Inverse ratio and proportion question

[physio]

I understood that for a direct proportion such as "if 6 mangoes cost $12, then how many dollars worth are 12 mangoes?", we have to first divide 12 by 6 to get the no. of divisions of 6 in 12 i.e. precisely 2 and then as 1 division is worth $12 we multiply the two divisions with $12 and that's how we get $24. What I don't understand is that for an inverse proportion such as "there are a certain no. of pills and for 10 patients the pills will last for 14 days, for 35 patients how long will the pills last?", the no. of days will obviously decrease and hence more the patients less the no. of days, but to calculate the answer we divide and not multiply as we do in direct proportion question such as the one above (we multiplied 2 with 12).

Why do we divide to find the answer in inverse proportion?

Why can't it be subtract or add or any of those things. What I mean is that what is the logic behind the last division?

Thanks!

 

[chiro]

Hey physio and welcome to the forums.

The simple answer is how you define the ratio and the units.

In the example above we have 6 mango's for 12 dollars. If we want to find mangos per dollar we divide 6:12 by 6 giving us 1:2 or, 1 mango per $2 dollars.

Now the key thing in a rate is that you have two quantities: the first one is the independent and the second is dependent.

In the above we have 1 mango every two dollars. If we have to standardize our dependent variable, we set it equal to 1 which means give 1:2 (one mango for every two dollars) we divide both parts by 2 which gives 0.5:1 or half a mango per dollar.

Now if we have to find dollars per mango we simply invert the mangos per dollar and set the dollar figure to one: specifically we have 0.5:1 for mangos per dollar so we have 1:0.5 dollars per mango and multiplying both sides by two gives us 2:1 or two dollars per mango.

The 1:0.5 is best visualized as a fraction like $$\frac{1}{0.5}$$

The logic behind this is that if we want something per something else we get a ratio that corresponds to how many a per b (in this case 6 mangos per $12 dollars) and then we obtain the right side of the ratio to be 1. If we want the opposite rate, we invert the rate (i.e. turn a:b into b:a) and then make the right hand side of the ratio to be 1.

It's exactly the same sort of thing when you are trying to convert say kilometres per hour to miles per hour or metres per second: we get our rate in the right form and then make the denominator of the fraction 1 to get "x things per unit".

So let's say we have 60 kilometres per hour: we write this as 60:1. Now let's say we want to find it per minute. We know that 60 minutes = 1 hour so we change this to 60:60 in terms of kilometres per minute. Now to find in terms of minutes we standardize the denominator to 1 which means dividing both numbers by 60 giving us 1:1 or one kilometre per minute. Same sort of thing can be done for say turning a kilometre into a metre: we have 60:1 kilometres per hour to 60*1000:1 metres per hour since 1 km = 1000 m. This gives us 60,000:1 or 60,000 metres per hour.