Burning ropes to measure time

[Agent Smith]

A question from another forum. Would like some clarification, tips and hints.

Two ropes, A and B. A burns in 50 minutes and B burns in 41 minutes (they burn at a uniform rate). Can we use the 2 ropes to measure 33 minutes?

I came up with an equation: [imath]50x + 41y = 33z[/imath]. We would like for [imath]z = 1[/imath]

The answer (what I think is the answer anyway): [imath]\frac{50}{2} = 25[/imath] and [imath]25 + 41 = 66[/imath] and [imath]\frac{66}{2} = 33[/imath]

For my equation: [imath]x = \frac{1}{4}[/imath] and [imath]y = \frac{1}{2}[/imath] and [imath]z = 1[/imath]

[imath]50 \times \frac{1}{4} + 41 \times \frac{1}{2} = 33 \times 1 = 33[/imath]

Correct?

We should've been given [imath]2[/imath] sets of equations as there are [imath]2[/imath] variables.

Comments/criticisms/etc.

 

[Aion]

Does your solution x=1/4 and y=1/2 mean that we measure 1/4 of rope A and half of B and burn these portions glued together or first burn rope A and then immediately burn the B portion?

I tried burning A at both ends and one end of B simultaneously. Then when A has burned up we have 16 minutes remaining of rope B. Now I burned the other end of rope B, which cuts the remaining time down to 8 min. Then the total time is 25 min + 8 min = 33 min.

 

[Agent Smith]

@Aion gracias for the correct solution.

Si, my "solution" involves cutting rope A into fourths and rope B into half. It's clumsy compared to your slick solution.

The only "advantage" I have is I have left over 3 segments, each 12.5 minutes long and 1 segment 20.5 minutes long.

Gracias.