Direct and Indirect Variation - HCHS Mathpod's Weblog



Direct and Indirect Variation

Variation, in general, will concern two variables, say height and

weight of a person, and how when one of these changes, the other might

be expected to change. We have direct variation if the two variables

change in the same sense, i.e. if one increases, so does the other.

We have indirect variation if one going up causes the other to go

down. An example of this might be speed and time to do a particular

journey, so the higher the speed the shorter the time.

Normally we let x be the independent variable and y the dependent

variable, so that a change in x produces a change in y. For example,

if x is number of motor cars on the road and y the number of

accidents, then we expect an increase in x to cause an increase in y.

(This obviously ceases to apply if number of cars is so large that

they are all stationary in a traffic jam.)

When x and y are directly proportional then doubling x will double the

value of y, and if we divide these variables we get a constant result.

Since if y/x = k then (2y/2x) = k where k is called the constant of

proportionality

We could also write this y = kx

Thus if I am given the value of x, I multiply this number by k to find

the value of y.

Example: Given that y and x are directly proportional and y = 2 when

x = 5, find the value of y when x = 15.

We first find value of k, using y/x = k

2/5 = k

Now use this constant value in the equation y = kx for situation when

x = 15

y = (2/5)*15

= 30/5 = 6

If you want to do this quickly in your head, you could say, x has been

multiplied by a factor 3 (going from 5 to 15), so y must also go up by

a factor of 3. That means y goes from 2 to 6.

Indirect Variation.

We gave an example of inverse proportion above, namely speed and time

for a particular journey. In this case if you double the speed you

halve the time. So the product speed x time = constant

In general, if x and y are inversely proportional then the product xy

will be constant.

xy = k

or y = k/x

Example: If it takes 4 hours at an average speed of 90 km/hr to do a

certain journey, how long would it take at 120 km/hr

k = speed*time = 90*4 = 360 (k in this case is the distance)

Then time = k/speed

= 360/120

= 3 hours.

To do this in your head, you could say that speed has changed by a

factor 4/3, so time must change by a factor 3/4.

PRACTICE PROBLEMS:

1. If y varies directly as x and y=8 when x=2, find y when x=6.

2. If y varies directly as x and y=-16 when x=-6, find x when y=-4.

3. If y varies inversely as x and y=16 when x=4, find y when x=3.

4. If y varies inversely as x and y=-18 when x=-6, find y when x=5.

5. The volume V of a gas varies inversely as its pressure P. If V = 80 cubic centimeters when P=2000 millimeters of mercury, find V when P=320 millimeters of mercury.

6. The length S that a spring will stretch varies directly with the weight F that is attached to the spring. If a spring stretches 20 inches with 25 pounds attached, how far will it stretch with 15 pounds attached?

7. How does the circumference of a circle vary with respect to its radius? What is the constant of variation?

8. A map of Alaska is scaled so that 3 inches represents 93 miles. How far apart are Anchorage and Fairbanks if they are 11.6 inches apart on the map?

9. Boyle’s Law states that when a sample of gas is kept at a constant temperature, the volume varies inversely with the pressure exerted on it. Write an equation for Boyle’s Law that expresses the variation in volume V as a function of pressure P.

10. Describe two real life quantities that vary directly with each other and two quantities that vary inversely with each other.

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