A proportion is an equation that equates two ratios. The cross product property of proportions states that the product of the means equals the product of the extremes, or ad = bc. This property can be used to solve word problems involving ratios and determine unknown values. For example, if a car gets 40 miles per gallon, and someone drives 5 miles to school and 5 miles back each day, it would take 1/4 gallon per day. For a full week it would take 5/3 gallons, or about 1.67 gallons, costing around $6.16 at $3.69 per gallon.

1. Proportions!
d
c
b
a

What is a proportion?
A proportion is an equation
that equates two ratios
The ratio of dogs and cats was 3/2
The ratio of dogs and cats now is 6/4=3/2
So we have a proportion :
4
6
2
3


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

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

4. Properties of a proportion?
d
c
b
a
 d
d
c
d
b
a



c
d
b
a


Let’s make sense of the Cross Product Property…
c
b
b
d
b
a




bc
ad 
For any numbers a, b, c, d:

5. Properties of a proportion?
4
6
2
3

If
Then
6
4
3
2

• Reciprocal Property
Can you see it?
If yes, can you think
of why it works?

6. How about an example?
5
2
=
𝑥
4
Solve for x:
5(4) = 2x
20 = 2x
10 = x
Cross Product Property

7. How about another example?
5
2
=
8
𝑥 Solve for x:
5x = 2(8)
5x = 16
x =
16
5
Cross Product Property
Can you solve it
using Reciprocal
Property? If yes,
would it be easier?

8. Can you solve this one?
5
𝑥 − 1
=
3
𝑥 Solve for x:
5x = (x-1)3
5x = 3x – 3
2x = -3
x =
Cross Product Property
−
3
2
Again, Reciprocal
Property?

9. Now you know enough about properties,
let’s solve the Mysterious problems!
40𝑚𝑖𝑙𝑒𝑠
1𝑔𝑎𝑙
=
(5 + 5)𝑚𝑖𝑙𝑒𝑠
40
1
=
10
𝑥
If your car gets 40 miles/gallon, how many gallons of gas do you need to
commute to school 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
1
4

10. So you use up 1/4 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:
1/4𝑔𝑎𝑙
1𝑑𝑎𝑦
=
𝑡_𝑔𝑎𝑙
5𝑑𝑎𝑦𝑠
1/4
1
=
𝑡
5
5
3
1 t
 t
3
)
5
(
1 
3
5

t Gal
What property
is this?

11. So you use up 5/3 gallons a week (which is about 1.67
gallons). Consider if the price of gas is 3.69 dollars/gal,
how much would it cost for a week?
Let s be the sum of cost for a week:
5 miles to school
5 miles to home
gallons
dollars
s
gallon
dollars
67
.
1
_
1
69
.
3

67
.
1
1
69
.
3 s

3.69(1.67) = 1s s = 6.16 dollars

12. So what do you think?
10 miles
You pay about 6 bucks a week just to get to school!
What about weekends?
If you travel twice as much on weekends, say drive
10 miles to the Mall and 10 miles back, how many
gallons do you need now? How much would it cost
totally? How much would it cost for a month?
5 miles
Think proportionally! . . . It’s all about proportions!