Notes : Depreciation, Provisions & Reserves - 1 Class 11 Notes | EduRev

Class 11 : Notes : Depreciation, Provisions & Reserves - 1 Class 11 Notes | EduRev

 Page 1


3/16/2011
1
8.1 Ratio and Proportion
Objectives
8.1 Ratio & Proportion
* Find and simplify the ratio of two 
numbers.
* Use proportions to solve real-life 
problems, such as computing the width 
of a painting.
Computing Ratios
? If a and b are two quantities that are measured 
in the same units, then the ratio of “a” to “b” 
is a/b.  The ratio of a to b can also be written 
as a:b.  Because a ratio is a quotient (meaning 
division), its denominator cannot be zero 
(can’t divide by zero).  Ratios are usually 
expressed in simplified form.  For instance, 
the ratio of 6:8 is usually simplified to 3:4. 
(You divided by 2)
8.1 Ratio & Proportion
Ex. 1:  Simplifying Ratios
? Simplify the ratios:
a. 12 cm b.  6 ft c.  9 in.
4cm 18 ft 18 in. 4 cm 18 ft 18 in.
OR
a. 12:4 b. 6:18 c. 9:18
The fraction and ratio are both ways to express 
ratios
8.1 Ratio & Proportion
Ex. 1:  Simplifying Ratios
? Simplify the ratios:
a. 12 cm b.  6 ft
4 m 18 in
Solution:  To simplify the ratios with unlike 
units, convert to like units so that the units 
divide out.  Then simplify the fraction, if 
possible.  
8.1 Ratio & Proportion
Ex. 1:  Simplifying Ratios
? Simplify the ratios:
a. 12 cm
4m 4 m
12 cm 12 cm 12 . 3
4 m 4·100cm     400 100
3:100
8.1 Ratio & Proportion
Ex. 1:  Simplifying Ratios
? Simplify the ratios:
b.  6 ft
18 in 18 in
6 ft 6·12 in 72 in. 4 4
18 in 18 in. 18 in. 1    
8.1 Ratio & Proportion
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3/16/2011
1
8.1 Ratio and Proportion
Objectives
8.1 Ratio & Proportion
* Find and simplify the ratio of two 
numbers.
* Use proportions to solve real-life 
problems, such as computing the width 
of a painting.
Computing Ratios
? If a and b are two quantities that are measured 
in the same units, then the ratio of “a” to “b” 
is a/b.  The ratio of a to b can also be written 
as a:b.  Because a ratio is a quotient (meaning 
division), its denominator cannot be zero 
(can’t divide by zero).  Ratios are usually 
expressed in simplified form.  For instance, 
the ratio of 6:8 is usually simplified to 3:4. 
(You divided by 2)
8.1 Ratio & Proportion
Ex. 1:  Simplifying Ratios
? Simplify the ratios:
a. 12 cm b.  6 ft c.  9 in.
4cm 18 ft 18 in. 4 cm 18 ft 18 in.
OR
a. 12:4 b. 6:18 c. 9:18
The fraction and ratio are both ways to express 
ratios
8.1 Ratio & Proportion
Ex. 1:  Simplifying Ratios
? Simplify the ratios:
a. 12 cm b.  6 ft
4 m 18 in
Solution:  To simplify the ratios with unlike 
units, convert to like units so that the units 
divide out.  Then simplify the fraction, if 
possible.  
8.1 Ratio & Proportion
Ex. 1:  Simplifying Ratios
? Simplify the ratios:
a. 12 cm
4m 4 m
12 cm 12 cm 12 . 3
4 m 4·100cm     400 100
3:100
8.1 Ratio & Proportion
Ex. 1:  Simplifying Ratios
? Simplify the ratios:
b.  6 ft
18 in 18 in
6 ft 6·12 in 72 in. 4 4
18 in 18 in. 18 in. 1    
8.1 Ratio & Proportion
3/16/2011
2
Ex. 2:  Using Ratios
? The perimeter of 
rectangle ABCD is 60 
centimeters.  The ratio 
of AB: BC is 3:2 Find
B
C
of AB: BC is 3:2.  Find 
the length and the 
width of the rectangle
w
l
A D
8.1 Ratio & Proportion
Ex. 2:  Using Ratios
? SOLUTION:  Because 
the ratio of AB:BC is 
3:2, you can represent 
the length of AB as 3x
B
C
the length of AB as 3x 
and the width of BC as 
2x.  
w
l
A D
8.1 Ratio & Proportion
Solution:
Statement
2l + 2w = P
2(3x) + 2(2x) = 60
Reason
Formula for perimeter of a rectangle
Substitute l, w and P
6x + 4x = 60
10x = 60
x = 6
Multiply
Combine like terms
Divide each side by 10
So, ABCD has a length of 18 centimeters and a width of 12 cm.  
8.1 Ratio & Proportion
Ex. 3:  Using Extended Ratios
? The measures of the angles 
in ?JKL are in the 
extended ratio 1:2:3.  
Find the measures of the 
l
K
2x°
angles.
? Begin by sketching a 
triangle.  Then use the 
extended ratio of 1:2:3 to 
label the measures of 
the angles as x°, 2x°, and 
3x°.
J
L
x°
3x°
8.1 Ratio & Proportion
Solution:
Statement
x°+ 2x°+ 3x° = 180°
6x = 180
Reason
Triangle Sum Theorem
Combine like terms
x = 30 Divide each side by 6
So, the angle measures are 30°, 2(30°) = 60°, and 3(30°) = 90°.  
8.1 Ratio & Proportion
Ex. 4:  Using Ratios
? The ratios of the side 
lengths of ?DEF to 
the corresponding 
side lengths of ?ABC
3 in.
F
C
side lengths of ?ABC 
are 2:1.  Find the 
unknown lengths.
8 in.
D E
A B
8.1 Ratio & Proportion
Page 3


3/16/2011
1
8.1 Ratio and Proportion
Objectives
8.1 Ratio & Proportion
* Find and simplify the ratio of two 
numbers.
* Use proportions to solve real-life 
problems, such as computing the width 
of a painting.
Computing Ratios
? If a and b are two quantities that are measured 
in the same units, then the ratio of “a” to “b” 
is a/b.  The ratio of a to b can also be written 
as a:b.  Because a ratio is a quotient (meaning 
division), its denominator cannot be zero 
(can’t divide by zero).  Ratios are usually 
expressed in simplified form.  For instance, 
the ratio of 6:8 is usually simplified to 3:4. 
(You divided by 2)
8.1 Ratio & Proportion
Ex. 1:  Simplifying Ratios
? Simplify the ratios:
a. 12 cm b.  6 ft c.  9 in.
4cm 18 ft 18 in. 4 cm 18 ft 18 in.
OR
a. 12:4 b. 6:18 c. 9:18
The fraction and ratio are both ways to express 
ratios
8.1 Ratio & Proportion
Ex. 1:  Simplifying Ratios
? Simplify the ratios:
a. 12 cm b.  6 ft
4 m 18 in
Solution:  To simplify the ratios with unlike 
units, convert to like units so that the units 
divide out.  Then simplify the fraction, if 
possible.  
8.1 Ratio & Proportion
Ex. 1:  Simplifying Ratios
? Simplify the ratios:
a. 12 cm
4m 4 m
12 cm 12 cm 12 . 3
4 m 4·100cm     400 100
3:100
8.1 Ratio & Proportion
Ex. 1:  Simplifying Ratios
? Simplify the ratios:
b.  6 ft
18 in 18 in
6 ft 6·12 in 72 in. 4 4
18 in 18 in. 18 in. 1    
8.1 Ratio & Proportion
3/16/2011
2
Ex. 2:  Using Ratios
? The perimeter of 
rectangle ABCD is 60 
centimeters.  The ratio 
of AB: BC is 3:2 Find
B
C
of AB: BC is 3:2.  Find 
the length and the 
width of the rectangle
w
l
A D
8.1 Ratio & Proportion
Ex. 2:  Using Ratios
? SOLUTION:  Because 
the ratio of AB:BC is 
3:2, you can represent 
the length of AB as 3x
B
C
the length of AB as 3x 
and the width of BC as 
2x.  
w
l
A D
8.1 Ratio & Proportion
Solution:
Statement
2l + 2w = P
2(3x) + 2(2x) = 60
Reason
Formula for perimeter of a rectangle
Substitute l, w and P
6x + 4x = 60
10x = 60
x = 6
Multiply
Combine like terms
Divide each side by 10
So, ABCD has a length of 18 centimeters and a width of 12 cm.  
8.1 Ratio & Proportion
Ex. 3:  Using Extended Ratios
? The measures of the angles 
in ?JKL are in the 
extended ratio 1:2:3.  
Find the measures of the 
l
K
2x°
angles.
? Begin by sketching a 
triangle.  Then use the 
extended ratio of 1:2:3 to 
label the measures of 
the angles as x°, 2x°, and 
3x°.
J
L
x°
3x°
8.1 Ratio & Proportion
Solution:
Statement
x°+ 2x°+ 3x° = 180°
6x = 180
Reason
Triangle Sum Theorem
Combine like terms
x = 30 Divide each side by 6
So, the angle measures are 30°, 2(30°) = 60°, and 3(30°) = 90°.  
8.1 Ratio & Proportion
Ex. 4:  Using Ratios
? The ratios of the side 
lengths of ?DEF to 
the corresponding 
side lengths of ?ABC
3 in.
F
C
side lengths of ?ABC 
are 2:1.  Find the 
unknown lengths.
8 in.
D E
A B
8.1 Ratio & Proportion
3/16/2011
3
Ex. 4:  Using Ratios
? SOLUTION:
? DE is twice AB and DE = 
8, so AB = ½(8) = 4
? Use the Pythagorean 
Theorem to determine
3 in.
F
C
5in
Theorem to determine 
what side BC is.
? DF is twice AC and AC = 
3, so DF = 2(3) = 6
? EF is twice BC and BC = 
5, so EF = 2(5) or 10 
8 in.
D E
A B 4 in
a
2
+ b
2
= c
2
3
2
+ 4
2
= c
2
9 + 16 = c
2
25 = c
2
5 = c
8.1 Ratio & Proportion
6in
Using Proportions
? An equation that 
equates two ratios is 
called a proportion.  
For instance if the
? = ?
Means Extremes
For instance, if the 
ratio of a/b is equal to 
the ratio c/d; then the 
following proportion 
can be written: 
? = ?
The numbers a and d are the 
extremes of the proportions.  
The numbers b and c are the 
means of the proportion.
8.1 Ratio & Proportion
Properties of proportions
1. CROSS PRODUCT PROPERTY.  The 
product of the extremes equals the product of 
the means.
If
? = ?, then ad = bc
8.1 Ratio & Proportion
Properties of proportions
2. RECIPROCAL PROPERTY.  If two ratios 
are equal, then their reciprocals are also 
equal.
If ? = ?, then    = ?
b
a
To solve the proportion, you find the 
value of the variable.
8.1 Ratio & Proportion
Ex. 5:  Solving Proportions
4
x
5
7
=
Write the original 
proportion.
Reciprocal prop.
Multiply each side by 4
Simplify.
x
4
7
5
= 44
x =
28
5
8.1 Ratio & Proportion
Ex. 5:  Solving Proportions
3
y + 2
2
y
=
Write the original 
proportion.
y y
Cross Product prop.
Distributive Property
Subtract 2y from each 
side.
3y = 2(y+2)
y =
4
3y = 2y+4
8.1 Ratio & Proportion
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