Question 1201507
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Answer: <font color=red size=4>120</font>



Explanation:



I'll break the instructions into three parts<ol><li>The ratio of Karen's beads to Sarah's beads is 7:5</li><li>When Sarah gives 48 beads to Karen, she has 1/3 of the number of beads Karen has. </li><li>During a game, Karen lost some beads to Sarah. As a result, Karen had 1/5 as many beads as Sarah. </li></ol>Part 1
Ratio of Karen's beads to Sarah's beads is 7:5
Scale each part by x to get 7x:5x
x is some positive whole number
7x = Karen's starting count
5x = Sarah's starting count


Part 2
Sarah gives 48 beads to Karen
Sarah's count of 5x drops to 5x-48
Karen's count of 7x increases to 7x+48
After these adjustments, we know that Sarah has 1/3 that Karen does.
In other words, Karen has triple that of Sarah.
Karen = 3*Sarah
7x+48 = 3*(5x-48)
7x+48 = 15x-144
7x-15x = -144-48
-8x = -192
x = -192/(-8)
x = 24
Then,
Karen's start count = 7x = 7*24 = 168
Sarah's start count = 5x = 5*24 = 120


Check so far:
Karen and Sarah start with 168 and 120 beads respectively.
Sarah gives 48 beads, so,
Karen: 168+48 = 216
Sarah: 120-48 = 72
Their ratio is Sarah/Karen = 72/216 = 1/3 which confirms we did things correctly so far.



Part 3
"During a game, Karen lost some beads to Sarah. As a result, Karen had 1/5 as many beads as Sarah. "


L = amount of beads lost
Karen's count goes from 168 to 168-L
Sarah's count goes from 120 to 120+L


Then,
Karen = (1/5)*Sarah
Sarah = 5*Karen
120+L = 5*(168-L)
120+L = 840-5L
L+5L = 840-120
6L = 720
L = 720/6
<font color=red>L = 120
Karen lost <u>120 beads</u> to Sarah</font>



Check:
Karen lost 120 beads, so she has 168-120 = 48 beads left.
Sarah gained those 120 beads, so she has 120+120 = 240 beads
Their ratio is Karen/Sarah = 48/240 = 1/5
The final answer is confirmed.
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