Question 1208259
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Answers:
total value at first = <font color=red>720 cents</font> = <font color=red>$7.20</font>
total number of coins at the end = <font color=red>66 coins</font>
Extras: started with 6 coins worth 20cents each, and 12 coins worth 50cents each.


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Work Shown


x = starting number of 20cent coins
2x = starting number of 50cent coins
Note that x is half of 2x.


A = 20x = starting value of just the 20cent coins
B = 50*(2x) = 100x = starting value of just the 50cent coins
C = A+B = 20x+100x = 120x = value of original group of coins (before any extras are added). Values are in cents.


Now add 8 copies of a 20cent coin and 40 copies of the 50cent coin.
x bumps up to x+8 to represent the new number of 20cent coins.
2x bumps up to 2x+40 to represent the new number of 50cent coins.
D = 20*(x+8) = 20x+160 = new value of just the 20cent coins
E = 50*(2x+40) = 100x+2000 = new value of just the 50cent coins
F = D+E = (20x+160)+(100x+2000) = 120x+2160 = new total value in cents.


The new value (F) is 4 times as much as before (C)
new = 4*old
F = 4*C
120x+2160 = 4(120x)
120x+2160 = 480x
2160 = 480x-120x
2160 = 360x
x = 2160/360
x = 6
There are 6 coins worth 20 cents each in the box at first.
There are also 12 coins worth 50 cents each in the box at first (since 2x = 2*6 = 12)
C = total starting value = 6*20+12*50 = <font color=red>720 cents</font> = <font color=red>$7.20</font>


Now add 8 copies of a 20cent coin and 40 copies of the 50cent coin.
6+8 = 14 copies of a 20cent coin 
12+40 = 52 copies of a 50cent coin
The box now has 14+52 = <font color=red>66 coins</font>
F = new total value = 14*20+52*50 = 2880 cents = $28.80


The jump from C = 720 cents to F = 2880 cents is "times 4" since 720*4 = 2880
Or you could note that F/C = 2880/720 = 4.
We have confirmed the answers are correct.



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If you only cared about the total value, and not the coin counts, then you can follow this method:


m = total value of the original group of coins (before any extras added)
value in cents


Adding 40 coins worth 50cents each and 8 coins worth 20cents each will give an additional 40*50+8*20 = 2160 cents
m bumps up to m+2160 which represents the new total value.
This expression is 4 times that of the original value.


new = 4*old
m+2160 = 4m
4m-m = 2160
3m = 2160
m = 2160/3
m = <font color=red>720 cents</font> is the starting total value
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