Bosco

At a cafeteria, Mary orders two pieces of toast and a bagel, which comes out to $3.15. Gary orders a bagel and a muffin, which comes out to $3.50. Larry orders a piece of toast, two bagels, and three muffins, which comes out to $8.15. How many cents does one bagel cost?     

                        T_oast + B_agel  = 315 ¢     ——>>     T = 315 – B     

                        B_agel + M_uffin = 350 ¢     ——>>     M = 350 – B     

                         T + 2B + 3M  =  815 ¢     

                         (315 – B)  +  2B  +  [3 • (350 – B)]  =  815     

                         315 – B + 2B + 1050 – 3B  =  815     

                          – 2B + 1365  =  815     

                          – 2B  =  – 550     

                               B  =  275 ¢     

_.     

In a store window, there was a box of berries having a total weight of 200 kg.  The berries were 99% water, by weight.  After two days in the sun, the water content of the berries was only 90%, by weight.  What was the total weight of the berries after two days, in kg?     

You're starting with 200 kg of berries, and 99% of that is water.     

Since they're 99% water, that means that there's 1% of solid stuff.     

1% of 200 kg means that there are 2 kg of solid stuff.     

Some of the water evaporates in the sun, but the solid stuff doesn't.     

So, after the evaporation, there still remains the same 2 kg of solid stuff.     

Since the new water content is 90%, that means that now the solid stuff is 10%.     

So 2 kg (the solid stuff) is 10% of the total weight.     

That means the new total weight is 20 kg.     

_.     

if 30 cents is 1 how much is 3$     

30 is to 1   as   300 is to 10     

_.     

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     

we demonstrated 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     

_.     

Find the volume of the square pyramid, given its slant height​     

The volume of a square pyramid is found using the formula: V = (1/3) * base area * height, or V = (1/3) * s² * h, where 's' is the side length of the square base and 'h' is the height of the pyramid.     

Not too hard, all we need is the height.  Drop a perpendicular from the apex to the center of the base.  That's the height.  From that center point, draw a line across the square base over to the bottom of the slant height.  That's one leg of a right triangle and the slant height is its hypotenuse.  From those two sides, we can find the third side which is the height.  Thank you, Pythagoras.     

Remember that in a right triangle c² = a² + b² which means that b² = c² – a²     

(height)²  =  30² – 16²  =  900 – 256  =  644   

height  =  sqrt(644)  =  25.377     

V  = (¹/₃) • (32)² • 25.377  =  8662.02 cm³     

_.     

There are 10 students in a class: 6 boys and 4 girls.  If the teacher picks a group of 3 at random, what is the probability that everyone in the group is a girl ?     

Probability first pick is a girl                    4 / 10     

Probability second pick is a girl              3 / 9     

Probability third pick is a girl                   2 / 8     

Probability all three happening                4/10 • 3/9 • 2/8  =  24 / 720  =  1 / 30     

_.     

a marrow plant is 0.5 m long.Every week it grows by 20%     

a)Find the value of the common ratio.     

b)calculate how long the plant is after 12 weeks.     

C)comment on the predicted lenght of the marrow after one year.     

a)  Find the value of the common ratio.     

     The common ratio is 1.20     

b)  Calculate how long the plant is after 12 weeks.     

     (5 m) • (1.20)¹²  =  44.58 m      

c)  Comment on the predicted length of the marrow after one year.     

     After 52 weeks it would be (5 m) • (1.20)⁵²  =  65,523.15 m     

     At 3.2808 ft/m, that comes to more than 214,968 feet long.     

One face of a cube-shaped package has an area of 7 5/6 square inches.  What is closest to the total surface area of the package?      

A gardener has a cylindrical planter that has a diameter of 1 1/3 feet and is 2 feet deep.  What is the approximate number of cubic feet of potting soil that he will need to fill the planter?     

A cube has six faces.  If the area of one face is 7 ⁵/₆ square inches,     

we multiply that times six to find the area of all six of the cube's faces.     

Note that 7 ⁵/₆ is the same as ⁴⁷/₆.  So, ⁴⁷/₆ • 6  =  47 ... that's sq in.     

So the answer is C. 48 which is the wrong amount, but it's the closest.     

The volume of a cylinder is the area of the base times the height.     

Diameter is 1 ¹/₃ (same as ⁴/₃) feet so the radius is half that, i.e., ²/₃.    

V  =  π r² h  =  3.14 • (²/₃)² • 2  =  (3.14) • (⁴/₉) • 2  =  2.79 that's sq ft.     

So the answer is A. 2.79 which at least they got it right this time.     

_.     

The sum of Patrick's age and Anthony's age is 49. Patrick's age is one more than twice as old as Anthony. What are their ages?     

          P + A  =  49          —>     A  =  49 – P     (equation 1)     

          P  =  2A + 1                                             (equation 2)     

          Substitute the "49–P" equivalent for A in (eq 2)     

          P  =  2 • (49 – P) + 1     

          P  =  98 – 2P + 1     

          3P  =  99     —>     P  =  33     

          Substitute 33 for P in either of the original equations     

          33 + A  =  49     

                  A  =  49 – 33     —>     A  =  16     

_.     

1. Find the area of triangle ABC if AB = 6, BC = 8, and angle ABC = 90 degrees.     
2. If angle ABC = 135 degrees, and angle ACB = 30 degrees, what is angle BAC?     

1.        A_triangle  =  one half the base times the height     

                     A  = (1/2) • 6 • 8  =  24     

2.        BAC  =  180 – (135 + 30)     

           BAC  =  15     

_.     

Alice and Bob each have a certain amount of money. If Alice receives n dollars from Bob, then she will have 4 times as much money as Bob. If, on the other hand, she gives n dollars to Bob, then she will have 8 times as much money as Bob. If neither gives the other any money, what is the ratio of the amount of money Alice has to the amount Bob has?     

     A = Alice's original amount of money     

     B = Bob's original amount of money     

     (A + n)  =  (4) • (B – n)     

     (A – n)  =  (8) • (B + n)     Just looking at these two equations, I have misgivings.     

                                              How can Alice with less have more times Bob with more?     

     A + n  =  4B – 4n     ——>  A  =  4B – 5n     (eq 1)     

     A – n  =  8B + 8n     ——>  A  =  8B + 9n     (eq 2)     got to get rid of those n's somehow     

multiply (eq 1) by 9              9A  =  36B – 45n     

multiply (eq 2) by 5              5A  =  40B + 45n     

add the two                        14A  =  76B     

divide both sides by 14          A  =  (76/14) • B     

divide both sides by B           A / B  =  76 / 14          76 / 14 will reduce to 38 / 7     

ratio                                       Alice         38     

                                             –––––  =   –––     

                                               Bob           7     

check answer:  

38 + n  =  28 – 4n     ——>     5n  =  – 10     ——>     n  =  – 2     

38 – n  =  56 + 8n     ——>  – 9n  =  + 18     ——>     n  =  – 2     

It works out, to my surprise.  It requires accepting the concept of negative money.  I'm okay with that.     

_.     

At a cafeteria, Mary orders two pieces of toast and a bagel, which comes out to $3.15. Gary orders a bagel and a muffin, which comes out to $3.50. Larry orders a piece of toast, two bagels, and three muffins, which comes out to $8.15. How many cents does one bagel cost?     

                                  T + B = 315 ¢     ——>      T = 315 – B     

                                  B + M = 350 ¢     ——>     M = 350 – B     

                                   T + 2B + 3M  =  815 ¢     

                                   (315 – B)  +  2B  +  [3 • (350 – B)]  =  815     

                                   315 – B + 2B + 1050 – 3B  =  815     

                                    – 2B + 1365  =  815     

                                    – 2B  =  – 550     

                                         B  =  275 ¢     

_.