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 nad twice as many bookmarks as before, and Lionel had 32 bookmarks more than Ken. How many bookmarks did Ken give to Mike?
To solve this problem, let's break it down step by step:
1) Let's define the variables:
- Let's say Lionel had x bookmarks.
- Ken had 1.5 times as many bookmarks as Lionel, so Ken had 1.5x bookmarks.
- Mike had 0.75 (3/4) times as many bookmarks as Ken, so Mike had 0.75(1.5x) = 1.125x bookmarks.
2) The problem states that Ken and Lionel gave Mike a number of bookmarks in the ratio 3:1. So, we can write the equation:
Ken/Mike = 3/1
(1.5x - K)/(1.125x) = 3/1
3) The problem also states that after receiving the bookmarks, Mike had twice as many as before. Therefore, we can write the equation:
Mike (after giving) = 2(Mike before)
1.125x + K = 2(1.125x)
4) Lastly, the problem states that Lionel had 32 bookmarks more than Ken. So, we can write the equation:
Lionel = Ken + 32
x = 1.5x - 32
Now, let's solve these equations step by step:
1. Solve the equation x = 1.5x - 32 for x:
Subtract 1.5x from both sides:
-0.5x = -32
Divide both sides by -0.5:
x = 64
2. Substitute the value of x in the equations to find other variables:
Ken = 1.5x = 1.5(64) = 96 bookmarks
Mike = 1.125x = 1.125(64) = 72 bookmarks
3. Calculate the number of bookmarks Ken gave to Mike by using the ratio:
Ken/Mike = 3/1
Ken = 3/4 * Mike
96 = 3/4 * Mike
Multiply both sides by 4/3 to solve for Mike:
Mike = (96 * 4/3) = 128
Therefore, Ken gave Mike 128 - 72 = 56 bookmarks.