SOLUTION: Hi Mrs Johnson and her 2 sons Bernard and raymond shared a sum of money. The ratio received by Mrs Johnson to that received by Bernard was 5 to 2 . The ratio of money received by

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Question 1208419: Hi
Mrs Johnson and her 2 sons Bernard and raymond shared a sum of money. The ratio received by Mrs Johnson to that received by Bernard was 5 to 2 . The ratio of money received by Bernard to that received by Raymond was 3 to 4 . After Mrs Johnson gave 20% of her share to Raymond and Bernard donated 1/3 of his share to charity. Raymond then had $4270 more than Bernard.
What was the sum of money shared by the three of them.
Found 4 solutions by josgarithmetic, ikleyn, math_tutor2020, MathTherapy:
Answer by josgarithmetic(39616)   (Show Source): You can put this solution on YOUR website!
If you would start three variables as j, Mrs. Johnson; b, Bernard, and r, Raymond, then the ratio information would give these initial money amounts in just a single variable:
Mrs. Johnson
Bernard
Raymond

---

20% from Mrs. Johnson to Raymond and one-third of Barnard's to charity

Mrs. Johnson
Bernard
Raymond


Raymond has 4270 more dollars than bernard.
----not too hard to solve.
Answer by ikleyn(52776)   (Show Source): You can put this solution on YOUR website!
.
Mrs Johnson and her 2 sons Bernard and raymond shared a sum of money.
The ratio received by Mrs Johnson to that received by Bernard was 5 to 2 .
The ratio of money received by Bernard to that received by Raymond was 3 to 4.
After Mrs Johnson gave 20% of her share to Raymond and Bernard donated 1/3 of his share to charity.
Raymond then had $4270 more than Bernard.
What was the sum of money shared by the three of them.
~~~~~~~~~~~~~~~~~~ 

From this statement, "The ratio received by Mrs Johnson to that received by Bernard was 5 to 2",
I choose to write 5x for Mrs. Johnson amount and 2x for the Bernard amount.

Here "x" is the common measure of these amounts, in dollars. 


From the second statement, "The ratio of money received by Bernard to that received by Raymond was 3 to 4"
we conclude that Raymond then received   =  dollars. 


After re-distribution and donation, Raymond has   =  =  dollars;

                                    Bernard has   =  = .


Now Raymond has $4270 more than Bernard

     -  = 4270  dollars.


The setup is complete. Now we should solve this equation and find x.

Multiply both sides by 3

    11x - 4x = 3*4270

        7x   =  12810

         x   =  12810/7 = 1830.


Thus the sum of money shared by the three of them was  5x + 2x +  =  = 17690 dollars.


ANSWER.  The sum of money shared by the three of them was  17690 dollars.

Solved completely, using only one single unknown variable,
and in a way as it should be done and as it is expected to be done.



Answer by math_tutor2020(3816)   (Show Source): You can put this solution on YOUR website!

Answer = $17,690

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.
BeforeAfter
Mrs Johnsonm0.8m
Bernardb2b/3
Raymondrr+0.2m

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 = 17690 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.

--------------------------------------------------------------------------

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 = 17690 is the total sum of money
Answer by MathTherapy(10551)   (Show Source): You can put this solution on YOUR website!
Hi
Mrs Johnson and her 2 sons Bernard and raymond shared a sum of  money. The ratio received by Mrs Johnson to that received by Bernard was 5 to 2 . The ratio of money received by Bernard to that received by Raymond was 3 to 4 . After Mrs Johnson gave 20% of her share to Raymond and Bernard donated 1/3 of his share to charity. Raymond then had $4270 more than Bernard. 
What was the sum of money shared by the three of them.

Ratio of Mrs. Johnson's share to Bernard's: 5:2
Ratio of Bernard's share to Raymond's: 3:4
Bernard's part 2, in the ratio, 5:2 needs to MATCH with his part 3, in the ratio 3:4
To do this, we need to MULTIPLY the 1st ratio by 3, and the 2nd by 2
The 1st ratio, 5:2 becomes: 3(5):3(2) = 15:6
The 2nd ratio, 3:4 becomes: 2(3):2(4) = 6:8
We now get the following 3-part ratio for Mrs. Johnson, Bernard, and Raymond: 15:6:8

Let the multiplicative FACTOR be x
Then Mrs. Johnson, Bernard, and Raymond get: 15x, 6x, and 8x, respectively, for a total of 29x (15x + 6x + 8x)
After receiving 20% () of Mrs. Johnson's share, or , Raymond then had 8x + 3x = 11x
After donating () of his share to charity, or , Bernard ended up with 6x - 2x = 4x
As Raymond had $4,270 more than Bernard in the end, we get: 11x - 4x = 4,270
                                                                  7x = 4,270
                                        Multiplicative factor, or 

Subsituting MULTIPLICATIVE FACTOR, 610 for x, AMOUNT SHARED by the 3 = 29(x) = 29($610) = $17,690
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