Question 1201507
<pre>
The ratio of Karen's beads to Sarah's beads is 7:5 . When Sarah gives 48 beads to Karen, she has 1/3 of the number of beads Karen has. During a game, Karen lost some beads to Sarah. As a result, Karen had 1/5 as many beads as Sarah. How many beads did Karen lose?  

Let the multiplicative factor be x
Then originally, Karen and Sarah had 7x and 5x beads, respectively

Sarah gave 48 beads to Karen, leaving Sarah with 5x  -  48 beads, while Karen had 7x + 48, then.
As Sarah’s amount at that point was {{{1/3}}} of Karen’s, we get: {{{matrix(1,3, 5x - 48, "=", (1/3)(7x + 48))}}}
                                                       15x - 3(48) = 7x + 48 ---- Multiplying by LCD, 3
                                                          15x - 7x = 1(48) + 3(48)
                                                                8x = 4(48)
                                       Multiplicative factor or {{{matrix(1,7, x, "=", 4(48)/8, "=", 4(6), "=", 24)}}}

Initial amount Karen and Sarah had: 7(24) = 168, and 5(24) = 120, respectively
After Sarah gives 48 beads to Karen, Sarah had 120  -  48 = 72 remaining, while Karen had 168 + 48 = 216 beads

                                      Let number Karen lost to Sarah be A
                                      Then we get:{{{matrix(1,3, 216  -  A, "=", (1/5)(72 + A))}}}
                                             1,080  -  5A = 72 + A ----- Multiplying by LCD, 5  
                                              -  5A  -  A = 72  -  1,080
                                                     - 6A = - 1,008
  <font color = red><font size = 4><b>Number of beads lost to Sarah</font></font></b>, or {{{highlight_green(matrix(1,5, A, "=", (- "1,008")/(- 6), "=", highlight(168)))}}}</pre>