Ratios and Proportions.pptx

1. Ratios and Proportions

2. There are many uses of ratios and proportions.
We use them in map reading, making scale
drawings and models, solving problems.

3. The most recognizable use of ratios and
proportions is drawing models and plans for
construction. Scales must be used to
approximate what the actual object will be
like.

4. A ratio is a comparison of two quantities by
division. In the rectangles below, the ratio of
shaded area to unshaded area is 1:2, 2:4, 3:6,
and 4:8. All the rectangles have equivalent
shaded areas. Ratios that make the same
comparison are equivalent ratios.

5. A ratio can be written in a variety of ways.
You can use ratios to compare quantities or describe
rates. Proportions are used in many fields,
including construction, photography, and medicine.
a:b a/b a to b

6. A ratio of one number
to another number is
the quotient of the
first number divided
by the second. (As long as
the second number ≠ 0)

7. 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?
= =
Divide both numerator and
denominator by their Greatest
Common Factor 6.

8. Since ratios that make the same comparison
are equivalent ratios, they all reduce to the
same value.
2 3 1
10 15 5
= =

9. 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
To compare them, we need to convert both
numbers into the same unit …either cm or m.
• Let’s try cm first!
Once we have the
same units, we can
simplify them.

10. Using ratios
The ratio of faculty members to
students in one school is 1:15.
There are 675 students. How
many faculty members are
there?
faculty 1
students 15
1 x
15 675
15x = 675
x = 45 faculty
=

11. Proportions
Two ratios that are equal

12. Properties of a proportion?
2x6=12 3x4 = 12
3x4 = 2x6
Cross Product Property
A proportion is an equation that
states that two ratios are equal

13. A proportion is an equation that states that
two ratios are equal, such as:

14. In simple proportions, all you need to do is
examine the fractions. If the fractions both
reduce to the same value, the proportion is
true.
This is a true proportion, since both fractions
reduce to 1/3.
5 2
15 6
=

15. 1) Are the following true proportions?
2 10
3 5
=
2 10
3 15
=

16. 2) Solve for x:
4 x
6 42
=

17. 3) Solve for x:
25 5
x 2
=

18. Now you know enough about properties,
let’s solve the Mysterious problems!
If your car gets 30 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

19. Solve the following problems.
4) If 4 tickets to a
show cost $9.00, find
the cost of 14 tickets.
5) A house which is
appraised for $10,000
pays $300 in taxes.
What should the tax
be on a house
appraised at $15,000.

20. Direct Proportion
Direct proportion is a mathematical comparison
between two numbers where the ratio of the
two numbers is equal to a constant value.
Example: if the number
of individuals visiting a
restaurant increases,
earning of the restaurant
also increases and vice
versa.

21. Direct Proportion
8 laps of a race track has a total of 12
km. What would the distance be
for 20 laps of the race track?

22. Indirect Proportion
Two quantities are said to be in indirect
proportion if an increase in one leads to a
decrease in the other quantity and a
decrease in one leads to an increase in
the other quantity.

23. Indirect Proportion
A worker takes 10 days to fit a
bathroom. How long would it
take 2 workers to fit a bathroom?

24. Similar Figures
The Big and Small of it

25. For Polygons to be Similar
corresponding angles must
be congruent,
and
corresponding sides must
be proportional
(in other words the sides
must have lengths that
form equivalent ratios)

26. Congruent figures have the same size and
shape. Similar figures have the same shape
but not necessarily the same size. The two
figures below are similar. They have the same
shape but not the same size.

27. Let’s look at the two
triangles we looked at
earlier to see if they are
similar.
Are the corresponding
angles in the two
triangles congruent?
Are the corresponding
sides proportional?
(Do they form equivalent
ratios)

28. Just as we solved for
variables in earlier
proportions, we can solve for
variables to find unknown
sides in similar figures.
Set up the corresponding
sides as a proportion and then
solve for x.
Ratios
x/12
and
5/10
x 5
12 10
10x = 60
x = 6

29. Determine if the two triangles are similar.

30. In the diagram we can
use proportions to
determine the height
of the tree.
5/x = 8/28
8x = 140
x = 17.5 ft

31. The two windows below
are similar. Find the
unknown width of the
larger window.

32. These two buildings are
similar. Find the height of
the large building.