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Chapter 18

Ratio, Proportion, & Variation

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Sect 18.1 : Ratio and Proportion

A ratio conveys the notion of “relative magnitude”.

Ratios are used to compare two quantities.

Remember the set of rational numbers?

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A ratio can be used to compare quantities of the same kind

(can be expressed using the same units).

Example 1:

The TI-84 calculator display screen is 94 pixels by 62 pixels.

Write this as a ratio in simplest form.

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Example 2:

Write as the ratio of whole numbers in simplest form.

5

3

5

to 2

1

10

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Example 3:

Write the ratios in simplest form.

a) The ratio of 75 s to 3 min

b) The ratio of 2.7 kg to 60.5 g

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A ratio can be used to compare denominate numbers with

different units. These ratios are also called rates.

Example

5 lb of fertilizer covers 1200 square feet. How many

square feet are covered per pound of fertilizer?

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An equation stating that two ratios are equal is called a

________________________.

If

a

b

c

, then ad bc .

d

The product of the ______________ equals the product of the

________________.

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Example 2:

To conserve energy and still allow for as much natural lighting as

possible, an architect suggest that the ratio of the area of a

window to the area of the total wall surface to be 5 to 12. Using

this ratio, determine the recommended area of a window to be

installed in a wall that measures 8 ft by 12 ft.

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Examples from the book:

P. 494 # 26, 40, 44

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Section 18.2: Variation

To estimate the number of phone calls expected per day

between two cities, one telecommunication company

used the following formula

C

0.02 P1 P2

d

2

,

which shows that the daily number of phone calls, C,

increases as the populations of the cities, P1 and P2, in

thousands, increase and decreases as the distance, d,

between the cities increases.

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Variation

Variation formulas show how one quantity changes in

relation to other quantities.

Quantities can vary directly, inversely, or jointly.

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Direct Variation

When you swim underwater, the pressure in your ears depends on

the depth at which you are swimming. The formula p 0.43 d

describes the water pressure, p, in pounds per square inch, at a

depth of d feet.

a) Find the water pressure in your ears at a depth of 10 ft.

b) Find the water pressure in your ears at a depth of 20 ft.

c) Find the water pressure in your ears at a depth of 40 ft.

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Direct Variation

In this example, the water pressure is a constant multiple

of your underwater depth. If your depth is doubled, the

pressure is doubled; if the depth is tripled, the pressure is

tripled; etc.

The pressure in your ears is said to vary directly as your

underwater depth.

p 0.43 d

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Direct Variation

When two quantities are in the same proportion and can be

expressed as an equation in the form y = k x, we say that

“y varies directly as x” or “y is directly proportional

to x”.

The (non-zero) number k is called the constant of

proportionality.

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Direct Variation Example

The number of gallons of water, W, used when taking a shower varies

directly as the time, t, in minutes, in the shower. A shower lasting 5

minutes uses 30 gallons of water. How much water is used in a

shower lasting 11 minutes?

a) Write the general equation:

b) Find the value of the constant of proportionality (k):

c) Write the specific equation:

d) Find the desired value:

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Direct Variation

Direct variation can involve variables to higher powers.

y kx

n

We say that y is directly proportional to the nth power of x.

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Direct Variation Example

The distance, s, that a body falls from rest varies directly as the

square of the time, t, of the fall. If skydivers fall 64 feet in 2

seconds, how far will they fall in 4.5 seconds?

a) Write the general equation:

b) Find the value of the constant of proportionality (k):

c) Write the specific equation:

d) Find the desired value:

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Inverse Variation

When two quantities are related by an equation in the

form

k

y

x

we say that “y varies inversely as x” or “y is inversely

proportional to x”.

The (non-zero) number k is called the constant of

variation.

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Inverse Variation

For example, for a fixed distance, the time it takes to travel

that distance is inversely proportional to the rate at which one

drives.

The distance from Plattsburgh to Albany is 150 miles.

T im e

150

R ate

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Inverse Variation Example

At a constant temperature, the pressure, P, of a gas container varies

inversely with the volume, V, of the container. The pressure of a gas

sample in a container whose volume is 8 cubic inches is 12 pounds per

square inch. If the sample expands to a volume of 22 cubic inches,

what is the new pressure of the gas?

a) Write the general equation:

b) Find the value of the constant of proportionality (k):

c) Write the specific equation:

d) Find the desired value:

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Combined Variation

In combined variation, direct and inverse

variation occur at the same time.

y

kx

y

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Combined Variation Example

One’s intelligence quotient, or IQ, varies directly as a person’s mental

age and inversely as that person’s chronological age. A person with a

mental age of 25 and a chronological age of 20 has and IQ of 125. What

is the chronological age of a person with a mental age of 40 and an IQ

of 80?

a) Write the general equation:

b) Find the value of the constant of proportionality (k):

c) Write the specific equation:

d) Find the desired value:

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Joint Variation

Joint variation is a variation in which a variable varies

directly as the product of two or more other variables.

To show that y varies jointly as x and z, we would write

y kxz

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Joint Variation

Newton’s famous formula for gravitation shows this:

F

G m1 m 2

d

2

The force, F, between two bodies varies jointly as the product

of their masses, m1 and m2, and inversely as the square of the

distance between them, d 2. (G is the gravitational constant.)

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Joint Variation Example 1

The volume of a cone, V, varies jointly as its height, h, and the square

of its radius, r. A cone with a radius measuring 6 feet and a height

measuring 10 feet has a volume of 120 cubic feet. Find the volume of

a cone having radius of 12 feet and a height of 2 feet.

a) Write the general equation:

b) Find the value of the constant of proportionality (k):

c) Write the specific equation:

d) Find the desired value:

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Joint Variation Example 2

The centrifugal force, C, of a body moving in a circle varies jointly with

the radius of the circular path, r, and the body’s mass, m, and inversely

with the square of the time, t, it takes to move about one full circle. A

6-gram body moving in a circle with radius 100 cm at a rate of 1 rev in

2 seconds has a centrifugal force of 6000 dynes. Find the centrifugal

force of an 18-gram body moving in a circle with radius 100 cm at a rate

of 1 revolution in 3 seconds.

a) Write the general equation:

b) Find the value of the constant of proportionality (k):

c) Write the specific equation:

d) Find the desired value:

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End of Section

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