Bosco

 Nora and Renata had an equal number of polished stones at first.     

 After Nora received 65 more stones and Renata lost 23 stones, Nora had three times as many stones as Renata.     

 How many stones did they have altogether at first?      

 N = Nora's original number of stones     

 R = Renata's original number of stones     

 N = R     

 (N + 65)  =  3 • (R – 23)     

 N = R so substitute it          (N + 65)  =  3 • (N – 23)     

                                               N + 65  =  3N – 69     

                                                   –2N  =  –134     

                                                       N  =  67    

 since N = R                                    R  =  67     

 check answer     

 does (67 + 65)  =  3 • (67 – 23) ?     

                  132  =  201 – 69     

                  132  =  132   True      

_.     

What is the solution set for (2x - 7)^2 = 25?     

                                            (2x – 7)²  =  25     

sqrt both sides                    +(2x – 7)  =  +5     

four possibilities                  +(2x – 7)  =  +5     —>     x  =  +6     

                                            +(2x – 7)  =  –5     —>     x  =  +1     

                                            –(2x – 7)  =  +5     —>     x  =  –6     

                                            –(2x – 7)  =  –5     —>     x  =  –1     

_.     

     A list of 10 positive integers, each less than 15, has an average of 6.     

     What is the maximum number of ones (the number 1, not the digit) that could be in the list?      

     The 10 integers average 6, so the total of the 10 integers is 60.     

     If you have 9 1's, can you make up the other 51 with 1 number?     No     

     If you have 8 1's, can you make up the other 52 with 2 numbers?   No     

     If you have 7 1's, can you make up the other 53 with 3 numbers?   No     

     If you have 6 1's, can you make up the other 54 with 4 numbers?   Yes - three 14's & one 12     

_.     

Two real numbers are chosen at random between 0 and 2. What is the probability that their sum is no more than 1?    

Assuming the numbers are integers, the following are all the possible choices.     

       0 0    0 1    0 2     

       1 0    1 1    1 2     

       2 0    2 1    2 2     

The highlighted numbers are the pairs whose sum is no more than 1.     

Sum no more than 1       3     

All possible choices        9          ——>>    probability is 3 / 9  or  1 / 3     

                                                                   or, as a percentage,  33.33%     

_.     

Jamie's office recycled a total of 10 kilograms of paper over 2 weeks. After 3 weeks, how many kilograms of paper will Jamie's office have recycled? Solve using unit rates.     

      10 kg     

     ––––––  =  5 kg/wk          Therefore   —>>   (5 kg/wk) • (3 wk)  =  15 kg     

        2 wk     

_.     

Buns are sold in a packs of 8 and hot dogs are sold in packs of 12. What is the least number of hot dogs you can make if you don’t want any leftovers? How many packs of each should you buy? If you are holding a party and you expect between 80 and 100 people, how many should you buy?     

The worst case (or best, depending on your viewpoint) is that you have to feed 100 people.     

You can buy 12 pks of buns and 8 pks of dogs, and feed 96 people, if there are 4 no-shows.     

But, to ensure you can feed all 100 people, you have to get 15 pks buns and 10 pks dogs to     

make 120 sandwiches.  Then 80 people get one sandwich and 20 people get two sandwiches.     

_.     

The lines -2x + y = k and 0.5x + y = 14 intersect when x = -8. What is the value of k?     

y  =  k + 2x          —> when x = –8  then                           —>   y = k – 16     

y  =  14 – 0.5x     —> when x = –8  then  y = 14 – (–4)     —>   y = 18     

                                                                                                   k – 16  =  18     

                                                                                                           k  =  18 + 16     

                                                                                                           k  =  34     

_.     

Each row of a seating arrangement seats 7 or 8 people. $47$ people are to be seated. How many rows seat exactly 8 people if every seat is occupied?     

In the 8-person rows, you can put 40 people into 5 rows     

and then, the remaining 7 people sit in one 7-person row.     

_.     

 The angles of a quadrilateral are xº, 2xº, 4xº, 8xº.  Find x     

The sum of the internal angles of a quadrilateral is 360^o.     

Therefore, (x + 2x + 4x + 8x)  =  360     

                                        15x  =  360     

                                            x  =  24     

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