Transcript 7.1 Ratios and Proportions - William H. Peacock, LCDR USN

Ratios and Proportions
Section 7.1
Objective

Use ratios and proportions.
Key Vocabulary





Ratio
Proportion
Extremes
Means
Cross products
What is a Ratio?
A ratio is a comparison of two quantities using
division.
 It can be expressed as a to b, a : b, or as a
fraction a/b where b ≠ 0.
 Order is important!



Part: Part
Part: Whole
Whole: Part
Ratio

Example:



Your school’s basketball team has won 7 games
and lost 3 games. What is the ratio of wins to
losses?
Because we are comparing wins to losses the first
number in our ratio should be the number of wins
and the second number is the number of losses.
games won 7 games 7


The ratio is
games lost 3 games 3
Ratio

Ratios are usually expressed in simplified
form.



For instance, the ratio of 6:8 is usually simplified
to 3:4.
Divide out common factors between the
numerator and the denominator.
A ratio in which the denominator is 1 is called a
unit ratio.
15 5
 (unit ratio)
 Example:
3 1
Example 1
The total number of students who participate in sports programs
at Central High School is 520. The total number of students in
the school is 1850. Find the athlete-to-student ratio to the
nearest tenth.
To find this ratio, divide the number of athletes by the total
number of students.
0.3 can be written as
Answer: The athlete-to-student ratio is 0.3.
Your Turn
The country with the longest school year is China with 251
days. Find the ratio of school days to total days in a year for
China to the nearest tenth. (Use 365 as the number of days in a
year.)
A. 0.3
B. 0.5
C. 0.7
D. 0.8
Example 2
Simplify the ratio.
a. 60 cm : 200 cm
b.
3 ft
18 in.
SOLUTION
a. 60 cm : 200 cm can be written as the fraction
60 cm
60 ÷ 20
200 cm = 200 ÷ 20
=
3
10
60 cm
.
200 cm
Divide numerator and denominator
by their greatest common factor, 20.
Simplify. 3 is read as “3 to 10.”
10
Example 2
b.
3 ft
= 3 · 12 in.
18 in.
18 in.
36 in.
=
18 in.
36 ÷ 18
=
18 ÷ 18
2
=
1
Substitute 12 in. for 1 ft.
Multiply.
Divide numerator and denominator
by their greatest common factor, 18.
Simplify.
2
is read as “2 to 1.”
1
Example 3
In the diagram, AB : BC is 4 : 1 and AC = 30.
Find AB and BC.
SOLUTION
Let x = BC. Because the ratio of AB to BC
is 4 to 1, you know that AB = 4x.
AB + BC = AC
4x + x = 30
5x = 30
x=6
Segment Addition Postulate
Substitute 4x for AB, x for BC, and 30 for AC.
Add like terms.
Divide each side by 5.
Example 3
To find AB and BC, substitute 6 for x.
AB = 4x = 4 · 6 = 24
ANSWER
BC = x = 6
So, AB = 24 and BC = 6.
Example 4
The perimeter of a rectangle is 80 feet. The ratio of the
length to the width is 7 : 3. Find the length and the width
of the rectangle.
SOLUTION
The ratio of length to width is 7 to 3. You can
let the length l = 7x and the width w = 3x.
2l + 2w = P
2(7x) + 2(3x) = 80
14x + 6x = 80
20x = 80
x=4
Formula for the perimeter of a rectangle
Substitute 7x for l, 3x for w, and 80 for P.
Multiply.
Add like terms.
Divide each side by 20.
Example 4
To find the length and width of the rectangle,
substitute 4 for x.
l = 7x = 7 · 4 = 28
ANSWER
w = 3x = 3 · 4 = 12
The length is 28 feet, and the width is
12 feet.
Your Turn:
1. In the diagram, EF : FG is 2 : 1 and EG = 24.
Find EF and FG.
ANSWER
EF = 16; FG = 8
2. The perimeter of a rectangle is 84 feet. The ratio of
the length to the width is 4 : 3. Find the length and
the width of the rectangle.
ANSWER
length, 24 ft; width, 18 ft
Your Turn:

The
perimeter
of
rectangle ABCD is 60
centimeters. The ratio
of AB: BC is 3:2. Find
the length and the
width of the rectangle
B
C
w
A
l
D
Solution:

Because the ratio of
AB:BC is 3:2, you can
represent the length of
AB as 3x and the width
of BC as 2x.
B
C
w
A
l
D
Solution:
Statement
2l + 2w = P
2(3x) + 2(2x) = 60
6x + 4x = 60
10x = 60
x=6
Reason
Formula for perimeter of a rectangle
Substitute l, w and P
Multiply
Combine like terms
Divide each side by 10
So, ABCD has a length of 18 centimeters and a width of 12 cm.
Extended Ratios


Ratios can also be used to compare 3 or more
quantities, these are called extended ratios.
The extended ratio a:b:c means that the ratio
of the first two quantities is a:b, the ratio of
the last two quantities is b:c, and the ratio of
the first and last quantities is a:c.
Example 5
In ΔEFG, the ratio of the measures of the angles is 5:12:13,
and the perimeter is 90 centimeters. Find the measures of the
angles.
5
___
5x
______
Just as the ratio 12 or 5:12 is equivalent to 12x or
5x:12x, the extended ratio 5:12:13 can be written as
5x:12x:13x. Write and solve an equation to find the value of
x.
Example 5
5x + 12x + 13x = 180
30x = 180
x =6
Triangle Sum Theorem
Combine like terms.
Divide each side by 30.
Answer: So, the measures of the angles are 5(6) or 30, 12(6)
or 72, and 13(6) or 78.
Your Turn
The ratios of the angles in ΔABC is 3:5:7. Find the
measure of the angles.
A. 30, 50, 70
B. 36, 60, 84
C. 45, 60, 75
D. 54, 90, 126
Important Ratios
Find the first 13 terms in the following sequence:
1, 1, 2, 3, 5, 8, …
This is called the Fibonacci Sequence!
Important Ratios
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!
Important Ratios
1 5
You approach the ratio
2
4
fx =
3
1+ 5
2
2
1
2
4
6
8
10
12
The Golden Ratio
1 5
This ratio,
, is called the Golden Ratio.
2
The Golden Ratio = 1.61803398…
The Golden Ratio


The ancient Greeks considered the Golden Ratio when used
in shapes the most aesthetically pleasing to the eye.
The Greeks used the Golden Ratio to do everything from
making a pentagram, to constructing a building, to combing
their hair.
Cartoon
What’s a Proportion?

A proportion is an equation stating that two
ratios are equal.

Example:
a c

b d
Proportions


If the ratio of a/b is equal
to the ratio c/d; then the
following proportion can
be written:
The values a and d are the
extremes. The values b and
c are the means. When the
proportion is written as a:b
= c:d, the extremes are in
the first and last positions.
The means are in the two
middle positions.
Means
Extremes
=
Properties of Proportions
CROSS PRODUCT PROPERTY
1. The product of the extremes equals the
product of the means.
If
then ad  bc.
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
To solve a proportion involving a variable,
simply set the two cross products equal to
each other. Then solve!
275  25
15 x
1525
275x
375 275x
1.36  x
More Proportion Properties
Example 6
A.
Original proportion
Cross Products
Multiply.
y = 27.3
Answer: y = 27.3
Divide each side by 6.
Example 6
B.
Original proportion
Cross Products
Simplify.
Add 30 to each side.
Divide each side by 24.
Answer: x = –2
Your Turn
A.
A. b = 0.65
B. b = 4.5
C. b = –14.5
D. b = 147
Your Turn
B.
A. n = 9
B. n = 8.9
C. n = 3
D. n = 1.8
Example 7
y+2
5
=
Solve the proportion
.
3
6
SOLUTION
y+2
5
=
3
6
5 · 6 = 3(y + 2)
30 = 3y + 6
30 – 6 = 3y + 6 – 6
24 = 3y
24 3y
=
3
3
8=y
Write original proportion.
Cross product property
Multiply and use distributive property.
Subtract 6 from each side.
Simplify.
Divide each side by 3.
Simplify.
Example 7
CHECK
Solve a Proportion
Check your solution by substituting 8 for y.
y + 2 8 + 2 10 5
=
=
=
6
6
3
6
Your Turn:
Solve the proportion.
3 = 6
1. x
8
ANSWER
4
5
15
= y
3
ANSWER
9
m+2
14
3.
=
5
10
ANSWER
5
2.
PROPORTIONS CAN BE USED
TO MAKE PREDICTIONS
Example 8
PETS Monique randomly surveyed 30 students from her
class and found that 18 had a dog or a cat for a pet. If there
are 870 students in Monique’s school, predict the total
number of students with a dog or a cat.
Write and solve a proportion that compares the number of
students who have a pet to the number of students in the
school.
Example 8
18 ● 870 = 30x
15,660 = 30x
522 = x
←
Students who have a pet
←
total number of students
Cross Products Property
Simplify.
Divide each side by 30.
Answer: Based on Monique's survey, about 522 students at
her school have a dog or a cat for a pet.
Your Turn
Brittany randomly surveyed 50 students and found that
20 had a part-time job. If there are 810 students in
Brittany's school, predict the total number of students
with a part-time job.
A. 324 students
B. 344 students
C. 405 students
D. 486 students
Equivalent Ratios

Simplify the following ratios:



4 to 8
10 to 8
8 to 10
4 = 4/4
8
8/4
=1
= 1 to 2
2
GCF = 4
Step 1 – Write the ratio as a fraction
Step 2 – Simplify the fraction (Find the greatest common factor (GCF) of
both numbers and divide the numerator and denominator by the GCF).
Step 3 – Write the equivalent ratio in the same form as the question
Equivalent Ratios can be formed by
multiplying the ratio by any number.

For example, the ratio 2 : 3 can also be
written as



4 : 6 (multiply original ratio by by 2)
6 : 9 (multiply original ratio by by 3)
8 : 12 (multiply original ratio by by 4)
The ratio 2 : 3 can be expressed as
2x to 3x (multiply the original ratio by any
number x)
Equivalent Proportions


Equivalent forms of a proportion all have the
same cross product.
The following proportions are equivalent.
a c b d a b c d
 ,  ,  , 
b d a c c d a b

Examples:
28 x 50 755 28 50 x 755

,

,

,

50 755 28 x
x 755 28 50
Assignment

Pg. 361 – 363 #1 – 55 odd