Ivan and Adeline are in a classroom with a chalkboard. They are standing on different halves of the board, and on each half, the number $2$ is written. When Ivan's teacher gives a signal, Ivan multiplies the number on his side of the board by $-2$ and writes the answer on the board, erasing the number he started with. Adeline does the same on each signal, except that she multiplies by $2$. The teacher gives 10 signals in total. How many times (including the initial number) do Ivan and Adeline have the same number written on the board (including at the beginning)?

The number of times Ivan and Adeline have the same number written on the board is 6.

Step-by-step explanation:

Consider the procedure as follows:

Consider the numbers on each half of the board:

          Ivan                            Adeline

             2                                     2

      2 × -2 = -4                        2 × 2 = 4

     -4 × -2 = 8                         4 × 2 = 8

      8 × -2 = -16                      8 × 2 = 16

   -16 × -2 = 32                      16 × 2 = 32

   32 × -2 = -64                    32 × 2 = 64

  -64 × -2 = 128                   64 × 2 = 128

  128 × -2 = -256               128 × 2 = 256

-256 × -2 = 512                256 × 2 = 512

  512 × -2 = -1024              512 × 2 = 1024

-1024 × -2 = 2048           1024 × 2 = 2048

Thus, the number of times Ivan and Adeline have the same number written on the board is 6.

step-by-step explanation:

a.   since the profit made in 2012 is equal to the profit made in 2014, the profit is           symmetric about the line y = 120,000.

step-by-step explanation:

given that megan owns a clothing boutique.

the profits of 2010, 11, are recorded and shown as follows:

years since     2010 0     1     2     3       4       5

profits made in 1000's   39       84     111     120   111     84

we find that profit gradually increases form 2010 to 2013 upto 120,   then starts decreasing.

the profits for 2012 and 2014 are the same while that of 2011 and 2015 are also the same.

this shows that there is symmetry about the line profit = 120000

hence answer is:

a.   since the profit made in 2012 is equal to the profit made in 2014, the profit is           symmetric about the line y = 120,000.

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yes

step-by-step explanation: