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Page 1: Direct variations and Indirect variations

Ratio and Proportion

Page 2: Direct variations and Indirect variations

What is a ratio?

Ratios are written with the : symbol.

A ratio is a comparison of two similar quantities.

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The ratio of 6 to 3 is…6Γ·3

63

6 :3

2 :1

(division form)

(fraction form)

(ratio form)

(simplest form)

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A Recipe for pancakes uses 3 cups of flour and 2 cups of milk. What is

the ratio of flour to milk?

πŸ‘ :𝟐

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There are 9 dogs, 5 are boys, and 4 are girls.

What is the ratio of boys to girls?What is the ratio

of boys to all dogs?What is the ratio

of all dogs to girls?

πŸ“ :πŸ’πŸ“ :πŸ—πŸ— :πŸ’

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There are 5 puppies, 2 are boys, and 3 are girls

What is the ratio of boys to girls?What is the ratio of girls to boys?What is the ratio of boys to all puppies?What is the ratio of girls to all puppies?

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What is a proportion?

A proportion is a statement of the equality of two

ratios.¿𝟐 :𝟏𝟏𝟎 :πŸ“

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What is a proportion?

A proportion is a statement of the equality of two

ratios.𝟏𝟎 :πŸ“=𝟐 :𝟏

Means

Extremes

…inner terms in a proportion

…outer terms in a proportionThe product of the means

is equal to the product of the extremes.

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Two ways to solve a proportion…

𝟏𝟐 :πŸ’=πŸ‘ :𝟏

𝟏𝟐¿𝟏𝟐

πŸπŸπŸ’ =

πŸ‘πŸ

𝟏𝟐¿𝟏𝟐

Means & Extremes: Cross-Product Propertyof Proportion:

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Find the fourth term in a proportion if the first three terms are 10, 16 and 20

respectively.

𝟏𝟎 :πŸπŸ”=𝟐𝟎 :π±πŸ‘πŸπŸŽΒΏπŸπŸŽπ±

Let x be the fourth term.

𝐱=πŸ‘πŸ

Check:𝟏𝟎 :πŸπŸ”=𝟐𝟎 :πŸ‘πŸ

πŸ‘πŸπŸŽΒΏπŸ‘πŸπŸŽ The fourth term is 32.

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𝟏𝟐𝐱 ΒΏπŸπŸ–πŸ–

Let x be the height of the

tree.

𝐱=πŸπŸ’ πŸ•πŸπ’™ =

πŸπŸπŸ’

See Keona’s Garden

πŸ•πŸ :𝐱=𝟏𝟐 :πŸ’

The tree is 24 feet high.𝟏𝟐𝐱 ΒΏπŸπŸ–πŸ–π±=πŸπŸ’

𝐱

OTHER WAY…

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In a Book Fair…If three books cost Php 300.00, how much would five books cost?

Answer: Five books wouldcost Php 500.00

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VARIATION CONSTANT – a quantity whose

value does not change.

VARIABLE – quantity whose value is changing.

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Direct Variation(Direct Proportion)

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Direct Variation:π’š=π’Œπ’™

Constant of VariationDependent Variable

Independent Variable

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Direct Variation:π’š=π’Œπ’™

y varies directly as x Y is directly proportional to

x

Read as:

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Step 1: Solve the Constant of Variation.

How much would it cost you to buy 7 similar books if 3 books cost Php300.00?

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Step 2: Substitute the value of k in y=kx.

How much would it cost you to buy 7 similar books if 3 books cost Php300.00?

Use this equation to solve the problem.

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Step 3: Substitute the other given to the new equation.

How much would it cost you to buy 7 similar books if 3 books cost Php300.00?

7 similar books would

costPhp 700.00

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Direct Variation:

As x increases, y increases As x decreases, y

decreases

Number of Books (x) 1 2 3 4 5 6 7Cost of Books

in Php (y)100 200 300 400 500 600 700

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Graphing Direct Variation:Number

of Books

(x)

Cost of Books in Php

(y)

1 1002 2003 3004 4005 5006 600

1 2 3 4 5 6 7 8 9 10

NUMBER OF BOOKS BOUGHT (x)

1000

900800700600500400300200100

COST

OF

BOO

KS IN

Php

(y)

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Solution:

If y varies directly as x, and y = 25 and x = 15, find y if x = 9.

The value of y is 15

if x = 9.

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Solution:

If m is directly proportional to p, and m = 10 and p = 2, find m if p = 8.

The value of m is 40

if p = 8.

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Solution:

The area of a square (A) varies directly as the square of its side (s). If the area of a

square is 16 while the side is 4, find the area whose side is 12.

The area of a

square is 144.

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Solution:

The cost of rice C varies directly as the weight of rice in kilograms w. If 2 kg of rice cost Php 72.00, find the cost of 11 kg rice.

11 kg of rice cost

Php 396.00

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Indirect Variation(Indirect Proportion)

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Inverse Variationy varies inversely as x ifk 0

such that xy=k or y kx

Just as with direct variation, a proportion can be set up solve problems of indirect variation.

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Find y when x=15, if y varies inversely as x and x=10 when y=12Solve by equation:

xyk1012k120k

xyk15y 120y 8

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If N varies inversely as M, and N = 5 when M= 3, find N when M= 10.

(10)n = 15 (3)(5) = k

15 = k

Solveby equation: mn = k10n = 15

n = 3/2n = 15/1010 = 10

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The measure of the central angle and the number of sides of regular polygons are tabulated below.

Number of sides

n4 5 6 8 9 10

Central angle ΞΈ

90Β° 72Β° 60Β°

45Β°

40Β° 36Β°a.What is the constant of variation?b.What is the value of ΞΈ where n = 36?

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a. What is the constant of variation?

nΞΈ = kIf n = 4 when ΞΈ = 90Β°, find k.(4)(90Β°) = 360Β°

k = 360Β°

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b. What is the value of ΞΈ where n = 36?

nθ = k(36) Θ = 360°

36 36Θ = 10°