Question 1199415: the ratio of the number of beads Kenny has to the number of beads Sarah has is 7:5.When Sarah gave 48 beads to Kenny, she has 1/3 of the number of beads Kenny has. (a) How many beads Kenny have after receiving the beads from Sarah? (b)How many beads must Kenny give back to Sarah so that she will have 5 times as many beads as him?
Answer by math_tutor2020(3816)   (Show Source):
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Part (a)
To start things off, Kenny has 7x beads and Sarah has 5x beads.
The x is some positive integer.
The ratio 7x:5x reduces to 7:5
Sarah then gives 48 beads to Kenny.
Let's update their counts
Kenny = 7x+48
Sarah = 5x-48
After this point, Sarah has 1/3 the number of beads Kenny has.
This can be rephrased to "Kenny has triple the number of beads Sarah has"
Kenny's Count = 3*(Sarah's Count)
7x+48 = 3*(5x-48)
7x+48 = 15x-144
7x-15x = -144-48
-8x = -192
x = -192/(-8)
x = 24
Then,
Kenny's initial count = 7x = 7*24 = 168
Sarah's initial count = 5x = 5*24 = 120
The ratio 168:120 reduces to 7:5 after dividing both parts by the GCF 24.
You can think of it like saying 168/120 = 7/5.
Now add 48 to Kenny's count while subtracting 48 from Sarah's count
Kenny = 168+48 = 216
Sarah = 120-48 = 72
Note that 72*3 = 216 to show that Kenny has triple the amount of beads compared to Sarah.
Answer to part (a) is 216 beads.
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Part (b)
Let y be the amount of beads Kenny must gift to Sarah so that she will have 5 times as many as he will have.
Refer to the values computed in part (a) earlier.
We found that:
Kenny ended up with 216 beads
Sarah ended up with 72 beads
If Kenny gives Sarah y number of beads, then,
Kenny now has 216-y beads
Sarah now has 72+y beads
Sarah's count = 5*(Kenny's count)
72+y = 5*(216-y)
72+y = 1080-5y
y+5y = 1080-72
6y = 1008
y = 1008/6
y = 168
Answer to part (b) is 168 beads.
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