Fractions are numbers that are used to represent parts of whole quantities.
The denominator of a fraction (the bottom) shows the number of equal parts in the whole.
The numerator of a fraction (the top) shows the number of parts being talked about or being used.

A pizza arrives from the Deli Shop and it is cut into 8 equally sized pieces. You eat the first piece.
It can be said that …

which says that one of the eight total pieces has been eaten.

To compare fractions:
a) convert the fractions to a common denominator. The converted fraction with the larger resulting numerator will be the larger fraction.

$frac1$

b) convert the fractions to decimal values (divide the top by the bottom) and compare the values.

$frac2$

Properties of Fractions

Given: a, b, c and d are real numbers, variables, or expressions such that b, c, d ≠ 0.

Property:

Example:

1. Signs and Fractions: $fracsign$

$fracsignE$

The placement of a negative sign can be in front of the fraction, in the numerator or in the denominator.

2. Equivalent Fractions:
$fracequivalent$

$fracequivalentE$
“Cross Multiply” – In a proportion, the product of the means equals the product of the extremes.

3. Adding & Subtracting:
Get a common denominator.
$fracadda$

$fracaddE$
You can use the product of the two denominators as the common denominator if you cannot find a smaller common denominator.

4. Multiplying:
$fracmult$

$fracmultE$
Just multiply through the top and multiply through the bottom.

5. Dividing:
$fracdivide$

$fracdivideE$
Invert (flip) the second term, and multiply.

6. Simplify: Search for the largest factor that will divide evenly into both the numerator and denominator. It may be the case that the fraction is already in its simplest form.

$fracsimpE$
Writing all of these steps is not necessary. Just be careful not to make a careless mistake.

You arrive at a party where 4 different pizzas (of the same size) are being served. Since you like all of the pizzas, you decide to assemble

one full pizza by combining parts of each of the available choices. You pick up 1/3 of the cheese pizza, 1/2 of the pepperoni pizza, and 1/8 of the sausage pizza. How much of the anchovy pizza will you need to pick up to complete your “new” whole pizza?

Solution:   Let x = fractional part of the anchovy
pizza needed to complete the pizza.
Add the portions together and set them equal to 1
(the whole new pizza). Solve for x.
$frac3$

You will only need 1/24 of the anchovy pizza to complete a whole new pizza. A very small slice!

Ratio:

A ratio is a comparison of two or more quantities.

The order of the numbers in a ratio is important.

The notation for ratio uses a colon, such as a : b
and is read “the ratio of a to b“.

The value of ratio a : b is expressed as the quotient of the first number, a, divided by the second number, b, where b cannot be zero. Yes, it looks like a fraction.

Since the value of a ratio can be written as a fraction, it can also be written in any form that is equivalent to that fraction (such as a decimal or a percent).

Notice that the order is important: 3 : 4 ≠ 4 : 3 because 3/4 ≠ 4/3.

A unit ratio is a two-term ratio where the second term is 1.

Example: 3 : 1 or or 3 to 1. All ratios can be reduced to a unit ratio.

Consider this situation: There are 16 girls and 14 boys in a math class.

Part-to-Part Comparison:
If we compare the number of girls to the number of boys, we obtain a ratio of
16 : 14 or 8 : 7
This is comparing one “part” of the class to another “part” of the class.

Part-to-Whole Comparison:
If we compare the number of girls to the total number of students, we obtain a ratio of 16 : 30 or 8 : 15
This is comparing one “part” of the class to the “whole” class.

It is always important to read questions carefully to determine
if the ratio will be between “two parts” or between “one part and the whole”.

Let’s pull these ideas together in this next example:

Given:

Ratio Comparison	Ratio	Reduced Ratio	Unit Ratio
Part-to-Part: red stars to blue stars	2 : 8 or	1 : 4 or	0.25 : 1 or
Part-to-Part: blue stars to red stars	8 : 2 or	4 : 1 or	4 : 1 or
Part-to-Whole: red stars to ALL of the stars	2 : 10 or	1 : 5 or	0.2 : 1 or
Whole-to-Part: All of the stars to red stars	10 : 2 or	5 : 1 or	5 :1 or
Part-to-Whole: blue stars to ALL of the stars	8 : 10 or	4 : 5 or	0.8 : 1 or
Whole-to-Part: All of the stars to blue stars	10 : 8 or	5 : 4 or	1.25 : 1 or

If you see a ratio such as

, it may be easier to understand what the ratio represents if you multiply both sides to remove the mixed number.

Terms used with ratios:
Continued Ratio: (the comparison for more than two quantities) a : b : c

Equivalent Ratios: (all reduce to the same value 1:5) 2:10, 3:15, 1:5 or

Visual Ways to Solve Ratio Problems: “Looking for Patterns!”

These visual representations of ratios are simply different ways to think about the same concept. They are all based on the idea that the value of a ratio can be expressed in various equivalent forms. In each situation (except for tape diagrams), we are looking to find the amount of BBQ sauce needed when using 10 cups of water.

Using Ratio Table – Addition Pattern
A marinade is created with 2 cups of water and 5 cups of BBQ sauce.

Start the table with the ratio from the problem. Then add the values from this starting entry in each column. The resulting ratios are all equivalent to the starting ratio.

Using Ratio Table – Multiply Pattern 1
A marinade is created with 2 cups of water and 5 cups of BBQ sauce.

To find the amount of BBQ sauce needed with 10 cups of water, just multiply 2 x 5 to get 10, and then multiply 5 x 5 to get the needed 25 cups of BBQ sauce.

Using Ratio Table – Multiply Pattern 2
A marinade is created with 2 cups of water and 5 cups of BBQ sauce.

Once you notice the multiplication pattern you can multiply 10 cups of water by that value (2.5) to find the needed cups of BBQ sauce (25).

Using Tape Diagrams
A marinade is created with 2 cups of water and 5 cups of BBQ sauce.
For this example, we are looking for the number of cups of water needed with 15 cups of BBQ sauce.

Start by preparing boxes for the given ratio 2 : 5. Include enough BBQ boxes to get to 15. Draw in the water boxes so the number of boxes is the same as the number of BBQ boxes. Answer: 6 cups of water.

Using Double Number Lines
A marinade is created with 2 cups of water and 5 cups of BBQ sauce.

A double number line is set up in the same manner as setting up the addition table. Also, the addition pattern and multiplication patterns will apply to this double number line.

Using a Coordinate Graph
A marinade is created with 2 cups of water and 5 cups of BBQ sauce.

Preparing a table first may make it easier to prepare a graph. The table will also clearly show that the first column will be the x-coordinates and the second column will be the y-coordinates. A coordinate graph will also illustrate the addition patterns and multiplication patterns seen previously.	The pattern across the graph will show +2 to each x-value and +5 to each y-value.
Notice that the plots follow along a straight line through the origin.

The Callahan family has two dogs and five cats.

Tape Diagram
Each box represents the same number of animals.

a) What is the ratio of the number of dogs to the number of cats? Answer: 2 : 5

b) What is the ratio of the number of cats to the number of dogs? Answer: 5 : 2

c) What is the ratio of the number of cats to the total number of cats and dogs? Answer: 5 : 7

From the chart at the right below:

a) What is the ratio of boys to girls in the class?
15 : 12 or 5 : 4

b) What is the ratio of girls to the total number of students in the class? 12 : 27 or 4 : 9

Students in Math Class

Boys	Girls	Total

15	12	27

The table below shows possible ways a bakery can meet a specific percentage of sales of muffins.
Which of the following ratios of number of muffins sold to number of muffins baked could be included in this table?

Choose:
1) 28 : 35
2) 24 : 40
3) 12 : 18
4) 28 : 40

Choice 1 is correct.

The ratio displayed in this table is 4 to 5 and 28 : 35 is the only choice that is equivalent to 4 : 5.

Specific Percentage of Sales

Muffins Sold	Muffins Baked

8	10
16	20
20	25
32	40

The table below shows a constant ratio of cubic inches of water to weight of water in pounds.

a) What is the ratio of water to weight?
The ratio is 200 : 8 or 25 : 1

b) What is the value of x?
x = 800
Since 800 : 32 is equivalent to 25 : 1.
or
Since 32 is 8 x 4, then x is 200 x 4.

Constant Ratio

Water (Cubic Inches)	Weight (Pounds)

200	8
500	20
x	32
1500	60

The double number line below shows a constant ratio of
cm : minutes to be 5 : 3. Complete the diagram.

a) How many minutes did it take for the the distance traveled to be 10 cm.? 6 minutes

b) When 12 minutes has passed, what was the distance traveled? 20 cm.

c) When 9 minutes has passed, what was the distance traveled? 15 cm.

During the school year, a teacher graded 180 bonus quizzes. The number of quizzes receiving A’s, B’s, and C’s were in the ratio of
5 : 3 : 1, respectively. How many bonus quizzes received a grade of A for the school year?

SOLUTION: (by Tape Diagram)	The tape diagram shows that 100 bonus quizzes received an A
SOLUTION: (by Algebra) Represent the 5 : 3 : 1 ratio as 5x : 3x : 1x and set = to total 180 5x + 3x + x = 180 9x = 180 x = 20	Since the 5x represents the number of A’s, substitute 20 for x to find the final answer. 5(20) = 100 100 bonus quizzes received an A

FYI only: Ratios vs Fractions: Are the values of ratios actually “fractions”?

In terms of ratios, we can say:

part-to-part
comparison
2 : 3
There are 2 green boxes for every 3
blue boxes.

part-to-whole
comparison
2 : 5
The ratio of green
boxes to all of the
boxes is 2 to 5.

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By the definition of fraction,
we can accurately say:

→ 2 of 5 boxes are green
But the definition of fraction is not accurate for the other situation:

→ 2 of 3 boxes are green
By definition, fractions represent parts of whole quantities.

Conclusion: While a part-to-part comparison may be represented by a quotient, that quotient does not meet the formal definition of a fraction which is a number representing a part of a whole quantity. The part-to-whole comparison, on the other hand, does meet this definition. It is customary to see ratios (in both comparisons) written as “fractions”, which allows the ratios to be manipulated in ways similar to working with fractions.
Are the values of ratios actually fractions? Sometimes! But don’t fret the difference.

Rates:

“I am traveling at a high rate of speed!!
I’m going 5m inches per minute where m equals 10.

So, at what rate of speed is this little ant
actually traveling in this matchbox cruiser?
Answer: 50 inches per minute

A rate is a ratio that compares two different kinds of numbers, such as miles per hour, or inches per second. A unit rate compares a quantity to its unit of measure.

A rate can express how long it takes to do something.

To drive 50 inches in one minute is to drive at the
rate of 50 in./min.

This fraction (50/1) expressing the rate has units of distance
in the numerator and units of time in the denominator.

When the denominator of the “rate” fraction is a “single unit of one” (such as 1 minute), the rate is referred to as a unit rate. The most common type of rate is “per unit of time“, such as “miles per hour” or “inches per minute”.

When working with rate, the words “per, for and each” are often used to separate the units.

Symbols such as / or @ may be used when discussing rates.

Unit rate – a rate with a denominator of 1.

Other Ways to Look at Rates

Rates • A rate is a ratio in which the two measurements have different units. • A rate refers to the frequency by which an event happens. • A rate refers to the changes in measurements or units.
• Rates are most often expressed in “per unit” form with a denominator of 1. Examples: 55 mph; $15 per hour; 350 calories per serving.

Solving a problem dealing with rate may involve dealing with a proportion.

Situation: The ant’s cruiser is powered by sugar water. If the cruiser goes 125 inches per tablespoon of sugar water, how far can it go on 5 tablespoons of sugar water?

The equation at the left shows two ratios being equal. This equation is called a proportion.

To solve, observe that the bottom values changed by a multiple of 5. Do the same to the top values. 25 x 5 = 125. ANSWER: 125 inches.

Find the unit rate: The ant’s cousin has a bigger matchbox cruiser that travels at a rate of 150 inches in 2 minutes. What is the unit rate traveled by this crusier?

Solution: To find the unit rate, the denominator needs to be 1.
Method 1: Observe that 2 minutes changing to 1 minute is division by 2. Do the same to the top: 150 divided by 2 = 75 Answer: 75 inches per 1 minute. Or just 75 inches per minute. To get unit rate, simply divide: 150 ÷ 2 = 75	Method 2: (More advanced method) Use “cross multiply” (in a proportion, the product of the means equals the product of the extremes) to solve. 2x =1350 x = 75 Answer: The unit rate is 75 inches per minute.

Find a missing term given rate: How long, in minutes, did it take the ant to cover 350 inches at a rate of 50 inches per minute?

Solution: Set up a proportion:	Method 1: Observe that 50 inches changing to 350 inches is a multiple of 7. Do the same to the denominator: 1 times 7 = 7. Answer 7 minutes.
Method 2: (A more advanced method.) Use “cross multiply” (in a proportion, the product of the means equals the product of the extremes) to solve. 50x = 350 x = 7 Answer: 7 minutes

Comparing unit rates: The sugar water fuel for the ant cruiser is sold in two different sizes. A 12 tablespoon container costs $0.62. A 8 tablespoon container costs $0.45. Which is the better buy?

Solution:
Convert each pricing to its unit rate ( 1 tablespoon) by using division.

The unit rate that is smaller (that costs less) is the better value.
ANSWER: Purchasing the container with 12 tablespoons of fuel is the better buy.

Distance = Rate x Time: The ant drives his matchbox cruiser from his home to his friend’s house traveling at the rate of 50 inches per minute. He then walks back to his home at the rate of 10 inches per minute. If the round trip took 9 minutes, how far is it from the ant’s home to his friend’s house?

Solution:
Let t = cruiser time
9 – t = walking time

Distance = Rate x Time
The distance driving the cruiser to the friend’s house is the same distance that the ant walks back home = round trip.

Hint: We need to first find the “time” which can then be used to find the “distance”.

Answer: Distance = 50(t) = 50(1.5) = 75 inches
As a check: 10(9 – t) = 10(9 – 1.5) – 10(7.5) = 75 inches.
The round-trip is the same distance in both directions.

Proportions

A proportion is a comparison of ratios.

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

A proportion can be written in two ways:

Both are read “6 is to 8 as 3 is to 4”.

In a proportion the first and last terms (6 and 4) are called the extremes.
The second and third terms (8 and 3) are called the means.

Determine if a Proportion is TRUE:
You can tell if a simple proportion is true by looking at each of the ratios (fractions). If the fractions both reduce to the same value, the proportion is true. The fractions are equivalent.

A TRUE proportion:
both fractions reduce to 1/3.

Find a Missing Part (a Variable) in a Proportion:

Simple Approach:
Remember that both fractions in a proportion must represent the same value. We can solve for a missing part of a proportion (the variable) by finding a common denominator for the two fractions. This will let you find the missing part.

While useful, this approach may not be sufficient when working with more complicated proportions.

To find x, use the common denominator of 30.
To change 5 to 30, multiply by 6.
The SAME must be done to the top,
(multiply 2 by 6) to keep the fractions equal.
ANSWER: x = 12

Universal Rule:
There is a rule (or algorithm) that can be followed to determine if two fractions are equal. The process multiplies from the top “across” to the bottom between the fractions in the proportion. If these products are equal, the fractions are equal. This method is very handy for setting up an equation to solve for a variable.

A precise statement of the rule is:
RULE: In a true proportion, the product of the means equals the product of the extremes.

Solve for x:

Using
RULE:

5 (x) = (2)(30)
5x = 60
x = 12

Cross Multiplication Algorithm (Rule):

An algorithm is a process or set of rules to be followed in calculations or problem solving.

Does this rule really work? Let’s take a closer look at what is happening.

This algorithm is simply the result of rewriting fractions to have the same denominator, and then examining their numerators.

Tidbit of Info: A proportion can be rewritten in different ways, yet remain true.
The following proportions are all equivalent (mathematically the same).
Check that this is true using “cross multiply”.

By Cross Multiply, all of these proportions can create
10y = 5x.

Method 1:
Perhaps you notice that each of these
ratios (fractions) can be simplified to 1/3, which
indicates that the proportion is TRUE.

Method 2: (using RULE)

(13)(24) = (39)(8)

312 = 312 (TRUE)

Method 1:
Getting a common denominator:

Method 2: (using RULE)

(x)(5) = (25)(2)
5x = 50
x = 10

Method 1: Getting a common denominator is not as easy in this problem as it was in Example 2. The common denominator will be 11x (where x represents the empty box). You can, in this problem, notice that 8 x 4 = 32. Multiplying the denominator by this same value will create a TRUE proportion: 11 x 4 gives the answer 44. If you want to use 11x as the common denominator, you will get:	Method 2: (using RULE)

The ratio of boys to girls in Spanish club is 4 to 5. If there are 25 girls in the club, how many boys are in the club?

Method 1:
Use the common denominator of 25:

x = 20 means there are 20 boys in the club.

Method 2:
Using the rule:

4•25 = 5•x
100 = 5x
x = 20 means there are 20 boys in the club.

When setting up a proportion in a word problem, it is important to remember to keep the same “units” in both numerators, and the same “units” in both denominators.
Such as:

The length of a stadium is 100 yards and its width is 75 yards. If 1 inch represents 25 yards, what would be the dimensions of the stadium drawn on a sheet of paper?

Method 1:
This problem can be solved by an intuitive approach, such as:
100 yards by 75 yards
100 yards = 4 inches (HINT: 100/25 created the 4)
75 yards = 3 inches (HINT: 75/25 created the 3)
Therefore, the dimensions would be 4 inches by 3 inches.

Method 2:
In the following proportions, notice that the inches are all on the top and the yards are all on the bottom for this solution. Other setups are possible.

Proportional Relationships

The key word is “equivalent (equal)” ratios.

3 Ways to Determine if Proportional Relationships Exist:

Verify that a given proportion is TRUE:
To determine if a simple given proportion is true, look at the fractions. If these ratios (fractions) both reduce to the same value, the proportion is true.

Double check: the “cross multiply” of 12 • 9 = 4 • 27 is true.

This is a TRUE proportion because
both fractions reduce to 1/3.
(and because 12 • 9 = 4 • 27).

Does the data displayed in a table show a proportional relationship?
First, remember that not ALL tables will display a proportional relationship.
To determine if a proportional relationship exists, you need to look for equivalent (equal) ratios within the table. Hopefully your table will be relatively small, as ALL values within the table will need to be checked.
All of the entries in the table at the right will create equal ratios.

(# of mules per bale of hay = 2 : 1)

Single Feedings

(Bales of hay per # of mules = 1 : 2)

Hint: You could plot the points from a table onto a coordinate grid to see if you have a proportional relationship. See graph information as it pertains to proportional relationships in the section below.

Does the data displayed in a coordinate graph show a proportional relationship?
First, remember that not ALL coordinate graphs will display a proportional relationship.
To determine if a proportional relationship exists look for the data to lie on a straight line passing through the origin.

When working with a coordinate graph, it is customary
to examine the ratio y/x (instead of x/y).
(dependent variable over independent variable)

If the x and y coordinates form proportional relationships, then there is some non-changing number (a constant) that when multiplied times x will create y.
In this example, that number is 3 (y = 3x), and is called the constant of proportionality.
The constant of proportionality is the unit rate
(without any labeling units).

Constant of Proportionality = 3/1 = 3
(the unit rate) y = 3x

Constant of proportionality
is the SLOPE of the line!

More about Constant of Proportionality:
• It is a positive number.
• It is also called the unit rate. (Find the y-value when x = 1.)
• It is what you multiply times x to get y.
• It is typically represented by the letter k. (y = kx or y/x = k)

Observing Proportionality on Coordinate Graphs:
On the graph at the right, each (x,y) coordinate indicates the number of jars of jam and the number of cups of sugar needed to produce that number of jars of jam.
(#JamJars, #CupsSugarNeeded)

The (0,0) coordinate establishes the fact that if there were no jars, there was no sugar needed.

The coordinate (1,½) establishes the unit rate .

The unit rate shows that the constant of proportionality for this graph is ½.
y = ½ x

This graph demonstrates a proportional
relationship because it is a straight line
passing through the origin.

Does the table shown at the right represent a proportional relationship?
Explain.

x	3	1	2	8
y	9	3	6	24

Solution:

TRUE! Since all of the ratios are equivalent, this table is a proportional relationship. The constant of proportionality is 3.

Does the table shown at the right represent a proportional relationship?
Explain.

x	5	25	16	35
y	1	5	3	7

Solution:

Since all of the ratios are NOT equivalent, this table is NOT proportional relationship.

	The circumference of a circle is proportional to its diameter and is represented by the equation . What is the constant of proportionality? What does it tell you about this relationship?
Solution: The constant of proportionality is π. [y = kx where k is the constant of proportionality]. It tells you that the unit rate is 3/1 and that the ratio of C/d will always be the same (constant) and will be π.

Does the graph shown at the right represent a proportional relationship? Explain.

Solution:
Since none of these ratios are equal to one another (and certainly not ALL equal to each other), this graph does not display a proportional relationship.