Quandaries
and Queries 

Given that y varies inversely as x and x varies directly as z.If z is doubled then y is halved.Why is that true.Please explain.I'm having trouble understanding the different types of variation 

Hi Abraham, If x varies directly as z it means that there is a constant k so that
k is a constant so its value is always the same, it doesn't depend on the values of x or z. Intuitively this means that if x doubles, so does z and if x triples so does z. Likewise if z doubles so does x and if z triples so does x. For example if the price of a can of juce is $1.10 then you can let x be the number of cans you buy and z what is costs you to buy them, then
so the cost varies directly as the number you buy. If y varies inversely as x then there is a constant c so that
Intuitively this means that if x doubles then y is reduced by ^{1}/_{2} and if x triples then y is reduced by ^{1}/_{3}. For example if you are going on a 200 mile drive then the time it takes, y, is inversely proportional to your average speed, x.
If your average speed is 50 mph then it will take 4 hours and if your average speed is 60 mph then the trip will take 3.33 hours. Hence in your original problem where y varies inversely as x and x varies directly as z, if z doubles so does x and hence y is reduced by ^{1}/_{2}. Penny 

