Question 1199415
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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 <font color=red><u>216</u></font> 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 <font color=red><u>168</u></font> beads.
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