Are you smarter than me? 10

Algebra Level 3

There are three kinds of liquids $X, Y,$ and $Z.$ Three jars $J_1, J_2,$ and $J_3$ contain 100 ml of liquids $X,Y,$ and $Z,$ respectively. By an operation we mean three steps in the following order:

stir the liquid in $J_1$ and transfer 10 ml from $J_1$ into $J_2;$
stir the liquid in $J_2$ and transfer 10 ml from $J_2$ into $J_3;$
stir the liquid in $J_3$ and transfer 10 ml from $J_3$ into $J_1.$

After performing the operation four times, let $x,y,$ and $z$ be the amounts of $X,Y,$ and $Z,$ respectively, in $J_1.$ Which of the following is correct?

x>y>z

z>x>y

x>z>y

y>x>z

1 solution

Ved Pradhan
Jun 17, 2020

It is not necessary to calculate the exact values if you can visualize the process happening. In this solution, the term $f(x)$ means a fraction of $x$ . $f(x)$ is always less than $x$ .

You start out with a lot of each liquid in each respective jar.

After the first move, in $J_2$ , we have $f(X) + Y$ .
After the second move, in $J_3$ , we have $f(f(X)) + f(Y) + Z$ .
After the second move, in $J_1$ , we have $f(f(f(X))) + f(f(Y)) + f(Z) + X$ .

This concept of a composition of taking fractions is very useful, because now, we can clearly see that $X > f(Z) > f(f(Y)) > f(f(f(X)))$ . Since $x=X+f(f(f(X)))$ , $y=f(f(Y))$ , and $z=f(Z)$ , it is clear that $\boxed{x>z>y}$ .

One thing I forgot to mention that's really important:

The fraction that the function $f(x)$ represents is not the same each time. For example, the first time, the fraction is $\frac{1}{10}$ , but the second time, it's $\frac{1}{11}$ . The good thing is that it doesn't matter! As long as $f(x)$ is significantly less than $x$ , it doesn't matter what fraction the function represents, or whether the fraction changes.

Ved Pradhan - 12 months ago

Are you smarter than me? 10

1 solution

1 pending report

Vote up reports you agree with