Bosco

If I give my brother 5 dollars, then we will have the same amount of money. If instead he gives me 24 dollars, then I'll have twice as much money as he will have. How much money does my brother currently have (in dollars)?     

Call B how much my brother has     

Call M how much I have     

(M – 5)  =  (B + 5)           ——>>  M  =  B + 10     (equation 1)     

(M + 24)  =  2 • (B – 24)                                        (equation 2)     

Substitute (B+10) in place     

of M in (equation 2)                     (B+10) + 24  =  2 • (B – 24)     

                                                             B + 34  =  2B – 48     

                                                                    82  =  B     

                                                                      B  =  82     

check answer     

from (eq 1)   M = 82 + 10          from (eq 2)   92+24  =?  2 • (82–24)     

                     M = 92                                           116  =?  116     true     

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Find the number of positive integers that satisfy both the following conditions:     

Each digit is a 1 or a 2     

The sum of the digits is 3     

     1 1 1     

     1 2     

     2 1          I found 3 integers that satisfy the conditions.     

_.     

Let a and b be real numbers such that a - b = 1 and a^3 - b^3 = 1.     

                  a   –  b    =  1     

                  a³  –  b³  =  1     

            c.   When a = 0, then b = –1      

                  When b = 0, then a = +1      

            a.   ab is either [0 • –1] or [0 • +1]  ...  0 either way     

            b.   a+b is either [0+(–1)] or [0+(+1)]   ...  –1 or +1      

_.     

factors of 1,000,000,000,000     

The prime factors of a trillion are 2 and 5.     

_.     

In a row of five squares, each square is to be colored either red, yellow, or blue, so that no two consecutive squares have the same color, and at lest three of the squares are red.  How many ways are there to color the five squares?     

In order not to have two reds next to each other,     

the three reds must occupy squares 1, 3, and 5.     

That way, there's another color in between them.     

That leaves squares 2 and 4 to fill, and two colors     

to do it with. There are only four ways.    

Blue Blue     Yellow Yellow     Blue Yellow     Yellow Blue      

So, the answer is four ways.     

_.     

How many ways can you arrange 2 boys and 3 girls in a line, if the 2 boys must stand next to each other?     

Strap the boys together and consider the pair as a single unit.  With the 3 girs, there are 4 units.     

The possible arrangements of 4 units is ₄P₄  =  24 ... but wait, we're not through yet.     

Now, have the two boys swap places and there are another 24 arrangements.   

So, altogether, there are 48 possible arrangements.     

_.     

      what is 9% of 25000       

      0.09 x 25,000  =  2250     

_.     

A spherical tank has a radius of 6 yards. It is filled with a liquid that costs $7.15 per cubic yard. What is the total value of the liquid in the tank? Use 3.14 for π . Enter your answer in the box to the nearest cent.     

     V_sphere  =  (⁴/₃) π r³     

     V  =  (⁴/₃) • 3.14 • (6 yd)³     

     V  =  (⁴/₃) • 3.14 • 216 yd³     

     V  =  904.32 yd³     

     Cost  =  (904.32 cuyd) •  (7.15 ^dollars/_cuyd)     

     Cost  =  6465.89 dollars     

_.     

A cylinder has a height of 13 feet and a radius of 8 feet. What is its volume? Use ​𝜋 ≈ 3.14 and round your answer to the nearest hundredth.     

          V_cyl  =  π r² h    where r is the radius of the circular base and h is the height     

          V  =  (3.14)(8²)(13)     

          V  =  3.14 • 64 • 13     

          V  =  2612.48 ft³     

_.     

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

How to get   3  .......     1, 3    3, 1                           2  ways     

How to get 12  .......     2, 6    6, 2    3, 4    4, 3       4  ways     

                                                                 total        6  ways     

Number of total possible combinations  6 x 6  =   36  ways     

                              So the probability is  6 / 36  =  1 / 6     

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