SOLUTION: 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

Question 1201507: 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?
Found 3 solutions by math_tutor2020, greenestamps, MathTherapy:
Answer by math_tutor2020(3816)   (Show Source): You can put this solution on YOUR website!

Answer: 120

Explanation:

I'll break the instructions into three parts
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.
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
L = 120
Karen lost 120 beads to Sarah

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.

Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!

The problem is poorly worded, leading to two different interpretations and thus two different answers.

The other tutor interpreted the problem as meaning that the ratio of Karen's to Sarah's beads was 7:5 before Karen lost some to Sarah. Another possible interpretation is that Karen lost some to Sarah AFTER Sarah gave 48 of her beads to Karen.

The other tutor showed a solution with his interpretation; here is a solution using the other interpretation.

Originally Karen had 7x beads and Sarah had 5x beads; the total number of beads is 12x.

After Sarah gave 48 beads to Karen, Sarah had 1/3 as many beads as Karen. Since the total number of beads is 12x, that means Sarah now has 3x beads and Karen has 9x beads.

So the number of beads Sarah gave to Karen, 48, is the difference between 7x and 9x (or the difference between 5x and 3x), which is 2x. So 2x = 48, x = 24, and the numbers of beads they started with was

Karen: 7x = 168
Sarah: 5x = 120

And after Sarah gives 48 = 2x beads to Karen, the numbers of beads they have was

Karen: 7x+2x = 9x = 216
Sarah: 5x-2x = 3x = 72

In the end, since Karen finished with 1/5 as many beads as Sarah, and since the total number of beads was 12x, Karen finished with 2x = 48 beads and Sarah finished with 10x = 240 beads.

So with this interpretation of the problem, the number of beads Karen lost to Sarah was the difference between 9x and 2x (or the difference between 3x and 10x), which is 7x = 168.

Second ANSWER: Karen lost 168 beads to Sarah.

Answer by MathTherapy(10552)   (Show Source): You can put this solution on YOUR website!

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 of Karen’s, we get: 15x - 3(48) = 7x + 48 ---- Multiplying by LCD, 3 15x - 7x = 1(48) + 3(48) 8x = 4(48) Multiplicative factor or 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: 1,080 - 5A = 72 + A ----- Multiplying by LCD, 5 - 5A - A = 72 - 1,080 - 6A = - 1,008 Number of beads lost to Sarah, or

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