Question

At first, the ratio of Dave's savings to Sam's savings was 5:4. After each of them donated $40 to charity, the ratio of Dave's savings to Sam's savings became 13:10. What was Dave's savings at first?

Answer 1

$300

Explanation:

Let Dave's saving be D and Sam's savings be S

• "At first, the ratio of Dave's savings to Sam's savings was 5:4":

`(D)/(S)=(5)/(4)\\4D=5S\\(4)/(5)D=S`

• "After each of them donated $40 to charity, the ratio of Dave's savings to Sam's savings became 13:10":

So we subtract 40 from each of them and then the ratio becomes 13 is to 10. Hence we can write:

`(D-40)/(S-40)=(13)/(10)\\10(D-40)=13(S-40)\\10D-400=13S-520\\-400+520=13S-10D\\120=13S-10D`

Plugging in
`(4)/(5)D` into S (as we found earlier), we can solve for D (our answer):

`120=13S-10D\\120=13((4)/(5)D)-10D\\120=(52)/(5)D-10D\\120=(2)/(5)D\\D=(120)/((2)/(5))=120*(5)/(2)=300`

So, Dave's savings at first, D, is $300.

Answer 2

ANSWER

Dave's savings at first was $300


EXPLANATION

Let the total savings of Dave and Sam be


`x \: dollars`


Since their savings is in the ratio,


`5:4`

The total ratio


`= 5 + 4 = 9`


Dave's Savings at first is


`= (5)/(9) (x)`

and Sam's savings at first is


`= (4)/(9) (x)`


When each of them donate $40 each, then


Dave's savings is now


`= (5x)/(9) - 40`
or


`= (5x - 360)/(9)`


and Sam's savings is now


`= (4x)/(9) - 40`

Or


`= (4x - 360)/(9)`


The ratio of their savings is now


`13 : 10`


We can now write the proportion,

`(5x - 360)/(9) : (4x - 360)/(9) = 13: 10`

This implies that,


`( (5x - 360)/(9) )/( (4x - 360)/(9) ) = (13)/(10)`


This simplifies to,


`(5x - 360)/(4x - 360) = (13)/(10)`


We cross multiply to get,


`10(5x - 360) = 13(4x - 360)`

We expand to get,


`50x - 3600 = 52x - 4680`


We group like terms to get,



`50x - 52x = - 4680 + 3600`

This simplifies to,


`- 2x = - 1080`

We divide through by -2, to get,


`x = 540`

Dave's savings at first is,


`= (5)/(9) * 540`


`= 5 * 60`


`= 300`

Therefore Dave's savings at first was $300