Hi Abraham,

If x varies directly as z it means that there is a constant k so that

x = k z.

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

z = $1.10 x

so the cost varies directly as the number you buy.

If y varies inversely as x then there is a constant c so that

y = c  ¹/_x.

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.

y = 200  ¹/_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

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