Bosco

A bag contains yellow, green, blue, and white marbles.  The ratio of yellow, green, blue, and white marbles is 1:2:3:4.  If the bag contains 180 marbles, then how many blue marbles are there?  

Consider:  "The ratio of yellow, green, blue, and white marbles is 1:2:3:4."  

Call the number of yellow marbles X.  

So the number of green marbles is 2X. 

And the number of blue marbles is 3X.  

And the number of white marbles is 4X.  

Add up all the marbles and you have     10X  =  180  

Therefore                                                   X  =  18 yellow marbles  

                                                                 2X  =  36 green marbles  

                                                                 3X  =  54 blue marbles  

                                                                 4X  =  72 white marbles  

_.

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 there still remains 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.  

_.

I'm gonna round some numbers drastically, so  

the products are gonna be appriximations, but  

this is just to give you an idea of my approach.  

use tan(30^o) to find BC = 6.9  

area ABC = (1/2)(12)(6.9) = 41.6  

use tan(30^o) to find CX = 4  

area BXC = (1/2)(6.9)(4) = 13.7  

area BXA = area ABC – area BXC  

area BXA = 27.9  

_.

I drove to the beach at a rate of 40 miles per hour.  If I had driven at a rate of 50 miles per hour instead, then I would have arrived 45 minutes later.  How many miles did I drive?  

You mean 45 minutes earlier.  Obviously, if you drive faster, you get there faster.  

This problem makes use of

the following relationship:                   Distance = Velocity x Time 

                                                           D  =  V • T  

case 1                                                 D  =  (40) • (T)  

case 2                                                 D  =  (50) • (T – 45)  

Since the Distance, D, is the  

same for both cases, let's set       

the "V•T"s equal to each other.             (50)(T – 45)  =  (40)(T)  

                                                               50T – 2250  =  40T  

Subtract 40T from both sides                  10T – 2250  =  0  

Add 2250 to both sides                                       10T  =  2250  

Divide both sides by 10                                           T  =  225   (this is time in minutes)  

Divide minutes by 60 to get hours          225 minutes  =  3.75 hours  

Plug this T back into original equation                     D  =  (40 mi/hr) • (3.75 hr)  =  150 miles  

_.

One ordered pair (a,b) satisfies the two equations ab^4 = 48 and ab = 72. What is the value of b in this ordered pair?  

To find b, consider                                                  ab⁴  =  48  

We will divide both sides by ab.  

Since ab=72, we will divide the left side  

by "ab" and the right side by its equal 72.  

                                                                              ab⁴         48  

                                                                             ——   =   ——  

                                                                              ab           72  

Note that ab⁴ = (ab) * (b³)  

Cancel ab out of the left side.  

Reduce 48/72 on the right side.  

                                                                               b³           2  

                                                                             ——   =   ——  

                                                                                1            3  

                                                                                 b   =   cube root of (2 / 3)  

_.

Five times a number is divided by 7 more than the number. If the result is 0, then what was the original number?  

                                                          5n  

                                                        –——  =  0     

                                                         n + 7  

                                                          5n  =  0  

                                                            n  =  0    

                   The only number that works is zero.  

_.

A wooden block is 4 inches long, 4 inches wide, and 1 inch high. The block is painted red on all six sides and then cut into sixteen 1 inch cubes. How many of the cubes each have a total number of red faces that is an even number?  

Draw a 4 x 4 grid on a sheet of paper, as a guide.  

You can mentally visualize the third dimension.  

Here's what I count:  

The 4 corner blocks                      4 red sides each  

The 8 outside edge blocks

excluding the corner blocks          3 red sides each  

The 4 inside blocks                       2 red sides each   

So, it looks like the cubes that have an even number  

of red sides are the 4 on the corners and the 4 inside.   

That comes to a total of 8 cubes that have an even number of red sides.  

No cube can ever have all 6 sides red, because the side(s) where a    

cube was butted up against another cube wouldn't have got painted.  

_.

Find the coefficient of y^4 in the expansion of (2y-5+3y^2)^6.

Use this website to expand the expression:     

https://www.symbolab.com/solver/expand-calculator

You'll find that it's 375 but go ahead and see for yourself.  

_.

Real numbers a and b satisfy  

a + ab = 250  

a - ab = -240  

Enter all possible values of a, separated by commas.  

You probably won't find a simpler problem than this in your whole math textbook.  

                                                       a + ab  =    250  

                                                       a – ab  =  –240  

Just add them together  

                                                      2a         =     10  

Divide both sides by 2  

                                                                a  =  5   

That's it.  

That's all the possible values.  

_.

A bag contains yellow, green, blue, and white marbles.  The ratio of yellow, green, blue, and white marbles is 1:2:3:4.  If the bag contains 180 marbles, then how many blue marbles are there?  

Consider:  "The ratio of yellow, green, blue, and white marbles is 1:2:3:4."  

Call the number of yellow marbles X.  

So the number of green marbles is 2X. 

And the number of blue marbles is 3X.  

And the number of white marbles is 4X.  

Add up all the marbles and you have     10X  =  180  

Therefore                                                   X  =  18 yellow marbles  

                                                                 2X  =  36 green marbles  

                                                                 3X  =  54 blue marbles  

                                                                 4X  =  72 white marbles  

_.