MATH 10-3 Lesson 9 Ratio, Proportion and Variation

8/9/2019 MATH 10-3 Lesson 9 Ratio, Proportion and Variation

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LESSON 9RATIO, PROPORTION and VARIATION


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RATIO


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Defnition o Ratio

A ratio is an indicated quotient o two quantities.

Every ratio is a raction and all ratios can be describedby means o a raction. The ratio oxand y is writtenasx : y. it can also be represented as .

Thus, .

y

x

y

xyx =:


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1. Express the following ratios as simplified fractions:

a) 5 : 20b) )8x2x(:)xx(

22++

EXAMPLE

2. !rite the following comparisons as ratios red"ced to lowest terms.

#se common "nits whene$er possible.

a) st"dents to 8 st"dents

b) days to % wee&s

c) 5 feet to 2 yards

d) 'bo"t 10 o"t of 0 st"dents too& (ath l"s

'ns. 1 : 2

'ns. : 21

'ns. 5 : *

'ns. 1 :


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PROPORTION


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Defnition o Proportion

A proportionis a statement indicatin theequality o two ratios.

Thus, , , are

proportions.

!n the proportionx : y = m : n, xand nare calledthe extremes,yand mare called the means.x

and mare the called the antecedents, yand narecalled the consequents.

!n the event that the means are equal, they arecalled the mean proportional.

n

m

y

x= n:m

y

x= n:my:x =


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1. +ind the mean proportional of

ans. ,52. -etermine the $al"e of x in the following proportion:

a) 2 : 5 x : 20

b)

EXAMPLE .25::225 xx =

1

x20

x=

ans. 8

ans.


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VARIATION


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DIRECT VARIATION


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DIRECT VARIATION

"any real#lie situations involve variables that arerelated by a type o equation called a variation.

$or e%ample, a stone thrown into a pond enerates

circular ripples whose circumerences anddiameters increase in si&e.

The equation C' de%presses the relationship

between the circumerence Co a circle and itsdiameter d. ! dincreases, then Cincreases. Thecircumerence Cis said to vary directly as thediameter d.


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DIRECT VARIATION

Defnition o Direct (ariationThe variableyvaries directl as the variablex, oryis directl proportional tox, i and only i

y ' kx

where k is a constant called the constant o!proportionalit or the variation constant"


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DIRECT VARIATION

Direct variation occurs in many daily applications. $ore%ample, suppose the cost o a newspaper is )* cents.

The cost Cto purchase nnewspapers is directly

proportional to the number n.

That is, C ' )*n. !n this e%ample the variationconstant is )*.

To solve a problem that involves a variation, wetypically write a eneral equation that relates thevariables and then use iven inormation to solve orthe variation constant.


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SOLVE A DIRECT VARIATION

The distance sound travels varies directly as thetime it travels. ! sound travels +-* meters in -seconds, fnd the distance sound will travel in )seconds.

olution/

0rite an equation that relates the distance dto thetime t.

1ecause dvaries directly as t, our equation is

d ' kt.


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SOL#TION

1ecause d ' +-*when t ' -, we obtain

+-* ' k - which implies

Thereore, the specifc equation that relates the dmeters sound travels in tseconds is d ' )t.

To fnd the distance sound travels in ) seconds,replacet with )to produce

d ' )2)3 ' +45)

cont/d


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SOL#TION

6nder the same conditions, sound will travel +45)meters in ) seconds. ee $iure +.+5.

cont/d

Figure 1.17


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DIRECT VARIATION

Defnition o Direct (ariation as the nth Power!y varies directl as t$e nt$ po%er ox,then

y ' kxn

where k is a constant.


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INVERSE VARIATION


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INVERSE VARIATION

Two variables also can vary inversely.

Defnition o !nverse (ariation

The variableyvaries inversel as the variablex,

oryis inversel proportional tox, i and only i



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INVERSE VARIATION

!n +44+, Robert 1oyle made a study o the compressibility oases. $iure +.+7 shows that he used a 8#shaped tube todemonstrate the inverse relationship between the volume o aas at a iven temperature and the applied pressure.

Figure 1.19


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INVERSE VARIATION

The 8#shaped tube on the let in $iure +.+7 showsthat the volume o a as at normal atmosphericpressure is 4* milliliters.

! the pressure is doubled by addin mercury 293,as shown in the middle tube, the volume o the asis halved to * milliliters.

Triplin the pressure decreases the volume o theas to :* milliliters, as shown in the tube at theriht in $iure +.+7.


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SOLVE AN INVERSE VARIATION

&ole's La% states that the volume Vo a sampleo as 2at a constant temperature3 varies inverselyas the pressure P. The volume o a as in a 8#shapedtube is 5) milliliters when the pressure is +.)atmospheres. $ind the volume o the as when thepressure is increased to :.) atmospheres.

olution/

The volume Vvaries inversely as the pressure P, so


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SOL#TION

The volume V is 5)milliliters when the pressure is+.)atmospheres, so

and k ' 25)32+.)3 ' ++:.)

Thus

cont/d


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SOL#TION

0hen the pressure is :.)atmospheres, we have

ee $iure +.:*.

cont/d

Figure 1.20


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INVERSE VARIATION

"any real#world situations can be modeled byinverse variations that involve a power.

Defnition o !nverse (ariation as the nth Power

!yvaries inversel as t$e nt$ po%er ox, then

where k is a constant and n ; *.


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(OINT AND COM&INED VARIATIONS


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ome variations involve more than two variables.

Defnition o 8oint (ariation

The variablezvaries )ointl as the variablesxand

yi and only i

z ' kxy



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SOLVE A (OINT VARIATION

The cost o insulatin the ceilin o a house varies+5) to insulate a :+**#square#oot ceilin withinsulation that is - inches thic=. $ind the cost oinsulatin a :-**#square#oot ceilin with insulationthat is 4 inches thic=.

olution/

1ecause the cost Cvaries


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SOL#TION

6sin the act that C ' +5)whenA ' :+** and T '-

ives us

+5)' k2:+**32-3 which implies

?onsequently, the specifc ormula or C is

cont/d


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SOL#TION

@ow, whenA ' :-** and T ' 4, we have

' **

The cost o insulatin the :-**#square#oot ceilinwith

4#inch insulation is >**.

cont/d


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Co*+ined variations involve more than one typeo variation.


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SOLVE A COM&INED VARIATION

The weiht that a hori&ontal beam with arectanular cross section can saely support varies


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SOL#TION

The eneral variation equation is

6sin the iven data yields

olvin or k produces k ' -*, so the specifc

ormula or

is


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SOL#TION

ubstitutin - or !, 4 or d, and +4 or l ives

MATH 10-3 Lesson 9 Ratio, Proportion and Variation

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