SOLUTION: Ali, Ben, Cain and Dan, shared a sum of money. Ali has 1/2 of the total amount of money that Ben, Cain and Dan have. Ben has 1/3 of the total amount of money that Ali, Cain and D

Algebra ->  Percentage-and-ratio-word-problems -> SOLUTION: Ali, Ben, Cain and Dan, shared a sum of money. Ali has 1/2 of the total amount of money that Ben, Cain and Dan have. Ben has 1/3 of the total amount of money that Ali, Cain and D      Log On


   



Question 1184444: Ali, Ben, Cain and Dan, shared a sum of money.
Ali has 1/2 of the total amount of money that Ben, Cain and Dan have.
Ben has 1/3 of the total amount of money that Ali, Cain and Dan have.
Cain has 1/7 of the total amount of money that Ali, Ben and Dan have.
(a) What fraction of the total sum of money does Dan have?
(b) Dan has $84. Find the sum of money shared by the 4 boys.

Found 2 solutions by greenestamps, Edwin McCravy:
Answer by greenestamps(13203) About Me  (Show Source):
You can put this solution on YOUR website!


(a) Ali has 1/2 of the total amount of money that Ben, Cain and Dan have.

Let x be the amount Ali has; then 2x is the total amount the other three have.
The total amount all of them have is then 3x.
That means Ali has 1/3 of the total they all have.

Similarly, Ben having 1/3 of the total that the other three have means Ben has 1/4 of the total amount.

And similarly Cain has 1/8 of the total amount.

Together, the fraction of the total amount that Ali, Ben, and Cain have is
1/3+1/4+1/8 = 8/24+6/24+3/24 = 17/24; that means Dan has 7/24 of the total amount.

ANSWER: Dan has 7/24 of the total amount.

(b) Dan's $84 is 7/24 of the total:
84+=+%287%2F24%29x
x=84%2F%287%2F24%29=84%2824%2F7%29=12%2A24=288

ANSWER: The total amount the four boys had was $288.


Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
Greenestamps did it the most elementary way. Think I'll do it using
matrices.  Maybe you've gotten that far.

>>>Ali, Ben, Cain and Dan, shared a sum of money.

>>>Let A,B.C.D be the fractions of their shares.  Then the sum of the
four fractions must be 1 (whole).

So A%2BB%2BC%2BD%22%22=%22%221

>>>Ali has 1/2 of the total amount of money that Ben, Cain and Dan have.

A%22%22=%22%22expr%281%2F2%29%28B%2BC%2BD%29

>>>Ben has 1/3 of the total amount of money that Ali, Cain and Dan have.

B%22%22=%22%22expr%281%2F3%29%28A%2BC%2BD%29

>>>Cain has 1/7 of the total amount of money that Ali, Ben and Dan have.

C%22%22=%22%22expr%281%2F7%29%28A%2BB%2BD%29

The system is

%22%22=%22%22%22%22=%22%22

The matrix is



Do row operations on that to get the row reduced echelon form (RREF)
of the matrix above.  Your TI-84 can handle that if you understand the
matrix feature.

The RREF matrix form of that is as below



So Ali got 1/3, Ben got 1/4, Cain got 1/8, and Dan got 1/7.

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

For (b)

Let $S = the total amount,

Then Dan's share was 7/24 of $S

So expr%287%2F24%29S=%22%2484%22
   7S=%22%242016%22
   S=%22288%22 

[Greenestamps was right!]]

Ali got 1/3 of $288 or $96
Ben got 1/4 of $288 or $72
Cain got 1/8 of $588 or $36
Dan got $84

Check:
The sum was $96
            $72
            $36
            $84
            ---
           $288 as we expected.  

Edwin