# math

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K bought some marbles and gave half of them to L. bought some stamps and gave half of them to K.

K used 5 stamps and L gave away 11 marbles. The ratio of the number of stamps to the number of marbles K had left then became 1:7 and the ratio of the number of stamps to the number of marbles L had left became 1:5.

How many stamps did L buy?

• math -

K's marbles started at km, and stamps at ks
L started with lm marbles and ls stamps.

after paragraph 1,

K has km/2 marbles and ks+ls/2 stamps
L has lm+km/2 marbles and ls/2 stamps

After spending and charity,

K has km/2 marbles and ks+ls/2-5 stamps
L has lm+km/2-11 marbles and ls/2 stamps

Finally,

(ks+ls/2-5)/(km/2) = 1/7
(ls/2)/(lm+km/2-11) = 1/5

Now, having four unknowns and only two equations, I must assume that K started with no marbles (ks=0), and L started with no stamps (lm=0). That leaves us with

(ls/2-5)/(km/2) = 1/7
(ls/2)/(km/2-11) = 1/5

Clear fractions and rearrange terms to get

7ls - km = 70
5ls - km = -22

ls=46
km=252

so, K bought 252 marbles, L bought 46 stamps

check:
after initial giving,
K and L each had 126 marbles and 23 stamps

Then K had 18 stamps and 126 marbles
and L had 23 stamps and 115 marbles

18/126 = 1/7
23/115 = 1/5

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