### Eva shares 5 cookie between 3 people!

Eva shared 1 cookie with each person, with 2 cookies left over. Then she cut one of the leftover cookies into 4 pieces, each piece 1/4. She shared 1/4 with each person, leaving 1 cookie and 1/4, and each person has 1 + 1/4 cookies so far.
Eva then cut the whole cookie and the 1/4 into eighths. She had 8/8 + 2/8 = 10/8. She shared 3/8 of a cookie with each person, with 1/8 left over. Each person then had 1 + 1/4 + 3/8 cookies. Eva then continued to cut each leftover piece into 4 pieces, sharing 1 with each person, with 1 left over!

Her method resulted in an infinite series!

Eva went on to do the same problem a different way, and each person got 1 + 2/3 cookies.
So 1 2/3 = 1 + 1/4 + 3/8 + 1/32 + 1/128 + ...
This got us into a discussion of fractions and decimal equivalents and adding the infinite series to see what it gets close to. All very interesting!

Betty, a fifth grade teacher (her son is working on the SAT test with Don also), got 5/3 for Eva's problem. I asked Betty to read Eva's solution to the problem and she realized that 1 2/3 = 1.666... an infinite repeating decimal, which is also an infinite series 1 + 6/10 + 6/100 + 6/1000 + ... Betty raised the question: If you can add 6/10 + 6/100 + 6/1000 + ... could you also multiply 6/10 x 6/100 x 6/1000 x ... ? A wonderful question! Betty proceeded to use a calculator to multiply 6/10 x 6/100 x 6/1000 and ended up with the number 2.16E-4. We had a big discussion about what this meant. 2.16E-4 means 2.16*10-4, or .000216 . Let's see what happens now!

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