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?
WRONG!!!
To solve this problem, let's break it down into steps:
Step 1: Assign variables
Let's assign variables to the unknown quantities in the problem:
- Let K represent the number of bookmarks Ken had initially.
- Let L represent the number of bookmarks Lionel had initially.
- Let M represent the number of bookmarks Mike had initially.
Step 2: Translate the given information into equations
According to the problem statement:
- Ken had 1 1/2 times as many bookmarks as Lionel: K = (3/2)L
- Mike had 3/4 as many bookmarks as Ken: M = (3/4)K
- The number of bookmarks Ken and Lionel gave Mike is in the ratio 3:1: (K-x)/(L-x) = 3/1, where x is the number of bookmarks Ken and Lionel gave Mike.
- As a result, Mike has twice as many bookmarks as before: M + x = 2M
- Lionel has 32 bookmarks more than Ken: L = K + 32
Step 3: Solve the equations
Now let's solve these equations to find the value of x, representing the number of bookmarks Ken gave to Mike.
From the first equation, we can express L in terms of K: L = 2/3K.
Substituting this value of L into the fifth equation, we get:
2/3K = K + 32
2K = 3K + 96
K = -96
However, since the number of bookmarks cannot be negative, this solution is not valid. So, we made an error in our calculations.
Let's go back and start again:
Step 1: Assign variables
- Let K represent the number of bookmarks Ken had initially.
- Let L represent the number of bookmarks Lionel had initially.
- Let M represent the number of bookmarks Mike had initially.
Step 2: Translate the given information into equations
According to the problem statement:
- Ken had 1 1/2 times as many bookmarks as Lionel: K = (3/2)L
- Mike had 3/4 as many bookmarks as Ken: M = (3/4)K
- The number of bookmarks Ken and Lionel gave Mike is in the ratio 3:1: (K-x)/(L-x) = 3/1, where x is the number of bookmarks Ken and Lionel gave Mike.
- As a result, Mike has twice as many bookmarks as before: M + x = 2M
- Lionel has 32 bookmarks more than Ken: L = K + 32
Step 3: Solve the equations
Let's substitute the value of L in terms of K into the third equation:
(K-x)/(2/3K-x) = 3/1
Cross-multiplying:
3(K-x) = 2(2/3K-x)
3K - 3x = 4/3K - 2x
Moving all the terms to the left side:
3K - 4/3K = 3x - 2x + 3x
(9K - 4K)/3 = 4x
5K/3 = 4x
Simplifying:
5K = 12x
Now let's substitute the value of L in terms of K into the second equation:
M = (3/4)K
Putting this value of M into the fourth equation:
(3/4)K + x = 2(3/4)K
3K/4 + x = 6K/4
3K + 4x = 6K
Moving all the terms to the left side:
6K - 3K = 4x - 3K
3K = 4x - 3K
6K = 4x
Now we have two equations:
5K = 12x
6K = 4x
Dividing both equations by K, we get:
5 = 12x/K
6 = 4x/K
Setting these two equations equal to each other:
12x/K = 4x/K
Cross-multiplying:
12xK = 4xK
Dividing both sides by xK:
12 = 4
This is not possible; therefore, there is no valid solution for this problem. It seems we made an error in our calculations.
There might be a mistake in the problem statement or the initial assumptions. I apologize for any confusion caused.
Number for Lionel ---- x ---------> 512
number for Ken = (3/2)x ---------> 768
"Mike had 3/4 as many bookmarks as Ken":
number for Mike = (3/4)(3/2)x = (9/8)x -------> 576
after exchange:
Mike now has (9/8)x + 4y -------> 1152
Ken now has (3/2)x - 3y ------> 336
Lionel now has x - y -----> 368
Does Mike have twice as many bookmarks as before ?
Is 1152 twice 576 ? YES!
Does Lionel have 32 bookmarks more than Ken ??
Well, Lionel now has 368 and Ken has 336, isn't that 32 more ????? YES
"Ken had 1 1/2 times as many bookmarks as Lionel" :
Number for Lionel ---- x
number for Ken = (3/2)x
"Mike had 3/4 as many bookmarks as Ken":
number for Mike = (3/4)(3/2)x = (9/8)x
"Ken and Lionel gave Mike a number of bookmarks in the ratio 3:1":
number Ken gave to Mike ---- 3y
number Lionel gave to Mike --- y , (note ratio of 3y : y = 3:1 )
Mike now has (9/8)x + 4y
Ken now has (3/2)x - 3y
Lionel now has x - y
"Mike nad twice as many bookmarks as before"
-----> (9/8)x + 4y = 2((9/8)x
times 8
9x + 32y = 18x or -9x + 32y = 0
9x - 32y = 0 **
"Lionel had 32 bookmarks more than Ken"
----> x-y - ( (3/2)x - 3y ) = 32
x - y - (3/2)x + 3y = 32
2x - 2y - 3x + 6y = 64
-x + 4y = 64 -----> -9x + 36y = 576 ***
add ** and ***
4y = 576
y = 144
in 9x - 32y = 0
9x - 4608 = 0
x = 512
Leaving it up to you to verify and state the conclusion,
(btw, it works!)