I really think teaching "tricks" like cross-multiplying before teaching how things actually work via the factor-label method really complicates things for a lot of people. When they are taught trick A for situation A and trick B for situation B and trick C for situation C, but not taught that all three tricks are just different applications of a very straightforward set of rules, they get caught in a trap of thinking there is some secret that they don't know, or that they have to memorize a new method for every new type of problem.
I have to second Roger's suggestion to try the factor-label method. It is applicable to ANY type of unit-based word problem and it makes the actual arithmetic very, very simple. The only part of it that can be difficult sometimes is figuring out where to start your set-up. But, often, using this method can lead to the answer just by following the units, even if you feel like you have no idea what you're doing.
1. A prescription calls for 100 mg of a drug that you have in a 250 mg/ 5 ml concentration. How many ml of the liquid do you need?
To solve this using the factor-label method, I would approach it in the following way. It seems very long, but that is only to make it extremely clear what you are doing, and once you've done it a few times, it comes second nature to do these steps in your head very quickly for most simple problems. Then, for very complex problems, doing it this long way makes it easy to keep track of.
Lay out all of your values with both the numbers (factors) and the units (labels).
100mg
250mg/ml
how many ml
1) Identify what is being asked for in the answer, including the units.
ml of liquid
2) Identify the known
amount given to you in the problem.
100mg
3) Anything left is a conversion factor, which you use to convert from the units given to you in the problem to the answer asked for in the problem.
250mg / 5ml. Or in plainer english: 250mg per 5 mL. Every 5 mL has 250mg, and every 5 mg has 5/250 mL.
4) Problem set-up. Start with your known amount.
100mg
Then pick an appropriate conversion factor that will CANCEL this unit and give you a new unit. To cancel a unit, the unit must be divided by itself. Remember that any number (or unit) divided by itself equals 1 and that multiplying or dividing by 1 does nothing and can be ignored.
250mg/ 5mL is the given conversion factor. But, we need to cancel the unit of
mg and this wouldn't be dividing mg by mg, it would be multiplying mg times mg. So you take the inverse, or "turn it upside-down".
250 mg/ 5 mL becomes 5 mL / 250mg
Now multiply your known amount times your conversion factor:
100mg * 5mL / 250mg
When you multiply units in this way, just treat them as if they were another number, or a variable if you are familiar with variables:
500 mg * mL / 250 mg
The unit mg appears in both the top and bottom, so you have mg/mg which you can cancel out and remove, which leaves:
500 mL / 250
Which gives you your answer, which should already be in the units the problem asked for (this is a good way to verify that you have used this method correctly):
2 mL
Sorry for writing such a ridiculously long post. I blame Adderall and a poor work ethic. I hope this can be at least somewhat helpful.
