Transcript Ratio and Proportions

Ratio and Proportions
Ratio of a to b
• The quotient a/b if a and b are 2 quantities
that are measured in the same units can
also be written as a:b.
* b cannot = 0, because the denominator
cannot be 0.
• Always write ratios in simplified form!
(reduce the fraction!)
Ratios
• A ratio is a comparison of numbers that can be
expressed as a fraction.
• If there were 18 boys and 12 girls in a class, you
could compare the number of boys to girls by
saying there is a ratio of 18 boys to 12 girls. You
could represent that comparison in three different
ways:
– 18 to 12
– 18 : 12
– 18
12
3
Ratios
• The ratio of 18 to 12 is another way to
represent the fraction
• All three representations
are
equal.
18
18
12
– 18 to 12 = 18:12 =12
• The first operation to perform on a ratio is
to reduce it to lowest terms
– 18:12 =
– 18:12 = 3:2
18
12
3
2
÷6
÷6
3
2
4
Example: simplify.
12 ft 1

24 ft 2
3 yd 3 ft 9 3

 
6 ft 1 yd 6 2
6 ft 12in 72 4

  4
18in 1 ft 18 1
Ratios
• A basketball team wins 16 games
and loses 14 games. Find the
reduced ratio of:
16
8
=
14
7
14
7
– Losses to wins – 14:16 =
=
16
8
– Wins to losses – 16:14 =
– Wins to total games played –
16:30 =16 = 8
30
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• The order of the numbers is critical
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Example: the perimeter of an isosceles
 is 56in. The ratio of LM:MN is 5:4.
find the lengths of the sides of the .
L
5x +5x+ 4x=56
5x
14x=56
x=4
M
N
4x
Ex: the measure of the s in a  are
in the extended ratio 3:4:8. Find the
measures of the s of the .
3x+4x+8x=180
15x=180
x=12
Substitute to find the angles:
3(12)=36, 4(12)=48, 8(12)= 96
Angle measures: 36o, 48o, 96o
Proportion
• An equation stating 2 ratios are =
• b and c are the means
• a and d are the extremes
a c

b d
Proportions
• A proportion is a statement that one
ratio is equal to another ratio.
–
–
–
–
Ex: a ratio of 4:8 = a ratio of 3:6
4:8 = 4 = 1 and 3:6 = 3 = 1
8
2
6
2
4:8 = 3:6
4 = 3
8
6
– These ratios form a proportion since
they are equal to other. 1 = 1
2
2
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Properties of Proportions
• Cross product property- means=extremes
1. If a  c then, ad=bc
b
d
• Reciprocal Property- (both ratios must be
flipped)
a
c
b
d
2. If  , then 
b
d
a
c
Proportions
• In a proportion, you will notice
that if you cross multiply the
terms of a proportion, those
cross-products are equal.
4
8
=
3
6
3
2
=
18
12 3 x 12 = 2 x 18 (both equal1236)
4 x 6 = 8 x 3 (both equal 24)
Proportions
N
12
=
3
4
4 x N = 12 x 3 Cross multiply the proportion
4N = 36
4N
4
36
=
4
Divide the terms on both sides
of the equal sign by the
number next to the unknown
letter. (4)
1N = 9
N=9
That will leave the N on the
left side and the answer (9) on
the right side
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Proportions
• Solve for N
• Solve for N
2 = N
5
35
5 x N = 2 x 35
15
N
= 3
4
5 n = 70
6
7
= 102
N
5N
5
4
N
= 6
27
=70
5
1N = 14
N = 14
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Proportions
• At 2 p.m. on a sunny day, a 5 ft
woman had a 2 ft shadow, while
a church steeple had a 27 ft
shadow. Use this information to
find the height of the steeple.
5
2
= H
27
• 2 x H = 5 x 27
• 2H = 135
• H = 67.5 ft.
height =
shadow
height
shadow
You must be careful to place
the same quantities in
corresponding positions in
the proportion
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Ratios
• The ratio of freshman to sophomores
in a drama club
is 5:6.
There are 18 sophomores in the drama
club.
How many freshmen are there?
Freshman
Sophomore
=
5
6
15 freshmen
=
x
18
Example
10(s-5)=4s
10s-50=4s
-50= -6s
25
s
3
s 5 s

4
10
The ratios of the side lengths of 
QRS to the corresponding side lengths
of  VTU are 3:2. Find the unknown
length.
V
Q
z
y
X
S
18cm
2cm
T
u
w
R
Example cont
x
3

x= 3cm
2
2
18 3

w 2
a2+b2=c 2
32+182=y2
9+324=y2
333=y2
22+122=z2
y  333
y ≈ 18.25cm
w= 12cm
4+144=z2
148=z2
z  148
z ≈ 12.17cm