Download 6.1 Ratio and Proportion

6.1 Ratio and Proportion
Objectives:
1. To recognize and use ratios and
proportions to solve problems
Example 1
Find the first 13 terms in the following
sequence:
1, 1, 2, 3, 5, 8, …
This is called the Fibonacci Sequence!
What is a Ratio?
A ratio is a comparison of two quantities,
usually by division.
a
• The ratio of a to b is a:b or b
• Order is important!
– Part: Part
– Part: Whole
– Whole: Part
– Units, sometimes important
Example 2
In a survey of American families, 150
families had a total of 360 children. What
is the ratio of children to families? On
average, how many children are there per
family?
Example 3
What happens when you take the ratios of
two successive Fibonacci numbers, larger
over smaller? What number do you
approach?
1 2 3 5 8 13 21
1 1 2 3 5 8 13
ETC!
The Golden Ratio
What happens when you take the ratios of two
successive Fibonacci numbers, larger over
smaller? What number do you approach?
4
fx =
3
1+ 5
2
2
1
2
4
6
8
10
12
The Golden Ratio
What happens when you take the ratios of two
successive Fibonacci numbers, larger over
smaller? What number do you approach?
It’s the Golden Ratio =
1.61803398…
Foxtrot
What’s a Proportion?
When two ratios are equal, it’s called a
proportion.
• What’s an example of a proportion? What
ratio is equal to ½?
• Proportions are often used in solving
problems involving similar objects.
Solving a Proportion
What’s the relationship between the cross
products of a proportion?
2.4150
360  2.4
150 1
They’re equal!
3601
Solving a Proportion
Cross Products Property
In a proportion, the product of the extremes
equals the product of the means.
Solving a Proportion
To solve a proportion involving a variable,
simply set the two cross products equal to
each other. Then solve!
275  25
1525
275x
15 x
375 275x
1.36  x
Example 4
Solve the proportion.
26  x
50 75
More Proportion Properties
Exercise 1
If you work for 2 weeks and earn $380, what
will you expect to earn in 15 weeks?
Exercise 1
If you work for 2 weeks and earn $380, what
will you expect to earn in 15 weeks?
2
15
=
380
𝑥
2𝑥 = 5700
𝑥 = $2,850
Exercise 2
Solve for y:
1
2

y 1 3y
Exercise 2
Solve for y:
1
2
=
𝑦 + 1 3𝑦
3𝑦 = 2 𝑦 + 1
3𝑦 = 2y + 2
𝑦=2
Exercise 3
Which is longer: a yardstick or a meter stick?
(Use the conversion factor 1 in. = 2.54 cm)
Exercise 3
Which is longer: a yardstick or a meter stick?
(Use the conversion factor 1 in. = 2.54 cm)
1 𝑦𝑑 = 36 𝑖𝑛 vs. 1 𝑚 = 100 𝑐𝑚
1 𝑖𝑛
𝑥 𝑖𝑛
=
2.54 𝑐𝑚 100 𝑐𝑚
100 = 2.54𝑥
39.37 (𝑖𝑛𝑐ℎ𝑒𝑠 𝑝𝑒𝑟 𝑚𝑒𝑡𝑒𝑟) = 𝑥
so a meter stick is longer
Exercise 4
The sides of a rose garden in the shape of a
right triangle are in the ratio of 8:15:17. If
the perimeter is 60 ft, what is the length of
the shortest side?
Exercise 4
The sides of a rose garden in the shape of a
right triangle are in the ratio of 8:15:17. If
the perimeter is 60 ft, what is the length of
the shortest side?
8𝑥 + 15𝑥 + 17𝑥 = 60
40𝑥 = 60
𝑥 = 1.5
Shortest side = 8(1.5) = 12 ft
The Greeks, Again!
The Greeks used the Golden Ratio to do
everything from making a pentagram, to
constructing a building, to combing their
hair.
The Golden Rectangle
If you make a rectangle with sides that have
the Golden Ratio, you’ve made a sparkly
Golden Rectangle.
s
l
The Golden Rectangle
This happens precisely when the ratio of the
long side to the short side is equal to the
ratio of the sum of the sides to the long
side.
l ls

s
l
s
l
Example 5
Assume that the smaller side of a Golden
Rectangle is 1. Use algebra to find the
exact value of phi, the Golden Ratio.
x x 1

1
x
s
1 5

2
l
Extra Credit Opportunities
1. Construct a Golden
Rectangle with a
compass and
straightedge and
explain how it
demonstrates the
Golden Ratio
2. Construct a
pentagram with a
compass and
straightedge and
explain how it
demonstrates the
Golden Ratio