampoline's circumference?     

Circumference of a circle  =  π times the diameter     

                                     C  =  π • 4 m      

                                     C  =  3.1418 • 4 m     

                                     C  =  12.5664 m     

rounded                        C  =  13 m     

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Find the area of triangle ABC if AB = 6, BC = 8, and angle ABC = 90 degrees.     

        Area_triangle  =  (1/2) x base x height     

                       A  =  (1/2) x 6 x 8     

                       A  =  24     

AB and BC are the legs (as distinct from the hypotenuse)     

because each of them has an end at the right angle.     

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River C is 200 miles longer than River D. If the sum of their lengths is 5,570 ​miles, what is the length of each​ river?     

C + D  =  5570     

C  =  D + 200     

D + C              =  5570     

D + (D + 200)  =  5570     

2D + 200         =  5570     

2D                   =  5570 – 200     

2D                   =  5370     

                   D  =  2685     

C                     =  D + 200     

C                     =  2685 + 200     

                   C  =  2885     

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Of the 80 spaces in the XYZ Industries' parking lot, 5% of them are reserved for handicapped parking. Of the non-handicapped spaces, 16 are reserved for company VIPS. How many spaces that are NOT reserved for handicapped parking are available to non-VIP employees?     

80 – (80)(5%)    =     

80 – (80)(0.05)  =     

80 – 4                =  76 for non-handicapped     

76 – 16              =  60 for non-VIP     

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Find all possible ways of numbering two 6-sided dice with positive integers (not necessarily distinct), so that when they are rolled, the 36 possible outcomes for the sum of the two rolls are the same as the 36 possible outcomes for the sum of two standard 6-sided dice.     

There's an infinite amount of integers to choose from, so     

you can re-number the first die any way you want to, just as     

long as you re-number the second die the same way as the     

first.  I don't like to use infinity as a number, but you never     

come to the last number, so I have to.     

Rolling one die twice is the same as rolling two dice once.     

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Find the vertex of the graph of the equation y = -2x^2 + 8x - 15 - 2x^2 + 12x - 7.     

                                             y  =  –2x² + 8x – 15 – 2x² + 12x – 7     

                                             y  =  –4x² + 20x – 22     

take 1^st derivitive                  y'  =  –8x + 20     

set 1^st deriv equal zero        –8x + 20  =  0     

                                                     –8x  =  –20     

                                                         x  =  2.5     <<—  this is the x-coordinate of the vertex     

The first derivitive is the slope of the curve at any point.  When the slope is zero, that     

means there's no slope, i.e., the curve is turning around and going the other direction.

plug 2.5 into equation            y  =  (–4)(2.5²) + (20)(2.5) – 22     

                                              y  =  –25 + 50 – 22     

                                              y  =  3                    <<—  this is the y-coordinate of the vertex     

so the vertex is located at     (2.5,  3)     

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A person is looking from a car to the top of a building. The angle of elevation to the top of the building from the car is 32°. The building is 200 feet away. How tall is the building? Round to the nearest foot.     

     tangent equals opposite over adjacent     

     tan(32)  =  x / 200     

     0.62487  =  x / 200     

     x  =  200 • 0.62487     

     x  =  124.97     to the nearest foot is 125     

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Ms. Rathman has 5 nickels, 3 dimes, and 2 quarter. In how many ways can she make change for 50¢?     

25  25     

25  10  10  5     

25  10  5  5  5     

25  5  5  5  5  5     

10  10  10  5  5  5  5           I count 5 ways.  If you find another, give me a call.     

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Amanda and her group of friends spent a total of 35.50 on 7 hot dogs and 6 cheese burgers. Greg and his group of friends spent 30.50 at the same snack bar on 5 hot dogsand 6 cheese burgers. For your answer WRITE THE EQUATIONS and state how much hot dogs cost and cheese burgers cost.     

                    7D + 6B  =  35.50     

                    5D + 6B  =  30.50     

                    2D           =  5.00     

                               D  =  2.50     

                    (7)(2.50) + 6B  =  35.50     

                         17.50 + 6B  =  35.50     

                                      6B  =  18.00     

                                        B  =  3.00     

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what will be the nature of physical quantity if a 75 kg man is running at 25 m/s^2     

               The phrase "is running" implies velocity, but m/s² is acceleration.     

               Continuing on that basis, mass times acceleration equals force     

               kilograms times meters per second squared equals newtons     

               In this example, it is 75 kg • 25 m / s²  =  1875 newtons     

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An aquarium tank is 1/6 full of water. When 2 gallons of water are added, the tank becomes 1/2 full. What is the total capacity of the aquarium tank, in gallons?     

                                     tank begins with         ........  1 / 6   full     

2 more gallons makes the tank be half full      ........  3 / 6  full     

this demonstrates that adding 2 gallons adds ........  2 / 6  of a tank     

therefore 1 gallon is 1 / 6 tank which means that the tank holds 6 gallons     

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Solve 3a + 4b = 5a - 6b + 24 - 3a for a in terms of b.     

                    3a + 4b  =  5a – 6b + 24 – 3a     

Move all the a's to one side, everything else to the other side.     

                    3a – 5a + 3a  =  – 6b + 24 – 4b     

                                       a  =  24 – 10b     

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