Arms of scale

[chawki]

1. The problem statement, all variables and given/known data
The arms of a scale are of equal length so that when weighing objects, one side will
hang lower if the load is heavier on that side.

2. Relevant equations
List the weights P1 through P4 in order starting from the heaviest and ending in the lightest based upon the following figure


3. The attempt at a solution
P1+P2=P3+P4
P1+P3>P2+P4
P4>P3
and then ? i have tried by substituting but nothing came out

[eumyang]

Here's a hint that may help: if you have four different weights, where placing two on either side of the scale balances it, then the lightest and heaviest weights must be paired together, and the two middle weights must be paired together. Take it from there.

[chawki]

Here's a hint that may help: if you have four different weights, where placing two on either side of the scale balances it, then the lightest and heaviest weights must be paired together, and the two middle weights must be paired together. Take it from there.

you mean added together?
P1+P3+P4=P2+P4+P3 ?

[eumyang]

No, that's not what I meant. Think of it this way. Suppose there were four weights, 10 kg, 20 kg, 30 kg, and 40 kg. In order to balance, the 10 kg and 40 kg must be placed together on one side, while the other 2 must be placed together on the other side. So I'm suggesting that you start with the idea that the heaviest and lightest weights are put together on one side, with the other two on the other side. So if P1 is the heaviest, for example, then P2 must be the lightest, and P3 & P4 are the two middle weights. You'll have a number of possible ordering of weights based on the first diagram alone. You'll need the other two diagrams to narrow it down.

[Mark44]

To add to what eumyang said, write an equation or inequality, as appropriate, for each of the three illustrations. When the arms of the scale are horizontal, write an equation that represents what the quantities are that are equal. When the scale is unbalanced, write an equation that represents that situation.

[chawki]

To add to what eumyang said, write an equation or inequality, as appropriate, for each of the three illustrations. When the arms of the scale are horizontal, write an equation that represents what the quantities are that are equal. When the scale is unbalanced, write an equation that represents that situation.

yes i already wrote but nothing

[eumyang]

You wrote these:
P1+P2=P3+P4
P1+P3>P2+P4
P4>P3

Look at the last two.
If P4 > P3 and P1+P3>P2+P4, what can you say about P1 and P2?
Is
P1 > P2
or
P1 = P2
or
P1 < P2?

[chawki]

You wrote these:
P1+P2=P3+P4
P1+P3>P2+P4
P4>P3

Look at the last two.
If P4 > P3 and P1+P3>P2+P4, what can you say about P1 and P2?
Is
P1 > P2
or
P1 = P2
or
P1 < P2?

I think P1>P2
then:
P3+P4-P2+P3>P2+P4
2P3>2P2
P3>P2

the order would be: P4,P3,P1,P2

[I like Serena]

yes i already wrote but nothing

You can add or subtract the equations from each other.
If you add for instance the first 2 equations, you get:

P1+P2=P3+P4
P1+P3>P2+P4
---------------- +
2xP1 + P2 + P3 > P2 + P3 + 2xP4
or
P1 > P4

Can you take it from there?

[eumyang]

I think P1>P2
then:
P3+P4-P2+P3>P2+P4
2P3>2P2
P3>P2

the order would be: P4,P3,P1,P2
Is this the order from heaviest to lightest? If so, I'm afraid this is incorrect. Let
P4 = 40, P3 = 30, P1 = 20, and P2 = 10,
and you can see that not all of the the equation and inequalities hold.

[chawki]

You can add or subtract the equations from each other.
If you add for instance the first 2 equations, you get:

P1+P2=P3+P4
P1+P3>P2+P4
---------------- +
2xP1 + P2 + P3 > P2 + P3 + 2xP4
or
P1 > P4

Can you take it from there?

wow that didnt come to my mind..we can add = with > :redface:

[chawki]

we get the order like this:
P1, P4, P2, P3