A
proportional relationship is one that maintains a constant ratio between the two values involved. For a quick example, say that 2 film tickets cost my $7. the ratio of tickets to dollars there would be 2/7. Naturally, if I wanted to buy twice as many tickets, it would cost me twice as much - I'd be paying $7 x 2 = $14 for 2 x 2 = 4 tickets, but the ratio - 2/7 - would be unchanged. That's a proportional relationship.
Here we're told that the relationship between a patient's body weight and the amount of medicine they receive is proportional, which means that the ratio between weight and medicine will stay the same, regardless of the weight of the patient. We're given one ratio: a 159 lb patient receives 212 mg of medicine, a ratio of 159/212. What we need to find is half of the other ratio; we're given that the patient weighs 129 lbs, but we aren't given the amount of medicine they receive. Let's call that amount
m. What we
do know is that these two quantities have a proportional relationship - the ratio between them never changes.
This means we can set the two ratios - 159/212 and 129/m - equal to each other:
The rest is fairly straightforward algebra:
So, we'll need
172 mg of medicine for a patient weighing 129 lbs.
(For that last calculation, I'd simply encourage you to use your calculator)
Alternatively:
If you want to avoid the hairy calculation at the end there, you can always try to reduce the ratio before moving on. In this case, with a little gruntwork, we can find that
Now we know that the ratio of medicine to body weight is (irreducably)
3 lbs to 4 mg of medicine.
A patient weighing 129 pounds weighs
43 times that much, so naturally, we'd need to give them
43 times more medicine, which works out to be 4 x 43 =
172 mg, as we'd expect.