SOLUTION: Liam and Noah shared a collection of 540 vintage stamps. Liam sold 2/5 fewer stamps than Noah. After the sale, Noah found that the number of stamps he had remaining was exactly 2/3

Algebra ->  Percentage-and-ratio-word-problems -> SOLUTION: Liam and Noah shared a collection of 540 vintage stamps. Liam sold 2/5 fewer stamps than Noah. After the sale, Noah found that the number of stamps he had remaining was exactly 2/3      Log On


   

Question 1210597: Liam and Noah shared a collection of 540 vintage stamps. Liam sold 2/5 fewer stamps than Noah. After the sale, Noah found that the number of stamps he had remaining was exactly 2/3 of the number of stamps he had sold. If Liam’s remaining stamps accounted for 4/7 of the total stamps the two boys possessed at the end, how many stamps did Liam have at first?
Found 4 solutions by greenestamps, josgarithmetic, KMST, MathTherapy:
Answer by greenestamps(13351) About Me  (Show Source):
You can put this solution on YOUR website!


FAULTY DATA!!

Choose a variable:
x = number Noah sold

The number Noah had left was 2/3 the number he sold:
(2/3)x = number Noah had left

Liam sold "2/5 fewer" than Noah -- i.e., he sold 3/5 AS MANY as Noah:
(3/5)x = number Liam sold

Then...

x + (3/5)x = (8/5)x = total sold

540-(8/5)x = total remaining

Liam's remaining stamps were 4/7 of the total remaining, so Noah's remaining stamps were 3/7 of the total remaining:

%282%2F3%29x=%283%2F7%29%28540-%288%2F5%29x%29

multiply by 21 to clear fractions

14x=9%28540-%288%2F5%29x%29
14x=4860-%2872%2F5%29x
70x=24300-72x
142x=24300

x = 24300/142 = 171 9/71

Obviously the answer should be a positive integer....

Answer by josgarithmetic(39818) About Me  (Show Source):
You can put this solution on YOUR website!
Only checked the description and question carefully, followed stepwise and made two equations but
have not yet tried to solve them. 
                Noah          Liam              TOTAL

START             n           540-n              540


SOLD SOME x       n-x         (540-n)-x+2x/5
Noah

Noah found that n-x=%282%2F3%29x.

Another described post-sale condition
%28540-n%29-x%2B%282%2F5%29x=%284%2F7%29%28n-x%2B%28540-n%29-x%2B%282%2F5%29x%29

Answer by KMST(5361) About Me  (Show Source):
You can put this solution on YOUR website!
The question is unclear, and I cannot find an interpretation that would yield a reasonable result (a whole number for all numbers of stamps involved in those transactions).

Liam and Noah shared a collection of 540 vintage stamps."
Apparently Noah has a number N of stamps, and Liam has a number L of stamps, and L%2BN=540

The third sentence seems clear, and we can translate it into an equation that leads us to
highlight%28x=%283%2F5%29N%29 for the number of stamps Noah sold.
The second sentence is a puzzle:
"Liam sold 2/5 fewer stamps than Noah." What could that mean?
A) I would think it could mean that the difference x-y is %282%2F5%29x
B) It could mean that the number of stamps Liam sold is y=%282%2F5%29x
C) It could mean that while Noah sold 3%2F5 of his stamps,
the ratio of the number of stamps Liam sold to the number he originally had is
3%2F5-2%2F5=1%2F5

I tried all three approaches and failed to find a whole number for all numbers of stamps involved.
However, if you do not calculate all numbers involved, you can find that Noah could start with N=324 stamps and Liam with L=216 stamps, from there on all the numbers of stamps would be fractional, but if Liam sells 1%2F5 of his stamps the non-integer numbers of remaining Liam's stamps and total stamps left are in exactly a 4%2F7 ratio.

The numbers of stamps left are %284%2F5%29L=4L%2F5 for Liam
and %282%2F5%29%28540-L%29=%281080-2L%29%2F5 for Noah
The total unsold is
4L%2F5%2B%281080-2L%29%2F5=%281080%2B2L%29%2F5
The ratio of Liam's unsold to total unsold is 4L%2F5%29%2F%28%281080%2B2L%29%2F5%29=4L%2F%281080%2B2L%29
All we have to do is solve 4L%2F%281080%2B2L%29=4%2F7 without thinking about all the transactions.
4L%2F%281080%2B2L%29=4%2F7-->L%2F%281080%2B2L%29=1%2F7-->7L%2F%281080%2B2L%29=1-->7L=1080%2B2L%29-->5L=1080%29-->L=1080%2F5%29-->highlight%28L=216%29

Answer by MathTherapy(10837) About Me  (Show Source):