Solving IF problems logically

Let's solve this problem logically:
A child is holding a 5.00 L helium-filled balloon. The temperature of the balloon is 294 K and the pressure inside the balloon is 0.966 atm. If the child accidentally lets go of the balloon and it floats up to a place high in the sky where the temperature is only 261 K and the pressure drops to 0.78 atm, what will be the new volume of the balloon?

We can organize the information from the problem in a table:

	initial	final
P	0.966 atm	0.78 atm
V	5.00 L	?
T	294 K	261 K

Note that in the table there is one (and only one) thing missing. In this case...the final volume.

To solve the problem logically, we will set up some math that looks like a factor label problem - but isn't. We'll start with the whatever is next to the missing piece. Here, that's the initial volume.

Then we'll add a fraction for each other variable in the problem, giving us this:

It is worth noting that temperatures are rarely recorded in Kelvin, but we'll deal with that later.

We'll use the first fraction to "deal with" the change in pressure. Looking at the IF table above, we can see that the pressure went down. We know (from Boyle's Law) that when the pressure goes down, the volume goes up. So, well arrange the pressure values in the first fraction with the larger value up.

Now, we'll use the other fraction to "deal with" the temperature change. We see that the temperature went down and we know (from Charles' Law) that when the temperature goes down, the volume goes down. So we'll put the larger temperature value down in the fraction.

Now, of course, we can solve the problem by doing the math as written:

Here's another example with much more annoying units.
A 5.40 L balloon holds 10.0 g of carbon dioxide at 34^oC and 1.06 atm. If 0.227 moles of carbon dioxide are added to the balloon, the pressure changes to 1500 mm Hg and the temperature drops to -17.2^oC, what will the new volume be?

	initial	final
P	1.06 atm	1500 mm Hg
V	5.40 L	?
T	34oC	-17.2oC
n	10.0 g CO2	+ 0.227 mols

Before we can solve this problem, we need to deal with these units.
First, temperature MUST be in Kelvin, so we'll need to change both temperatures by adding 273.15 to each value.

	initial	final
P	1.06 atm	1500 mm Hg
V	5.40 L	?
T	307 K	256.0 K
n	10.0 g CO2	+ 0.227 mols

The pressures must be in the same unit. Although either unit will work, we'll convert the mm Hg to atmospheres.

We also need to figure out the values for amount. First we'll convert the mass of CO₂ to moles

then we'll add the 0.227 moles. This will leave us with this data table:

	initial	final
P	1.06 atm	1.97 atm
V	5.40 L	?
T	307 K	256.0 K
n	0.227 mol	0.454 mol

Now, we're ready to solve the problem. We'll start the set up with volume (since that's what we're looking for in the end) and we'll need three fractions, since there are three other variables involved.

Pressure in this problem is rising. Rising pressure causes volume to go down, so the larger P value goes down.

The temperature is also decreasing. When the temperature goes down, the volume goes down, so the larger T value goes down.

Lastly, the amount is increasing. When the amount goes up, the volume goes up, so the larger amount value goes up.

and then we can solve the problem by doing the math, as written: