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A ratio is the comparison of two numbers written as a fraction.
For example:
Your school’s basketball team has won 7 games
and lost 3 games. What is the ratio of wins to
losses?
Because we are comparing wins to losses the first
number in our ratio should be the number of wins and
the second number is the number of losses.
The ratio is ___________
games won = _______
7 games = __
7
games lost
3 games
3
In a ratio, if the numerator and denominator are measured in
different units then the ratio is called a rate.
A unit rate is a rate per one given unit, like 60 miles per 1 hour.
Example: You can travel 120 miles on 60 gallons of gas.
What is your fuel efficiency in miles per gallon?
________
miles = ________
2 miles
Rate = 120
60 gallons 1 gallon
Your fuel efficiency is 2 miles per gallon.
An equation in which two ratios are equal is
called a proportion.
A proportion can be written using colon notation
like this
a:b=c:d
or as the more recognizable (and useable) equivalence of two
___
a = ___
c
fractions.
b
d
When Ratios are written in this order, a and d are the extremes,
or outside values, of the proportion, and b and c are the means,
or middle values, of the proportion.
___
a = ___
c
b
d
a:b::c:d
Extremes
Means
To solve problems which require the use of a proportion we can use
one of two properties.
The reciprocal property of proportions.
If two ratios are equal, then their reciprocals are equal.
The cross product property of proportions.
The product of the extremes equals the product of the means
Example:
2 6

x 9
9 2  6 x
18 6 x

6
6
3 x
Write the original proportion.
Use the cross product property.
Divide both sides by 6 to isolate the
variable, then simplify.
Three Kinds of Proportion
1. Direct Proportion
- a proportion when both antecedent
and the consequent increase or decrease.
Example: For every 15 boys who attended
the program, there were 10 girls. How many
girls were there if there were 60 boys who
attended the program?
Solution:
Proportion:
15 boys to 10 girls = 60 boys : x girls
15 : 10 = 60 : x
15x = 600
15
15
x = 40 girls
2. Indirect or Inversed Proportion
- A proportion when the antecedent
increases as the consequent decreases or vice
versa.
Example:
If 6 men can paint our house in
18 days, how long will it take 4 men
to do the same job?
Solution
men
6
4
6 : 4 = x : 18
4x = 108
4
4
x = 27 hours
hours
x
18
3. Partitive Proportion
- used to divide a number into
parts proportional to the given.
Example:
Four children decided to give their
grandmother a gift for her 70th birthday.
They decided to divide the expenses in
the ratio 2:3:4:5. How much will each one
pay if the cost of the gift is P1,400 only?
Solution
2 : 3 : 4 : 5 = 1,400
2n + 3n + 4n + 5n = 1,400
14n = 1,400
14
14
n = 100
P200 will be given by the first child
P300 will be given by the second child
P400 will be given by the third child
P500 will be given by the fourth child
Board Drill
Find the value of x:
4: 8 = x : 10
Board Drill
Find the value of x:
24: 48 = x : 8
Board Drill
Find the value of x:
15: x = 45 : 27
Board Drill
There are 24 flowers in
3 vases. If there are 6
vases, how many
flowers are there?
Board Drill
Two numbers are in the
ratio 5:7. If the sum of the
two numbers is 120, what
are the two numbers?
Board Drill
If 8 men can build the
extension of our house in 6
days, how many men are
needed to finish it in 4 days?