MATH10 ALGEBRA

RATIO, VARIATION AND PROPORTION

Week 3 Day 1 Ratio, Variation and Proportion (Algebra and Trigonometry, Young 2nd Edition, page 304-313)

TODAY’S OBJECTIVE

• Use ratio and proportion in solving problems involving them,• Identify the different types of variation,• Understand the difference between direct variation and

inverse variation,• Understand the difference between combined variation and

joint variation, and • Develop mathematical models using direct variation, inverse

variation, combined variation and joint variation.

At the end of the lesson the students are expected to:

Week 3 Day 1

Definition

RATIO

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

Thus,

Definition

RATIO

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

Thus, y

x

yx

y:x

Week 3 Day 1

1. Express the following ratios as simplified fractions:

a) 5 : 20

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

2. Write the following comparisons as ratios reduced to lowest terms. Use 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

EXAMPLEWeek 3 Day 1

Definition

PROPORTION

A proportion is a statement indicating the equality of two ratios.

Thus, , , are proportions.

In the proportion x : y = m : n, x and n are called the extremes, y and m are called the means. x and m are the called the antecedents, y and n are called the consequents.

In the event that the means are equal, they are called the mean proportional.

Definition

PROPORTION

A proportion is a statement indicating the equality of two ratios.

Thus, , , are proportions.

In the proportion x : y = m : n, x and n are called the extremes, y and m are called the means. x and m are the called the antecedents, y and n are called the consequents.

In the event that the means are equal, they are called the mean proportional.

n

m

y

x n:m

y

x n:my:x

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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

Week 3 Day 1

Definition

VARIATION

A variation is the name given to the study of the effects of changes among related quantities.

Variation describes the relationship between variables.

Definition

VARIATION

A variation is the name given to the study of the effects of changes among related quantities.

Variation describes the relationship between variables.

Week 3 Day 1

Direct Variation

When one quantity is a constant multiple of another quantity, we say that the quantities are directly proportional to one another .

Let x and y represent two quantities. The following are equivalent statements:

• y = kx, where k is a nonzero constant.

• y varies directly with x.

• y is directly proportional to x.

The constant k is called the constant of variation or the constant of proportionality.

Direct Variation

When one quantity is a constant multiple of another quantity, we say that the quantities are directly proportional to one another .

Let x and y represent two quantities. The following are equivalent statements:

• y = kx, where k is a nonzero constant.

• y varies directly with x.

• y is directly proportional to x.

The constant k is called the constant of variation or the constant of proportionality.

Definition page 304

Week 3 Day 1

Write an equation that describes each variation.

17. d is directly proportional to t. d=r when t=1.

19. V is directly proportional to both l and w. V=6h when w=3 qnd h=4.

24. W is directly proportional to both R and the square of I. W=4 when R=100 and I=0.25.

(Exercises page 309)

EXAMPLE Week 3 Day 1

1. In the United States, the costs of electricity is directly proportional to the number of kilowatt hours (kWh) used. If a household in Tennessee on average used 3098 kWh per month and had an average monthly electric bill of $179.99, find a mathematical model that gives the cost of electricity in Tennessee in terms of the number of kWh used. (Example 1 page 304)

2. Hooke’s Law states that the force needed to keep a spring stretched x units beyond its natural length is directly 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 force of 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 Week 3 Day 1

Direct Variation with Powers

Let x and y represent two quantities. The following are equivalent statements:

• , where k is a nonzero constant.

• y varies directly with the nth power of x.

• y is directly proportional to the nth power of x.

Direct Variation with Powers

Let x and y represent two quantities. The following are equivalent statements:

• , where k is a nonzero constant.

• y varies directly with the nth power of x.

• y is directly proportional to the nth power of x.

Definition page 305

nkxy

Week 3 Day 1

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” tall. How much does he weigh?

(Your Turn page 306)

EXAMPLE Week 3 Day 1

Inverse Variation

Let x and y represent two quantities. The following are equivalent statements:

• , where k is a nonzero constant.

• y varies inversely with x.

• y is inversely proportional to x.

The constant k is called the constant of variation or the constant of proportionality.

Inverse Variation

Let x and y represent two quantities. The following are equivalent statements:

• , where k is a nonzero constant.

• y varies inversely with x.

• y is inversely proportional to x.

The constant k is called the constant of variation or the constant of proportionality.

Definition page 306

xk

y

Week 3 Day 1

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 in a particular city is inversely proportional to the price of the house, find a mathematical equation that describes the demand for the houses as it relates to the price. How many potential buyers will there be for a $2 million 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)

Dem

and

(num

ber o

f pot

entia

l buy

ers)

Week 3 Day 1

Inverse Variation with PowersInverse Variation with Powers

Definition page 307

x.of power nth the toalproportion inversely is y or x,of power nth the with inversely

varies y that say we then ,xk

y equation the by related are y and xIf n

Week 3 Day 1

Joint Variation and Combined Variation

• When one quantity is proportional to the product of two or more other quantities, the variation is called joint variation.

Example: Simple interest which is defined as • When direct variation and inverse variation occur at the same time,

the variation is called combined variation.

Example: Combined gas law in chemistry,

Joint Variation and Combined Variation

• When one quantity is proportional to the product of two or more other quantities, the variation is called joint variation.

Example: Simple interest which is defined as • When direct variation and inverse variation occur at the same time,

the variation is called combined variation.

Example: Combined gas law in chemistry,

Definition page 307

VT

kP

tPrI

Week 3 Day 1

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 77⁰F). If the soda bottle is stored in a refrigerator, the temperature drops to approximately 279K (42⁰F). What is the pressure of the gas in the headspace once the bottle is chilled?

(Example 4 page 308)

EXAMPLE Week 3 Day 1

SUMMARY

Direct, inverse, joint and combined variation can be used to model the relationship between two quantities. For two quantities x and y we say that:

•

•

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

Combined variation occurs when one quantity is directly proportional to one or more quantities and inversely proportional to one or more other quantities.

kx.yif x to alproportion directly is y

.xk

yif x to alproportion inversely is y

Week 3 Day 1

Week 3 Day 1

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