Share and Share Alike
When Karen inherited an amount of money, she divided it amongst her four children in the following sequence. To Alice, she game $2 plus 1/3 of the remainder; to Brent, she game $2 plus 1/3 of the new remainder; to Curtis, she game $2 plus 1/3 of the new remainder; and finally to Debra, she game $2 plus 1/3 of the new remainder.
Karen then divided what was left equally amongst the four children. At the end, she discovered that the two girls together had received $35 more than the boys together.
Questions: What was Karen's inheritance? How much money did each child receive?
If you are up to a harder task:
When Karen inherited an amount of money, she divided it amongst her four children in the following sequence. To Alice, she game $n plus 1/m of the remainder; to Brent, she game $n plus 1/m of the new remainder; to Curtis, she game $n plus 1/m of the new remainder; and finally to Debra, she game $n plus 1/m of the new remainder. Karen then divided what was left equally amongst the four children. At the end, she discovered that the two girls together had received $p more than the boys together.
Can you determine a general expression for Karen's inheritance and how much money each child received...and what if there were q childrfen, half of each sex?
Source: W. Fleming's College Algebra, 1988, p. 95
Hint: Life will get quite complicated if you set x to be Karen's original amount and then you try to set up algebraic equations for each step...try to think what other problem solving techniques might prove helpful.
Solution Commentary: You will know when you have the correct solution...so let's focus on technique. Did you discover that this was a good problem for working backwards...or even guess and check...or even playing with on a spreadsheet? Some great patterns are lurking within.... For example, if you assume that Karen had $0 after giving Debra her share, you should be able to quickly figure out the original amount....plus certain numbers keep occurring.
