1) The document discusses ratios, proportions, and how to solve problems involving them. It defines key terms like ratio, equivalent ratios, and proportion. 2) It explains how to set up and solve proportions using cross products. For example, to solve the proportion 2:6 = 4:x, you would write 2/6 = 4/x, cross multiply to get 12 = 4x, and then solve for x. 3) It also discusses strategies for solving word problems involving ratios and proportions, like setting up the information as a double number line or using proportional thinking to relate the quantities. For example, to find the cost of 5 gallons if 2 gallons is $5.40, you can think

1. Pre-Algebra
Summer Workshop
Part 4: Ratios and Proportions

2. 2
RATIOS AND PROPORTIONS
:
Proportional Relationships
Key Terms:
A ratio is a relationship of two numbers. A ratio is denoted A∶B to indicate the order of the numbers. The number A
is first and the number B is second.
Two ratios A∶B and C∶D are equivalent ratios if they both have values that are equal.
A ratio relationship between two types of quantities, such as 15 miles per 2 hours, can be described as a rate 7.5
miles/hour.
The numerical part of the rate is called the unit rate and is simply the value of the ratio, in this case 7.5. This means
that in 1 hour the vehicle travels 7.5 miles. The unit for the rate is miles/hour, read miles per hour.
A ratio can be expressed as a fraction or as a decimal. 2:5 is the same as 0.4 or
Introduction to Ratios
In certain situations, comparison by division makes better sense than comparison by taking the
difference. The comparison by division is called the ratio of the two numbers.
Comparing any two quantities in terms of ‘how many times’ is known as ratio, and it is denoted by a
sign.
 Suppose you have a group of fruits and you want to find the ratio of apples to oranges.
Remember finding a ratio is when you want to compare two quantities. In this case we are
comparing the quantities of apples to the quantities of oranges.
1: 2 2: 4
are equivalent ratios

3. 3
RATIOS AND PROPORTIONS
Counting:
One way you can find out the ratio is by counting. There are 6 apples and 9 oranges. Therefore,
you can say that the ratio of apples to oranges is 6 to 9 or use the notation “:” to represent the
word “to” and simply write 6:9.
Reducing:
Remember we want to find how many apples there are to oranges. To efficiently compare the
quantities, it is important to reduce. Both “6” and “9” are divisible by “3”. You can reduce ratios
just like you reduce fractions.
You can also separate the groups or apples and oranges into three equal groups.
Each group has two apples and three oranges. For every two apples, there are three oranges.
The ratio of apples to oranges is 2:3.
 What if you wanted to find the ratio of oranges to apples? You would simply swap the numbers
around. For every three oranges, there are two apples. The ratio of oranges to apples is 3:2.
6: 9
2: 3
6 divided by 3 is 2 9 divided by 3 is 3

4. 4
RATIOS AND PROPORTIONS
Ratios and Fractions
Let’s review:
A ratio is a relationship or comparison between two quantities.
A fraction expresses a number that is part of a whole.
The idea of ratios as fractions can be confusing. This is because some ratios are fractions and some are
not.
 Can you think of a ratio that can never be a fraction?
Part-to-part ratio: This ratio compares a part of a whole to another part of a whole.
Suppose there is a group of 1 girl and 2 boys. We would write the ratio of girls to boys as 1:2
meaning for every 1 girl there are 2 boys.
Looking at the picture, we can see that there is a total of 3.
If you represent it as an array, you can see 1 girl and 2 boys make a total of 3.
Girl Boy Boy
The ratio of girls to boys in the group is 1 to 2 or 1:2.
part part
1: 2
part : part

5. 5
RATIOS AND PROPORTIONS
part
You can compare that to the fraction by using an array as well.
In this case the ratio 1:2 cannot be thought of as the fraction because it is a part to part relationship
not a part to whole relationship.
part
whole
whole

6. 6
RATIOS AND PROPORTIONS
 Can you think of a ratio that can be a fraction?
Part-to-whole ratio: This ratio compares the part of a whole to the whole.
Suppose there is a class of 10 boys with 21 students. If you wanted to find the ratio of boys to
students, you would simply count the boys in the class and then count the students. In this case
it is given. The ratio would be written as 10:21.
You can draw an array for this ratio.
boy boy boy boy boy boy boy boy boy boy
part
whole
part whole
whole
part
can be written as a fraction

7. 7
RATIOS AND PROPORTIONS
Equivalent Ratios
Equivalent ratios are just like equivalent fractions. If two ratios have the same value, then they are
considered equivalent. To find an equivalent ratio, first you need to understand the ratio. Then, you can
increase or reduce the fraction to generate equivalent ratios.
 Understand the ratio: As you learned, ratios can sometimes be written in fraction form.
Suppose you are given the ratio 2: 4 it can also be written as .
 Increase or reduce the fraction: As stated in part one of this workshop, fractions can be
increased by multiplying both the numerator and denominator by the same number. You can
reduce or simplify the fraction if they are both divisible by the same number.
×
×
=
×
×
=
×
×
=
÷
÷
=
 Convert fraction into ratio:
To convert a fraction to a ratio, you will make the numerator of the fraction the left side of the ratio
and the denominator the right side of the fraction.
2
4
=
4
8
=
6
12
=
8
16
=
1
2
2: 4 = 4: 8 = 6: 12 = 8: 16 = 1: 2
*An easier way to find equivalent fractions is to simply increase both sides of the ratio by a certain number.
Remember, reducing or increasing a ratio does not change the value of it.

8. 8
RATIOS AND PROPORTIONS
Solving Ratio in Table Format
Ratios can be represented in table format. Given a ratio, you will be able to solve for equivalent ratios.
 Suppose you are given the following ratio 24:40. Based off of the ratio 24:40, you are asked
to fill missing values in table. In order to find missing values in a table, you need to
understand the previous concept of equivalent ratios.
Numerator Denominator
12
24 40
80
Start with the ratio you are given and work from there. You were given the ratio 24:40. You
can find the missing values by comparing it to the given ratio.
To find the value of the denominator for the given numerator 12, you compare it to the
numerator of the given ratio. 12 is half of 24 or 24 divided by 2 is 12. To find the missing
denominator that corresponds with 12, simply divide 40 by 2 as well.
40 ÷ 2 = 20
Numerator Denominator
12 20
24 40
80
÷ 2 ÷ 2

9. 9
RATIOS AND PROPORTIONS
To find the missing numerator for the given denominator 80, you compare it to the
denominator of the given ratio. 80 is double the amount of 40 or 40 multiplied by 2 is 80. To
find the missing numerator that corresponds with 80, simply multiply 24 by 2 as well.
24 × 2 = 48
Numerator Denominator
12 20
24 40
48 80
Proportions
A proportion occurs when two ratios are forced to be equal to each other. Or you can also think of a
proportional relationship as having the same ratio.
Since a ratio is made up of two numbers, and a proportion compares 2 ratios, you’ll be dealing with 4
numbers here, 2 for each ratio. The first ratio will be between the two numbers represented by .
As a ratio, it will be , or : . The second ratio will be between the two numbers represented by
. As a ratio, it will be , or : .
A proportion can be stated as: . This can also be written as : : : : , or in fractional
form as = or / = / .
Here are some examples of proportions:
× 2
× 2

10. 10
RATIOS AND PROPORTIONS
Means and Extremes
As we learned in the previous section, a proportion has two ratios or four terms/numbers. Those four
numbers are considered either the “means” or “extremes”.
 Means: inside terms; the word "mean" comes through French from Latin medius meaning
"middle."
 Extremes: outside terms; the word "extreme" comes from Latin extremus meaning "outside"; so it
means "outermost."
The means are the inside terms, and , and the extremes are the outside terms, and , in the
proportion we might write in any of these ways:
∶ ∶ : ∶
∶ = ∶
/ = /
=
Whichever way we write it, except the last, and are on the inside, and and are on the outside.
Another way to remember this is:
 Means:
 Extremes:

11. 11
RATIOS AND PROPORTIONS
Proportion Validity
To understand whether two ratios are proportions, it is important to understand the concept of equivalent
ratios.
 Let’s say you were asked to determine if a proportion is true or false. How would you go about
finding the answer?
For example:
The problem is asking you to determine if 3:12 or is equal to 5:35 or . As you already know,
for two ratios to have a proportional relationship, they have to be equivalent.
The easiest way to determine if the ratios are equivalent is to put them in simplified forms (reduce).
Remember: to reduce ratios (or fractions), test their divisibility with the prime numbers 2, 3, 5, 7,
and continue until the numerator and denominator have no common factors.
Let’s start with . Both 3 and 12 are divisible by 3, so can be reduced. To simplify the ratio,
divide both the numerator and denominator by 3.
Now you can reduce the second ratio the same way. In this case, both 5 and 35 are divisible by 5.
Is the proportion true or false?
3
12
=
5
35
3 ÷ 3
12 ÷ 3
=
1
4
5 ÷ 5
35 ÷ 5
=
1
7

12. 12
RATIOS AND PROPORTIONS
Now you can use the simplified forms of the ratios to determine if they have a proportional
relationship.
It is clear that and are not the same ratios and are not equivalent. Therefore, the proportion =
is clearly false.
Solving Simple Proportions
Cross Products Property: Also called the Means-Extremes Product Property, the product of extremes
and product of the means are called cross products. Therefore, = .
1
4
≠
1
7
3
12
≠
5
35
Therefore, the proportion is
FALSE.

13. 13
RATIOS AND PROPORTIONS
9 2 = 3
Because of the Cross Products Property, solving proportions is simply a matter of stating the ratios as
fractions, setting the two fractions equal to each other, cross-multiplying, and solving the resulting
equation.
 Suppose you had to find the unknown value in the proportion: 2 ∶ = 3 ∶ 9.
2 : = 3 : 9
First, you convert the colon-based notation ratios to fractional form:
2
=
3
9
Next, cross multiply to solve for :
Solve for x:
9 2 = 3
18 = 3
18
3
=
3
3
6 =
= 6
Remember to solve for x, you can use The Balance Scale Method and isolate x. When you
solve an algebraic equation, the goal is to isolate the variable (which is usually x) and get x
alone on one side of the equation and some number on the other side.
Cross multiply and simplify.
Isolate x by dividing. Maintain
balance by dividing both sides
of the equation by 3.

14. 14
RATIOS AND PROPORTIONS
Here are more examples:

15. 15
RATIOS AND PROPORTIONS
Solving Word Problems
The following word problem involves a proportion:
 If 2 gallons of gasoline costs $5.40, how much would 5 gallons cost?
The general idea in these types of problems is that you have two quantities that change at the
same rate. Here are the two quantities: we have (1) gasoline, measured in gallons, and (2)
money, measured in dollars. We know both quantities (2 gallons costs $5.40), and are asked to
find the missing quantity (in this case, the cost for 5 gallons).
The first step you always want to take is organizing the information. The long line —— means
“corresponds to”.
Solving this problem involves proportional thinking:
You can write the problem as a proportion and solve.
You would solve this proportion using the cross-products property.
If 2 gallons of gasoline costs $5.40, how much would 5 gallons cost? 5 gallons would cost $13.50.
2 gallons —— 5.40 dollars
5 gallons —— dollars
2
5.40
=
5
2
5.40
=
5
2 = 5 5.40
2 = 27
2
2
=
27
2
= 13.50
cross multiply
Divide both sides of the
equation

16. 16
RATIOS AND PROPORTIONS
 Proportional Thinking: Once you understand proportions, you can also solve these types of
problems more easily.
Another way to solve this problem is to think about the proportional relationship. Remember,
proportions are equal. To solve this word problem, think about 2 gallons in relation to 5 gallons.
You know that the amount of gallons is increased by 2.5. With this understanding, you can
simply multiply the 5.40 dollars by 2.5 to maintain the proportional relationship and solve for x.
5.4 × 2.5 = 13.5. Therefore, the price of 5 gallons is $13.50.
2
5.40
=
5
2 × 2.5 = 5
5.4 × 2.5 =