MHB Finding the Cost of a Gift: Solving a Fractional Payment Dilemma

[Johnx1]

Devi, Sam and Nora shared the cost of a gift for their friend. The amount Devi and Sam paid was 3/10 of the cost and the amount Sam and Nora paid was 7/10 of the cost. Devi paid 5.85 dollars and Nora paid 13.80 dollars more than Devi. How much did sam pay?

My work:

How much Devi paid = D
How much Sam paid = S
How much Nora paid = N
Total money the spent = C

We Know D + S = 3/10C

We Know S + N = 7/10C

We know D spent 5.85. So, D = 5.85

We know N spend 13.80 more than D. So N = D + 13.80Then I did elimination:

19.65 + S = 7/10C
- 5.85 + S = 3/10C
------------------------------
= 13.80 = 4/10C

So, C =34.50.We are looking for how much sam paid, so then I did this below.

5.85 + Sam = 3/10(34.5)

Sam = 4.50

 

[HOI]

Do you have a question?! If you are asking if this is correct, it is not that hard to check:

You have arrived at the solution that Devi paid 5.85, Sam paid 4.50, Nora paid 19.65, and the total paid was 5.85+ 4.50+ 19.65= 30.

"The amount Devi and Sam paid was 3/10 of the cost"
Devi and Sam together paid 5.85+ 4.50= 10.35. That is NOT 3/10 of 30.

"the amount Sam and Nora paid was 7/10 of the cost."
Sam and Nora together paid 4.50+ 19.65= 24.15. That is NOT 7/10 of 30.

Here is how I would do it, letting D, S, and N be the amount each paid:

"The amount Devi and Sam paid was 3/10 of the cost" so D+ S= (3/10)(D+ S+ N). We can write that as 10(D+ S)= 3(D+ S+ N), 10D+ 10S= 3D+ 3S+ 3N, 7D+ 7S- 3N= 0.

"The amount Sam and Nora paid was 7/10 of the cost" so S+ N= (7/10)(D+ S+ N). We can write that as 10(S+ N)= 7(D+ S+ N), 10S+ 10N= 7D+ 7S+ 7N, 3S+3N- 7D= 0.

"Devi paid 5.85 dollars and Nora paid 13.80 dollars more than Devi." Well this makes the previous two equations almost trivial! (If this is consistent- we have effectively four equations in three unknowns. Solutions to three of the equations might not work in the fourth.)
D= 5.85 and N= 5.85+ 13.80= 19.65 so the previous two equations become:
7D+ 7S- 3N= 40.95+ 7S- 58.95= 0 so 7S= 18. S= 18/7= 2.57... (that is a repeating decimal.)
3S+ 3N- 7D= 3S+ 58.95- 40.95= 3S+ 18= 0. S= -6 which not only does not match the previous value, it makes no sense as an amount paid. This is a bad problem- the given informarion is not consistent!

 

[Johnx1]

Do you have a question?! If you are asking if this is correct, it is not that hard to check:

You have arrived at the solution that Devi paid 5.85, Sam paid 4.50, Nora paid 19.65, and the total paid was 5.85+ 4.50+ 19.65= 30.

My mistake, I should have asked in the question if I was heading the right path with the algebraic expression I created? :-)

In the book, Sam answer was $4.50.

Thank you for clearing up the problem.

 

[HOI]

The problem says that "Devi paid 5.85 dollars and Nora paid 13.80 dollars more than Devi." So Nora paid 13.80+ 5.85= 19.65. If Sam paid 4.50 then the total cost was 5.85+ 19.65+ 4.50= 30. 7/10 of 30 is 21 but Sam and Nora paid 19.65+ 4.50= 24.19, NOT 21! That is incorrect.