Question 1208419
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Answer = <font color=red>$17,690</font>


Work Shown


m = amount Mrs Johnson starts with
b = amount Bernard starts with
r = amount Raymond starts with


The ratio of Mrs Johnson's starting amount to Bernard's starting amount is 5:2, so,
m:b = 5:2
m/b = 5/2
2m = 5b
b = 2m/5
b = 0.4m


The ratio of Bernard's starting amount to Raymond's starting amount is 3:4, so,
b:r = 3:4
b/r = 3/4
4b = 3r
4*0.4m=3r
1.6m=3r
r = 1.6m/3


The equations we'll use are
b = 0.4m
r = 1.6m/3
We'll come back to them later.


Mrs Johnson donates 20% of her money to Raymond. 
Raymond's amount r increases to r+0.2m
Bernard's amount drops from b to b-(1/3)b = 2b/3 since he donates 1/3 of his money to charity. Bernard keeps 2/3 of his money. 


A table might be helpful.
<table border = "1" cellpadding = "5"><tr><td></td><td>Before</td><td>After</td></tr><tr><td>Mrs Johnson</td><td>m</td><td>0.8m</td></tr><tr><td>Bernard</td><td>b</td><td>2b/3</td></tr><tr><td>Raymond</td><td>r</td><td>r+0.2m</td></tr></table>
Then,
Raymond's new amount = (Bernard's new amount) + 4270
r+0.2m = 2b/3+4270
1.6m/3+0.2m = 2b/3+4270 ...................................... Plug in r = 1.6m/3
1.6m/3+0.2m = 2*0.4m/3+4270 .................................. Plug in b = 0.4m
3*( 1.6m/3+0.2m ) = 3*( 2*0.4m/3+4270 )
1.6m+0.6m = 0.8m+12810
1.6m+0.6m-0.8m = 12810
1.4m = 12810
m = 12810/1.4
m = 9150 = amount Mrs Johnson started with
b = 0.4m = 0.4*9150 = 3660 = amount Bernard started with
r = 1.6m/3 = 1.6*9150/3 = 4880 = amount Raymond started with
m+b+r = 9150+3660+4880 = <font color=red>17690</font> is the total amount


Check:
Let's verify each ratio mentioned.
m/b = 5/2
9150/3660 = 5/2
2.5 = 2.5
This equation is confirmed to be true. 
Now check the other ratio equation.
b/r = 3/4
3660/4880 = 3/4
0.75 = 0.75
Both ratio equations are confirmed.
Now we need to check what happens when some of the money changes hands.
20% of 9150 = 0.20*9150 = 1830
Mrs Johnson gives $1830 to Raymond. His amount starts at $4880 and increases to $4880 + $1830 = $6710
Bernard's amount starts at $3660. He donates 1/3 of that to charity and keeps the remaining 2/3 of it.
(2/3)*3660 = 2440
After the money is done changing hands, Bernard and Raymond will have $2440 and $6710 respectively.
Notice how 6710-2440 = 4270 which shows Raymond has $4270 more compared to Bernard.
We have verified each requirement. The answer is fully confirmed.


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Another way to solve


Write the ratios 5:2 and 3:4 side by side. 
Circle the inner values 2 and the 3. We can scale the ratios so that those circled values are equal to one another.
Triple both parts of 5:2 to get 15:6
Double both parts of 3:4 to get 6:8


5:2 and 3:4 update to 15:6 and 6:8 respectively.
We can then write the triple ratio 15:6:8 to quickly determine their share.
Mrs Johnson gets 15 parts out of 15+6+8 = 29 parts. She gets 15/29 of the money.
Bernard gets 6 parts out of 29. He gets 6/29 of the money.
Raymond gets 8 parts out of 29. He gets 8/29 of the money.


x = total sum of money
(15/29)x = amount Mrs Johnson gets
(6/29)x = amount Bernard gets
(8/29)x = amount Raymond gets


Mrs Johnson gives 20% of her money to Raymond.
20% of (15/29)x = (20/100)*(15/29)x = (3/29)x
Raymond's new amount would be (8/29)x+(3/29)x = (11/29)x


After donating 1/3 of his money to charity, Bernard's new amount would be (2/3)*(6/29)x = (4/29)x


Then,
Raymond's new amount = (Bernard's new amount) + 4270
(11/29)x = (4/29)x + 4270
29*(11/29)x = 29*( (4/29)x + 4270 )
11x = 4x + 123830
11x-4x = 123830
7x = 123830
x = 123830/7
x = <font color=red>17690</font> is the total sum of money
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