Direct variations and Indirect variations

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Transcript of Direct variations and Indirect variations
Ratio and Proportion
What is a ratio?
Ratios are written with the : symbol.
A ratio is a comparison of two similar quantities.
The ratio of 6 to 3 is…6÷3
63
6 :3
2 :1
(division form)
(fraction form)
(ratio form)
(simplest form)
A Recipe for pancakes uses 3 cups of flour and 2 cups of milk. What is
the ratio of flour to milk?
𝟑 :𝟐
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?
𝟓 :𝟒𝟓 :𝟗𝟗 :𝟒
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?
What is a proportion?
A proportion is a statement of the equality of two
ratios.¿𝟐 :𝟏𝟏𝟎 :𝟓
What is a proportion?
A proportion is a statement of the equality of two
ratios.𝟏𝟎 :𝟓=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.
Two ways to solve a proportion…
𝟏𝟐 :𝟒=𝟑 :𝟏
𝟏𝟐¿𝟏𝟐
𝟏𝟐𝟒 =
𝟑𝟏
𝟏𝟐¿𝟏𝟐
Means & Extremes: CrossProduct Propertyof Proportion:
Find the fourth term in a proportion if the first three terms are 10, 16 and 20
respectively.
𝟏𝟎 :𝟏𝟔=𝟐𝟎 :𝐱𝟑𝟐𝟎¿𝟏𝟎𝐱
Let x be the fourth term.
𝐱=𝟑𝟐
Check:𝟏𝟎 :𝟏𝟔=𝟐𝟎 :𝟑𝟐
𝟑𝟐𝟎¿𝟑𝟐𝟎 The fourth term is 32.
𝟏𝟐𝐱 ¿𝟐𝟖𝟖
Let x be the height of the
tree.
𝐱=𝟐𝟒 𝟕𝟐𝒙 =
𝟏𝟐𝟒
See Keona’s Garden
𝟕𝟐 :𝐱=𝟏𝟐 :𝟒
The tree is 24 feet high.𝟏𝟐𝐱 ¿𝟐𝟖𝟖𝐱=𝟐𝟒
𝐱
OTHER WAY…
In a Book Fair…If three books cost Php 300.00, how much would five books cost?
Answer: Five books wouldcost Php 500.00
VARIATION CONSTANT – a quantity whose
value does not change.
VARIABLE – quantity whose value is changing.
Direct Variation(Direct Proportion)
Direct Variation:𝒚=𝒌𝒙
Constant of VariationDependent Variable
Independent Variable
Direct Variation:𝒚=𝒌𝒙
y varies directly as x Y is directly proportional to
x
Read as:
Step 1: Solve the Constant of Variation.
How much would it cost you to buy 7 similar books if 3 books cost Php300.00?
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.
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
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
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)
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.
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.
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.
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
Indirect Variation(Indirect Proportion)
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.
Find y when x=15, if y varies inversely as x and x=10 when y=12Solve by equation:
xyk1012k120k
xyk15y 120y 8
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
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 θ
90° 72° 60°
45°
40° 36°a.What is the constant of variation?b.What is the value of θ where n = 36?
a. What is the constant of variation?
nθ = kIf n = 4 when θ = 90°, find k.(4)(90°) = 360°
k = 360°
b. What is the value of θ where n = 36?
nθ = k(36) Θ = 360°
36 36Θ = 10°