MATHEMATICS

1. Ratios and Proportions

2. What is a Ratio?
• A ratio is a comparison of two numbers.
• Ratios can be written in three different ways:
a to b
a:b
Because a ratio is a fraction, b can not be zero
b
a
Ratios are expressed in simplest form

3. How to simplify ratios?
• The ratios we saw on last
slide were all simplified.
How was it done?
b
a
Ratios can be expressed
in fraction form…
This allows us to do math
on them.
The ratio of boys and girls in the
class is
The ratio of the rectangle is
The ratio of cats and dogs in my
house is
11
12
b
a 1
4
1
2

4. How to simplify ratios?
• Now I tell you I have 12 cats and 6 dogs. Can you
simplify the ratio of cats and dogs to 2 to 1?
6
12 =
6
/
6
6
/
12 =
1
2
Divide both numerator and
denominator by their
Greatest Common Factor 6.

5. How to simplify ratios?
A person’s arm is 80cm, he is 2m tall.
Find the ratio of the length of his arm to his total height
m
cm
2
80

cm
cm
200
80
200
80 
5
2
To compare them, we need to convert both
numbers into the same unit …either cm or m.
• Let’s try cm first!

height
arm

Once we have the
same units, we can
simplify them.

6. How to simplify ratios?
• Let’s try m now!
height
arm
m
cm
2
80

m
m
2
8
.
0

Once we have the
same units, they
simplify to 1.
20
8

5
2

To make both numbers
integers, we multiplied both
numerator and denominator by
10

7. Now, on to proportions!
d
c
b
a

What is a proportion?
A proportion is an equation
that equates two ratios
The ratio of males and females was 3/2
The ratio of males and females now is 6/4=3/2
So we have a proportion :
4
6
2
3


8. Properties of a proportion?
4
6
2
3

2x6=12 3x4 = 12
3x4 = 2x6
Cross Product Property

9. Properties of a proportion?
d
c
b
a

• Cross Product Property
ad = bc
means
extremes

10. How about an example?
6
2
7 x
 Solve for x:
7(6) = 2x
42 = 2x
21 = x
Cross Product Property

11. How about another example?
x
12
2
7
 Solve for x:
7x = 2(12)
7x = 24
x =
7
24
Cross Product Property

12. Can you solve this one?
x
x
3
1
7


Solve for x:
7x = (x-1)3
7x = 3x – 3
4x = -3
x =
Cross Product Property
4
3


13. Now you know enough about properties,
let’s solve the some problems!
gal
x
miles
gal
miles
_
)
5
5
(
1
30 

x
10
1
30

If your motorcycle gets 30 miles/gallon, how many
gallons of gas do you need to commute to school
which is 5 miles away everyday?
5 miles to school
5 miles to home
Let x be the number gallons we need for a day:
Can you solve it
from here?
x = Gal
3
1

14. So you use up 1/3 gallon a day. How many gallons would
you use for a week?
5 miles to school
5 miles to home
Let t be the number of gallons we need for a week:
days
gal
t
day
gal
5
_
1
3
/
1

5
1
3
/
1 t

5
3
1 t
 t
3
)
5
(
1 
3
5

t Gal

15. How about another example?

16. How about this one!!
A piece of string that is 63 inches long is cut into 3
parts such that the lengths of the parts of the string
are in the ratio of 5 to 6 to 10. Find the length of
each part.

17. You try this one!!
A clothing store sells T-shirts in only three colors: red,
blue and green. The colors are in the ratio of 3 to 4 to
5, respectively. If the store has 20 blue T-shirts, how
many T-shirts does it have altogether?
Solution: Ratio 3:4:5
Red = 3( ) = red T-shirts
Blue = 4 ( ) = 20 blue T-shirts
Green = 5 ( ) = green T-shirts
5
5
5 15
25
60 T-shirts in all