Bosco

Tim is 4 years older than one-third of Juan's age.     

In 10 years, Juan will be 6 years older than twice as old as Tim is now.     

How old is each now?      

  T  =  4 + (1/3) J           —>>     3T  =  12 + J     (eq 1)     

  J + 10  =  6 + (2) T      —>>     2T  =    4 + J     (eq 2)     

  subtract (eq 2) from (eq 1)         T  =  8     

  substitute 8 for T in (eq 1)        24  =  12 + J     

                                                     J  =  12     

  check answers     

  8  =?  4 + 4         12 + 10  =?  6 + 16     

  8  =  8   true                22  =  22   true     

_.     

A quiz has four multiple-choice questions; each question is either true or false. How many different solution keys are possible?     

Each question has two possible answers, True or False     

Q1 has 2 possible answers – so, if there's only one question, there are 2 possible solution keys     

Q2 has 2 possible answers – so, with the previous question, there are 4 possible solution keys     

Q3 has 2 possible answers – with the previous 2 questions, that makes 8 possible solution keys     

Q4 has 2 possible answers – with the other 3 questions, there can be a possible 16 solution keys     

_.     

  Select the three expressions below that result in a negative solution. Group of answer choices     

  A)  (-7) + (-9)       B)   -3 - 6       C)   9 + (-8)       D)  (-2) - (-8)       F)  9 + (-12)      

  A)   =  – 16     

  B)   =  – 9     

  C)   =  + 1     

  D)   =  + 6     

  F)   =  – 3     

What happened to E) ?     

_.     

    A rectangular piece of paper with length 35cm and width 16cm has two semicircles cut out of it, as shown below.     

    Find the area of the paper that remains. Use the value 3.14 for π.     

    A  =  (35 • 16) – (2)(¹/₂ • 3.14)(8²)     

    A  =  560 – 200.96     

    A  =  359.04 cm²     

_.     

Ben twice chooses a random integer between 1 and 50 inclusive, twice (and he may choose the same integer both times). What is the probability that at least one of the numbers Ben chooses is a multiple of 8?     

There are 6 multiples of 8 between 1 and 50:  namely 8, 16, 24, 32, 40, and 48     

There are 50 numbers between 1 and 50.     

The probability of choosing a multiple of eight with the first number is 6 / 50.     

The probability of choosing a multiple of eight the second time is also 6 / 50.     

So the total probability is 12 / 50, which, denoted as a percentage, is 24% likely.     

This is only a probability; extrapolating it up to 100% does not create a certainty.     

_.     

Find the last two digits of the following sum: 5 + 10 + 15 + ... + 100.     

S  =  ( 20 / 2 )( 5 + 100 )     

S  =  10 • 105     

S  =  1050                           so the last two digits are 50     

The formula for the sum of a finite arithmetic series is S = (n/2)(a₁ + a_n),     

where 'n' is the number of terms, 'a₁' is the first term, and 'a_n' is the last term.     

Given that 2x = 9 and 27y = 2 find the value of xy.     

 2x = 9       ——>>     x  =  9 / 2     

 27y = 2     ——>>     y  =  2 / 27     

                                            9         2     

                                 xy  =  –––  •  –––     

                                            2         27     

                                            18                             1     

                                 xy  =  ––––     reduces to   –––     

                                             54                             3     

_.     

Two adjacent sides of a rhombus form a 45-degree angle. If each side of the rhombus measures 4 cm, what is the area of the rhombus, in square centimeters? Express your answer in simplest radical form.     

Area_rhombus  =  base • height     

                 A  =  4 cm • 2.8284 cm     

                 A  =  11.3136 cm²     —–>>  Want it in radical form?  Well 0.7071 is half the square root of 2.     

                                                                h  =  4 • ^sqrt(2)/₂     

                                                                h  =  2 • sqrt(2)     

                                                                A  =  4 cm • (2 • sqrt(2)) cm     

                                                                A  =  8 • sqrt(2) cm²     

How do I know the height?  Drop a perpendicular from the top obtuse angle to the side.     

The resulting figure is a right triangle, in which you know one side and all of the angles.     

Calculate the perpendicular, and that's the height.  Draw it and you'll see the following:     

                sin(45^o)  =  h / 4     

                           h  =  4 • sin(45^o)     

                           h  =  4 • 0.7071     

                           h  =  2.8284     

_.     

   The quadratic equation 2x^2 + bx + 18 = x^2 + 14 has a double root. Find all possible values of b.     

                                          2x² + bx + 18 = x² + 14     

                                            x² + bx + 4   =  0     

   A double root of a quadratic equation means that both roots are the same.     

   The alternative is that the two roots of the quadratic equation are different.     

   For the equation to have a double root, b has to be twice the square root of 4.     

                                            b  =  2 • sqrt(4)     

                                            b  =  2 • + 2     

                                      so, b could be  + 4     

                                      or,  b could be  – 4     

   If  1/4 of 2^(30) is 4x, then what is the value of x?     

      4x  =  2³⁰ / 4     

        x  =  2³⁰ / 16     This will reduce.  Is that what you want?     

                                  2³⁰ / 16  =  2³⁰ / 2⁴     

                                            x  =  2²⁶     

.     

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)     

_.     

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     

_.     

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.     

_.     

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     

_.