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The hypotenuse of an isosceles right triangle is 14 . How long is each leg of the triangle?     

     Imagine two more sides to make that triangle     

     into a square and that hypotenuse is a diagonal.     

     diagonal_square  =  side • sqrt(2)     

                      side  = 14 / sqrt(2)     the "side" here is the leg of the triangle     

                        leg  =  14 • sqrt(2) / 2     

                         leg  =  7 • sqrt(2)     

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A catering assistant has 2 litres of orange juice.     
How many 200 ml glasses can she fill with the juice?     

2 liters = 2000 milliliters     

               2000 milliliters / 200 ^milliliters/_glass  =  10 glasses     

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The average of 10 numbers is 85. If the numbers 70 and 78 are removed from the set of numbers,     

what is the average of the remaining numbers?     

Since the average of the 10 numbers is 85, the sum of the ten numbers is 850.  (i.e., 85 = ⁸⁵⁰/₁₀ ).   

Subtract 148 (i.e., remove 70 and 78) from that sum and the sum of the remaining 8 numbers is 702.     

The new average is given by 702 divided by 8.  Using a calculator, find that 87.75 is that new average.     

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The quadratic equation 2x^2 + bx + 18 = x^2 + 2 has a double root. Find all possible values of b.     

                                      2x² + bx + 18  =  x² + 2     

                                        x² + bx + 16  =  0     

To have a double root, b has to be double the square root of 16.     

                                      b  =  2 • sqrt(16)     

                                      b  =  2 • + 4     

                                      b could be + 8     

                                 or, b could be – 8     

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    In what quadrant does the point (-5, -8) lie?     

    Since x = –5 it's left of the origin    

    Since y = –8 it's below the origin     

    Left and Down puts the point in the 3^rd Quadrant     

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The x-intercepts of the parabola y = x^2 + bx + c are (-3,0) and (-5,0) Find the equation of the parabola, and submit your answer in y = x^2 + bx + c form.     

     y = x² + bx + c     We know that when x = –5, y = 0     

                                                 and when x = –3, y = 0      so substitute those in the equation     

     (–5)²  +  (–5) b  +  c  =  0     –>   25 – 5b + c  =  0     

     (–3)²  +  (–3) b  +  c  =  0     –>     9 – 3b + c  =  0     

     subtract                                            – 2b        =  – 16     

                                                                         b  =  8         so substitute that in the equation     

                                                          25 – 40 + c  =  0     

                                                                          c  =  15     

     therefore we get      y  =  x² + 8x + 15     I wasn't a 100% sure of this answer so I drew it     

                                                                      on Desmos and the resulting curve looked good.     

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[T]he points of this 3-by-3 grid are equally spaced horizontally and vertically. How many different sets of three points of this grid can be the three vertices of a triangle?     

     9C3  =        84     

     minus             8     

     answer         72     

I based this answer on the reasoning that a     

triangle can be drawn using any three points     

in the grid, except when those three points lie     

in a straight line.     

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   How many positive integers are there whose digits strictly decrease from left to right, and the sum of the digits is 6?     

   60     51       42       321     

            510     420     3210     

   I thought of these seven . . . maybe you can come up with some more.     

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Adam has 16.45 kg of flour, and he uses 6.4kg to make hot cross buns .The remaining flour is exactly enough to make 15 batches of scones.How much flour,in kg will be in a batch of scones?     

          16.45     original amount of flour     

            6.40     used to make hot cross buns     

          10.05     remaining flour     

          10.05                         enough to make 15 batches of scones     

          10.05 / 15  =  0.67     what it takes to make one batch of scones     

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A bag contains 10 counters. 6 of them are white. a counter is taken at random and not replaced. A second counter is taken out of the bag at random. Calculate the probability that both two counters are white.     

Probability first counter is white                6 / 10     

Probability second counter is white           5 / 9     

Probability both conditions occurring       (6 / 10) • (5 / 9)  =  30 / 90     

                                                                               reduces to 1 / 3    or  33 ¹/₃ %       

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Alex chooses a number at random from the set {1, 2, 3, 4, 5}.  Winnie also chooses a number at random from the same set.  (They can choose the same number.)  What is the probability that the product of their numbers is at least 4?     

The following are the 25 pairs that could occur.     

1 1    1 2    1 3    1 4    1 5     

2 1    2 2    2 3    2 4    2 5     

3 1    3 2    3 3    3 4    3 5     

4 1    4 2    4 3    4 4    4 5     

5 1    5 2    5 3    5 4    5 5     

The highlighted pairs are the 20 that multiply to at least four.     

The probability of choosing a highlighted pair is 20 / 25 which is 80%.     

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The correct answer is 250.

The 450 juniors at East HS drink a total of 1500 cartons of milk per 5-day week. If the average senior drinks milk at the same rate as the average junior, what is the average number of total cartons of milk the 600 seniors drink each school day?     

1500 cartons for 5 days works out to 300 cartons for 1 day     

spread among 450 juniors, so each junior drinks ²/₃ of a carton per day.     

The seniors drink milk at the same rate; however, there are 600 of them,     

so the senior class drink ²/₃ x 600  =  400 cartons for 1 day     

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