math 10 week3 ratio

8/6/2019 MATH 10 Week3 Ratio

1/19

MATH10

ALGEBRA

,ATIO VARIATION ANDROPORTION

, Week 3 Day 1 Ratio Variation and Proportion ( ,Algebra and Trigonometry Young 2nd ,Edition


2/19

ODAY SOBJECTIVE

Use ratio and proportion in solving problemsinvolving them,

Identify the different types of variation,

Understand the difference between directvariation and inverse variation,

Understand the difference between combined

variation and joint variation, and Develop mathematical models using direct

variation, inverse variation, combined variationand joint variation.

A t th e e n d o f th e le sson th e stu d e n ts a re:expected to

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efinitionRATIOA atio .is an indicated quotient of two quantities

Every ratio is a fraction and all ratios can be.described by means of a fraction The ratio of xand

y is written as : ,x y it can also be represented as

,Thus

efinitionRATIO

A atio .is an indicated quotient of two quantitiesEvery ratio is a fraction and all ratios can be

.described by means of a fraction The ratio of xandy is written as : ,x y it can also be represented as

,Thus

y

x

y

xy:x =

Week 3 Day 1


4/19

.1 Express the following ratios as simplified

:fractions)a :5 20

)b)8x2x(:)4x4x( 22 ++

2. Write the following comparisons as ratios reduced tolowest .terms U .se common units whenever possible)a 4 students to 8 students

)b 4 days to 3 weeks

)c 5 feet to 2 yards )d About 10 out of 40 students took Math Plus

EXAMPLE

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efinitionPROPORTIONA proportionis a statement indicating the equality

.of two ratios

, , ,Thus

.are proportions

In the proportion : = : ,x y m n x and nare called theextremes, yand mare called the .meansxand mare the

called the ,antecedents yand nare called theconsequents.

,In the event that the means are equal they are calledthe .e a n p r o po rt io n a l

efinitionPROPORTION

A proportionis a statement indicating the equality.of two ratios

, , ,Thus

.are proportions

In the proportion : = : ,x y m n x and nare called theextremes, yand mare called the .meansxand mare the

called the ,antecedents yand nare called theconsequents.

,In the event that the means are equal they are calledthe .e an p r o p o r t i on al

n

m

y

x= n:m

y

x= n:my:x =

Week 3 Day 1


6/19

.1 Find the mean proportional of

.2 Determine the value of x in the following

:proportion)a : = :2 5 x 20

)b

.25::225 xx =

4

1

x20

x=

EXAMPLE

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DefinitionVARIATIONA variation is the name given to the study of the

.effects of changes among related quantities.Variation describes the relationship between variables

DefinitionVARIATIONA variation is the name given to the study of the

.effects of changes among related quantities .Variation describes the relationship between variables

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irect VariationWhen one quantity is a constant multiple of another

,quantity we say that the quantities are d i r e c t l yr o p o r t i o n a l .to one another.Let x and y represent two quantities The following are

:equivalent statements = , .y kx where k is a nonzero constant

y aries directly .with x y is irectly proportional .to x

The constant k is called the onstant of variationorthe .onstant of proportionality

irect VariationWhen one quantity is a constant multiple of another

,quantity we say that the quantities are d i r e c t l yr o p o r t i o n a l .to one another.Let x and y represent two quantities The following are

:equivalent statements

= , .y kx where k is a nonzero constant

y aries directly .with x y is irectly proportional .to x


Definition page 304

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. Write an equation that describes each variation

.17 dis directly proportional to t. =d rwhen = .t 1

.19 Vis directly proportional to both land w. =V 6hwhen=w 3 qnd = .h 4

.24 Wis directly proportional to both Rand the square ofI. =W 4when =R 100 and = . .I 0 25

(Exercises page 309)

EXAMPLE

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10/19

.1 In th e U n ite d ,S ta te s th e co sts o f e le ctricity is d ire ctly( )p ro p o rtio n a l to th e n u m b e r o f kilo w a tt h o u rs kW h

.u sed If a h ou se h o ld in Ten n e ssee on a ve ra g e u se d3 0 9 8 kW h p er m on th an d h ad an ave rag e m on th ly

. ,e le ctric b ill o f $ 1 7 9 9 9 fin d a m a th e m a tica l m o d e l th a tg ive s th e co st o f e le ctricity in Te n n e sse e in te rm s o f

.th e n u m b e r o f k W h u se d ( )Example 1 page 304 . 2 Hooke s Law states that the force needed to keep a

spring stretched x units beyond its natural length isdirectly proportional .x Here the constant of

.proportionality is called a spring constant)a . Write Hooke s Law as an equation

)b If a spring has a natural length of 10 cm and a forceof 40 N is required to maintain the spring,stretched to a length of 15 cm find the spring

.constant)c What force is needed to keep the spring stretched to

a length of 14cm? ( &Exercise 23 page 191 from Algebra. , & ,Trig by Stewart Redlin Watson 2nd )edition

EXAMPLE

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irect Variation with Powers.Let x and y represent two quantities The following are


, .where k is a nonzero constant

y aries directly with the n h power of x. y is irectly proportional to the n h powerof .x

irect Variation with Powers.Let x and y represent two quantities The following are


, .where k is a nonzero constant

y aries directly with the n h power of x. y is irectly proportional to the n h powerof .x

Definition page 305

nkxy=

Week 3 Day 1


12/19

.1 ( )A brother and sister have weight pounds that varies as( )the cube of the cube of height feet and they share the

.same proportionality constant The sister is 6 feet. .tall and weighs 170 pounds Her brother is 6 4 tallHow much does he weigh?

( )Your Turn page 306

EXAMPLE

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InverseVariation.Let x and y represent two quantities The following are


, where k .is a nonzero constant

y aries inversely with x. yis nversely proportional to .x


InverseVariation.Let x and y represent two quantities The following are


, where k .is a nonzero constant

y aries inversely with x. yis nversely proportional to .x


Definition page 306

x

ky=

Week 3 Day 1

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14/19

.1 The number of potential buyers of a house decreases as(the price of the house increases see the graph on the

).below If the number of potential buyers of a house ina particular city is inversely proportional to the

,price of the house find a mathematical equation thatdescribes the demand for the houses as it relates to.the price How many potential buyers will there be for

a $2million house? ( )Example 3 page 306

EXAMPLE

200 400 600 800

200

600

400

800

1000

( , )100 1000

( , )200 500

( , )400 250

( , )600 167

( )Price of the house in thousands of dollars

(

)

D e

m a

n d

n u

m b

e r

o f

p o

t e

n t

i a

l

bu

y e

r s

Week 3 Day 1

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Inverse ariation with Powersnverse ariation with Powers

Definition page 307

x.ofpowernththeto

alproportioninverselyisyorx,ofpowernththewithinversely

variesythatsaywethen,x

kyequationthebyrelatedareyandxIf

n=

Week 3 Day 1

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oint Variation and Combined Variation When one quantity is proportional to the product of two or

,more other quantities the variation is called oint.ariation :Example Simple interest which is defined as When direct variation and inverse variation occur at the

,same time the variation is called ombined variation . : ,Example Combined gas law in chemistry

oint Variation and Combined Variation When one quantity is proportional to the product of two or

,more other quantities the variation is called oint.ariation :Example Simple interest which is defined as When direct variation and inverse variation occur at the

,same time the variation is called ombined variation . : ,Example Combined gas law in chemistry

Definition page 307

V

T

kP=

tPrI=

Week 3 Day 1

EXAMPL W k 3 D 1


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.1 The gas in the headspace of a soda bottle has a volume

. , ( ),of 9 0 ml pressure of 2 atm atmospheres and a( ).temperature of 298K standard room temperature of 77F,If the soda bottle is stored in a refrigerator the

( ).temperature drops to approximately 279K 42F What isthe pressure of the gas in the headspace once the

bottle is chilled? ( )Example 4 page 308

EXAMPLE

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SUMMARY

Direct, inverse, joint and combined variation can

be used to model the relationship between twoquantities. For two quantities xand ywe say that:

Joint variation occurs when one quantity is directlyproportional to two or more quantities.

Combined variation occurs when one quantity isdirectly proportional to one or more quantities andinversely proportional to one or more otherquantities.

kx.yifxtoalproportiondirectlyisy =

.x

k

yifxtoalproportioninverselyisy=

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CLASSWORK

HOMEWORK

# , , , , , , , , , -s 22 32 33 36 37 39 40 42 43 47 page 309 313

# , , , -s page 20 27 46 53 page 309 310

math 10 week3 ratio

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