Question 1184054
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Ken had 1 1/2 times as many bookmarks as Lionel. Mike had 3/4 as many bookmarks as 
Ken. Ken and Lionel gave Mike a number of bookmarks in the ratio 3:1. As a result, Mike 
had twice as many bookmarks as before, and Lionel had 32 bookmarks more than Ken. 
How many bookmarks did Ken give to Mike?

Let original number Lionel had, be L
Then original number Ken had was, 1.5L
Also, Mike’s original number = {{{matrix(1,3, (3/4)(1.5L), "=", 4.5L/4)}}}

Since Ken and Lionel gave Mike an amount in a ratio of 3:1, then Ken and Lionel gave Mike 3x, and x, respectively,
with x being the MULTIPLICATIVE factor of the total number given to Mike. In all, Mike received 4x (3x + x) from
Ken and Lionel 
After Ken gave Mike 3x, Ken was left with 1.5L - 3x
After Lionel gave Mike x, Lionel was left with L - x
After receiving a total of 4x from Ken and Lionel, Mike then had: {{{4.5L/4 + 4x}}}

After receiving 4x from Ken and Lionel, Mike had twice his previous total,
so we get: {{{matrix(2,3, 2(4.5L/4), "=", (4.5L/4) + 4x,
2(4.5L/4) - (4.5L/4), "=", 4x )}}}
               2(4.5L) - 4.5L = 4(4x) ------ Multiplying by LCD, 4
                         4.5L = 16x
                   4.5L - 16x = 0 ------- eq (i)

Now, since Ken was left with 32 LESS than Lionel, after giving to Mike, we then have: 
                    1.5L - 3x = L - x - 32
            1.5L - L - 3x + x = - 32
                     .5L - 2x = - 32 ----- eq (ii)
                   4.5L - 18x = - 288 ---- Multiplying eq (ii) by 9 --- eq (iii)
                           2x = 288 ------ Subtracting eq (iii) from eq (i)
Multiplicative factor of number given to Mike, or {{{matrix(1,5, x, "=", 288/2, "=", 144)}}}

                              <font color = red.<font size  = 4><b>Ken gave Mike,</font></font></b> 3x = 3(144) = <font color = red.<font size  = 4><b>432</font></font></b></pre>