Bosco

Given that M(1,-6) is the midpoint of AB and A(-2,-1) is one endpoint, what is the sum of the coordinates of point B?     

Visualize or, better yet, sketch the following on an x-y coordinate grid.     

From endpoint A (–2,–1) to midpoint M (+1,–6) you went right 3 and down 5.     

To get from the midpoint to endpoint B, move the same directions and amounts.     

For x, from +1 go right 3 to +4.  For y, from –6 go down 5 to –11.  B is at (+4,–11).     

Adding the coordinates of B   ——>>   (+4) + (–11)  =  –7 is their sum.     

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I realized that I had overlooked something obvious.     

Reversing the order or the digits makes three more.     

     0  3  6     

     1  3  5     

     2  3  4     

     3  3  3     

     4  3  2     

     5  3  1     

     6  3  0     

That makes the new total, seven of them.     

_.     

Find the number of arithmetic sequences such that:     
* The arithmetic sequence contains three terms     
* All the terms are integers in \{0, 1, 2, \dots, 10\}     
* The sum of the terms is 9.     

     0  3  6

     1  3  5     

     2  3  4     

     3  3  3    Is r = 0 legit?     

     That's all I can think of, four of them.     

_.     

Jim has between 1 and 3 tens. Cradle has between 4 and 6 ones. What number can they make?     

a. 13     

b. 25     

c. 46     

0 ten and no/some ones will total either   0,   4,  5,  or   6     

1 ten and no/some ones will total either 10, 14, 15, or 16     

2 ten and no/some ones will total either 20, 24, 25, or 26     

3 ten and no/some ones will total either 30, 34, 35, or 36     

Looks like B is the only answer, that is, 25.     

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In isosceles triangle RST below, what is the value of y?     

The sum of all three angles of a triangle is always 180^o     

After you subtract 38^o for T, the two Y's have 142^o total     

The value of a single Y is half of 142^o, thus each Y = 71^o     

_.     

Ben had 1/5 as many books as Jann. After Jann gave Ben 4 books, Ben had 1/3 as many books as Jann. How many books did Jann have in the end?     

B = (¹/₅) • J                  —>>  J = 5B                          (eq 1)     

(B+4) = (¹/₃) • (J–4)     —>>  3 • (B+4) = J–4   

                                               3B + 12  = J – 4     

                                                          J  = 3B + 16     (eq 2)     

set (equation 1) equal to (equation 2) in terms of B     

                               5B  =  3B + 16     

                               2B  =  16     

                                 B  =  8     

referring back to (equation 1)     J  =  5B     

                                                   J  =  5 • 8     

                                                   J  = 40               this is how many books Jann started with     

Jann gives away 4 books            40 – 4  =  36     this is how many books Jann ended up with     

_.     

The ratio of the number of magnets Aspilla and Barry had was 2 : 3. When Aspilla bought 4 more magnets and Barry bought 13 more magnets, the ratio became 1 : 2. Find the number of magnets Barry had in the end.     

A / B = 2 / 3                    crossmultiply ——>>    3 • A = 2 • B                   ——>>  3A = 2B                 (eq 1)     

(A+4) / (B+13) = 1 / 2     crossmultiply ——>>    2 • (A+4) = 1 • (B+13)    ——>>  2A + 8 = B + 13     (eq 2)     

multiply (eq1) by 2          6A = 4B                     (eq 3)     

multiply (eq 2) by 3         6A + 24 = 3B + 39     (eq 4)     

per (eq 3) 6A = 4B so substitute 4B into (eq 4)     

                                          4B + 24 = 3B + 39     

                                          4B – 3B = 39 – 24    

                                                    B = 15                      btw, this means that A = 10     

                                                                                     but the problem didn't ask for A     

add the 13 that B bought     B + 13 = 28     

_.     

There are 70 cookies in a box. The number of raisin cookies was 1/4 the number of plain cookies. Josephine ate 3 raisin cookies and 3 plain cookies. Find the difference between the number of raisin and plain cookies in the box in the end.     

Consider the ratio 1/4.     

That is, for every 5 cookies, 1 is a raisin and 4 are plain.     

So divide 5 into 70 to get 14 ... let's call them "portions".     

The raisin cookies get 1 portion, so there are 14 of them.     

The plain cookies get 4 portions, so there are 56 of them.     

The difference in the number of cookies is (56 – 14) = 42      

So, as long as Josephine eats the same number of raisin cookies     

as she eats plain cookies, the difference will always be 42 cookies.     

_.     

Your weekly paycheck is 10 percent less than your coworker's. Your two paychecks total 780. Find the amount of each paycheck.     
Your coworker's:                and your's:     

Your two paychecks total 780, so Yours = 390 ... i.e., half of 780.     

Then, 390 = 0.90 • CoWorker's     ——>>     CoWorker's = 390 / 0.90     

                                                                       CoWorker's = 433.33     

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