Equivalent Fractions

When fractions have different numbers in them, but have the same value, they are called equivalent fractions. 

Let's take a look at a simple example of equivalent fractions: the fractions ½ and 2/4. These fractions have the same value, but use different numbers. You can see from the picture below that they both have the same value. 

 How can you find equivalent fractions? 

Equivalent fractions can be found by multiplying or dividing both the numerator and the denominator by the same number. 

How does this work? 

We know from multiplication and division that when you multiply or divide a number by 1 you get the same number. We also know that when you have the same numerator and denominator in a fraction, it always equals 1. For example: 

So as long as we multiply or divide both the top and the bottom of a fraction by the same number, it's just the same as multiplying or dividing by 1 and we won't change the value of the fraction. 

Multiplication example: 

Since we multiplied the fraction by 1 or 2/2, the value doesn't change. The two fractions have the same value and are equivalent. 

Division example: 

You can also divide the top and bottom by the same number to create an equivalent fraction as shown above. 

Cross Multiply 

There is a formula you can use to determine if two fractions are equivalent. It's called the cross multiply rule. The rule is shown below: 

This formula says that if the numerator of one fraction times the denominator of the other fraction equals the denominator of the first fraction times the numerator of the second fraction, then the fractions are equivalent. It's a bit confusing when written out, but you can see from the formula that it's fairly simple to work out the math. 

If you get confused on what to do, just remember the name of the formula: "cross multiply". You are multiplying across the two fractions like the pink "X" shown in the example below. 




Comparing Fractions 

How can you tell if one fraction is bigger than another? 

In some cases it's pretty easy to tell. For example, after working with fractions for a while, you probably know that ½ is bigger than ¼. It's also easy to tell if the denominators are the same. Then the fraction with the larger numerator is bigger. 

However, sometimes it's difficult to tell which is bigger just by looking at two fractions. In these cases you can use cross multiplication to compare the two fractions. Here is the basic formula: 

Here is an example: 


Key Things to Remember

• Equivalent fractions may look different, but they have the same value.
• You can multiply or divide to find an equivalent fraction.
• Adding or subtracting does not work for finding an equivalent fraction.
• If you multiply or divide by the top of the fraction, you must do the same to the bottom.
• Use cross multiplication to determine if two fractions are equivalent.

Introduction to Fractions

What is a fraction? 

A fraction represents part of a whole. When something is broken up into a number of parts, the fraction shows how many of those parts you have. 

Pictures of Fractions 

Sometimes the best way to learn about fractions is through a picture. See the pictures below to see how the whole of a circle can be broken up into different fractions. The first picture shows the whole and then the other pictures show fractions of that whole. 

 Numerator and Denominator 

When writing a fraction there are two main parts: the numerator and the denominator. The numerator is how many parts you have. The denominator is how many parts the whole was divided into. 



 
Fractions are written with the numerator over the denominator and a line in between them. 

Types of Fractions 

There are three different types of fractions: 

1. Proper Fractions - A proper fraction is one where the numerator is less than the denominator. Note that a proper fraction is always less than one. 




 

2. Improper Fractions - An improper fraction is one where the numerator is greater than the denominator. Note that an improper fraction is always greater than one. 
 

3. Mixed Fractions - A mixed fraction had both a whole number part and a fractional part. 

Reciprocals 

A reciprocal is a fraction where the numerator and denominator are reversed. It can also be looked at as 1 over the number. When you take a number or fraction and multiply it by its reciprocal, the answer is always 1. 

Equivalent Fractions 

Sometimes fractions may look different and have different numbers, but they are equivalent or have the same value. 

One of the simplest examples of equivalent fractions is the number 1. If the numerator and the denominator are the same, then the fraction has the same equivalent value as 1. 


Here are some equivalent fractions for 3/4. The equivalent fractions are all multiples of 3/4. Take 15/20 for example. 3x5 = 15 and 4x5 = 20. 

 
Decimals 

When decimal points are used in numbers, the number to the right of the decimal point is a type of fraction. Depending on the place value it can be 1/10, 1/100, 1/1000 or some other factor of 10. 

Examples: 

0.3 = 3/10
0.42 = 42/100 

Percents 

Another type of fraction is the percent. The "percent" is a fraction with a denominator of 100. When you say 50% it is the same as saying 50/100. 

Solving Algebra Equations with Multiplication and Division

This page assumes you know about variables, basic algebraic equations, and how to solve them using addition and subtraction. 

In addition to using addition and subtraction to solve equations, we can also use multiplication and division. 

Main Rule 

The main rule we need to remember is that when we divide or multiply one side of the equation we have to do the same thing to other side of the equation. We also have to make sure that we divide or multiply the ENTIRE side of the equation and not just a part of it. 

Simple Example 

We'll take a simple example first: 

If 2x = 6, what does x = ? 

We can tell by just looking at this that x = 3, however, we can also solve for it. By learning to solve for x, we can then apply this method to more difficult problems where we can't tell the answer just by looking at the equation. 

Solving for x 

2x = 6 

We want to get x by itself on one side of the equation. We can do this by dividing 2x by 2 or multiplying by ½. 

2x (1/2) = 6 (1/2)
(2/2) x = 6/2
x = 3 

Let's try a more difficult problem. This time we will need to add and subtract as well. 

3x - 6 = 15 

It's easiest to do the addition and subtraction steps first with this kind of equation. 

add 6 to both sides
(3x - 6) + 6 = (15) + 6 
3x = 21 

divide both sides by 3
(3x)1/3 = (21)(1/3) 

x = 7 

Now we should check our answer by plugging in x = 7 back into the original equation: 

3x - 6 = 15
3(7) - 6 = 15
21 - 6 = 15
15 = 15 

Another Example Problem with 2 Variables 

Solve for x in the following equation: 

4x + 3y -12 = 24 - y + 2x 

Add 12 to both sides 

(4x + 3y -12) + 12 = (24 - y + 2x) + 12
(4x + 3y) = (36 - y + 2x) 

Subtract 2x from both sides so there is no x on the right side 

(4x + 3y) - 2x = (36 - y + 2x) - 2x
(2x + 3y) = (36 - y) 

Subtract 3y from both sides so that 2x is alone on one side 

(2x + 3y) - 3y = (36 - y) - 3y
(2x) = (36 - 4y) 

Divide both sides by 2 so that we get x all alone 

(2x)1/2 = (36 - 4y)1/2 

x = 18 - 2y 

Note that we divided both 36 and 4y by 2 on the right side. 

Let's check our answer using the original equation: 

4x + 3y -12 = 24 - y + 2x
4(18 - 2y) + 3y -12 = 24 - y + 2(18 - 2y)
72 - 8y + 3y - 12 = 24 - y + 36 - 4y
60 - 5y = 60 - 5y 

Things to Remember

• Always perform the same operation to both sides of the equation.
• When you multiply or divide, you have to multiply and divide by the entire side of the equation.
• Try to perform addition and subtraction first to get some multiple of x by itself on one side.
• Always double check you answer by plugging it back into the original equation.

Solving Algebra Equations with Addition and Subtraction

The Equation 

One of the basic concepts of algebra is the equation. The main thing to know about an equation is that everything on one side of the equal sign (=) must equal everything on the other side of the equal sign. 

Variables 

Variables are things that can change or have different values. In algebra, we are usually trying to find the value of one or more variables. In algebraic equations, the variable is represented by a letter. 

On this page our variables will be represented by the letters "x" and "y". 

Simple Equation 

Here is a simple equation with x as the variable: 

x + 5 = 7 

What does x = ? 

x = 2 because 2 + 5 = 7. 

Solving an Equation 

In the equation above we could just tell by looking at it that x = 2, however, this isn't always the case. Sometimes we have to work harder in order to solve for the equation. 

We can sometimes solve an equation by adding or subtracting the same number to both sides of the equation. We know this is okay, because as long as we perform the same operation to both sides of the equation, then the equation doesn't change. 

Let's try solving for this simple example by adding or subtracting to both sides: 

x + 5 = 7 

We want to find out what x equals, so we need to get x by itself on one side of the equation. If we subtract 5 from the left side, x will be by itself. Following our earlier rule, we need to do the same to right side. 

(x + 5) - 5 = (7) - 5 

x = 2 

Another Example: 

Solve for x: 

x - 2y + 7 = y + 15 

We need to get x by itself, so lets start by subtracting 7 from each side: 

(x - 2y + 7) - 7 = (y + 15) - 7
x - 2y = y + 8 

Now we need to get rid of the - 2y, we can do this by adding 2y to each side: 

(x - 2y) + 2y = (y + 8) + 2y 

x = 3y + 8 

Now we should double check this answer by plugging it back into the original equation: 

x - 2y + 7 = y + 15 

Substitute 3y + 8 for x 

3y + 8 - 2y + 7 = y + 15 

3y - 2y + 8 + 7 = y + 15 

y + 15 = y + 15 

Here we have learned how to solve an equation by adding and subtracting to each side, but what if we have something like 2x = 4? To solve for that equation, we need to multiply and divide from each side. 

Things to Remember

• Always perform the same operation to both sides of the equation.
• You can add and subtract numbers from both sides of the equation to solve for x or y.
• Always double check you answer by plugging it back into the original equation.

Ratios, Fractions, and Percentages

Here's an example of using fractions to help reduce the ratio: 

Reduce the ratio 6:72 to its simplest form 

6:72 can be written as the fraction 6/72
6/72 can be reduced to 3/36 by dividing both the numerator and denominator by 2
3/36 can be further reduced to 1/12 by dividing both the numerator and denominator by 3
1:12 is the simplest form of the ratio 

Proportions 

We haven't used this term yet, but a proportion is when ratios are equal to each other. Similar to when we have reduced ratios to their simplest form using fractions, we have created ratios that are proportional. 

The above example shows a proportion where: 

6/72 = 1/12 

In this case 6 is to 72 as 1 is to 12. These ratios are proportional and say the same thing. 

Percentages 

Proportions are often written as percentages. 

Here is a simple example: 

The following ratios are all proportional: 

5:50 6:60 10:100 

They all can be reduced to another proportion 1:10. This can be written as a percentage of 10 %. All of the above ratios can be written as 10 %. 

Note: in order for a percentage to make sense, the second number or term in the ratio needs to be a total number or the total set number. This is a bit confusing, so we'll describe this concept more in the next section. 

Are ratios the same as fractions? 

We often write ratios as fractions, especially to help us to do the math, but are they the same as fractions? Generally ratios are best written as fractions when the second term, called the consequent term, is the total of the set. 

For example, if we have 8 apples and 12 oranges, our ratio of apples to fruit is 8:20. Written as a fraction this would be 8/20 or 2/5. This means that two fifths of our fruit is apples. This makes sense. 

Note: this ratio can also be written as a percentage; 40% of the fruit is apples. 

Next let's compare the ratio of apples to oranges which is 8:12. This can be written as the fraction 8/12 and reduced to 2/3. But this fraction doesn't tell us much or make a lot of sense beyond the ratio of apples to oranges. We have 2/3 of what? Doesn't really mean a lot. 

You can't really write this as a percentage either. It would be rounded to 67%, but 67% of what? You need the consequent, or second term, to be the total or the number of fruit. 

Ratios

A ratio is a way to show a relationship or compare two numbers of the same kind. 

We use ratios to compare things of the same type. For example, we may use a ratio to compare the number of boys to the number of girls in your class room. Another example would be to compare the number of peanuts to the number of total nuts in a jar of mixed nuts. 

There are different ways we use to write ratios, and they all mean the same thing. Here are some of the ways you can write the ratios for the numbers of B (Boys) and G (Girls): 

the ratio of B to G
B is to G
B:G 

Note that when writing the ratio you place the first term first. This seems obvious, but when you see the question or ratio written as "the ratio of B to G" then you write the ratio B:G. If the ratio was written "the ratio of G to B" then you would write it as G:B. 

Ratio Terminology 

In the example above, B and G are terms. B is called the antecedent term and G is called the consequent term. 

Example problem: 

In a classroom with 15 total kids there are 3 kids with blue eyes, 8 kids with brown eyes, and 4 kids with green eyes. Find the following: 

The ratio of blue eyed kids to kids in the class? 

The number of blue eyed kids is 3. The number of kids is 15.
Ratio: 3:15 

The ratio of brown eyed kids to green eyed kids? 

The number of brown eyed kids is 8. the number of green eyed kids is 4.
Ratio: 8:4 

Absolute values and reducing ratios 

In the examples above we used the absolute values. In both cases these values could have been reduced. Just like with fractions, ratios can be reduced to their simplest form. We'll reduce the above ratios to their simplest form to give you an idea as to what this means. If you know how to reduce fractions, then you can reduce ratios. 

The first ratio was 3:15. This can also be written as the fraction 3/15. Since 3 x 5 =15, this can be reduced, like a fraction, to 1:5. This ratio is the same as 3:15. 

The second ratio was 8:4. This can be written as the fraction 8/4. This can be reduced all the way to 2:1. Again, this is the same ratio, but is reduced so that it is easier to understand. 

Order of Operations

Skills needed:
Multiplication
Division
Addition
Subtraction 

In math problems it's important to do the operations in the right order. If you don't, you may end up with the wrong answer. In math, there can be only one correct answer, so mathematicians came up with rules to follow so we can all come up with the same correct answer. The correct order in math is called the "order of operations". The basic idea is that you do some things, like multiplication, before others, like addition. 

For example, if you have 3 x 2 + 7 = ? 

This problem could be solved two different ways. If you did the addition first you would get: 

3 x 2 + 7
3 x 9 = 27 

If you do the multiplication first, you get: 

3 x 2 + 7
6 + 7 = 13 

The second way is correct as you should do the multiplication first. 

Here are the rules in the Order of Operations:

• Do everything inside of brackets first.
• Next, any exponents or roots (if you don't know what these are, don't worry about them for now).
• Multiplication and division, performing them left to right
• Addition and subtraction, performing them left to right

Let's do a few examples: 

40 + 1 - 5 x 7 + 6 ÷ (3 x 2) 

First we do the brackets: 

40 + 1 - 5 x 7 + 6 ÷ 6 

Now we do the multiplication and division, left to right: 

40 + 1 - 35 + 1 

Now addition and subtraction, left to right: 

The answer = 7 

Note: even on the last step if we had added 35 + 1 first then we would have done 41 - 36 = 5. This is the wrong answer. So we need to do the operations in order and left to right. 

Another order of operations example: 

6 x 12 - (12 x 7 - 10) + 2 x 30 ÷ 5 

We do the math inside the brackets first. We do the multiplication in the brackets first: 

6 x 12 - (84 - 10) + 2 x 30 ÷ 5 

Finish the brackets: 

6 x 12 - 74 + 2 x 30 ÷ 5 

Multiplication and division next: 

72 - 74 + 60 ÷ 5
72 - 74 + 12 

The answer is 10. 

How to remember the order? 

There are different ways to remember the order. One way is to use the word PEMDAS. This can be remembered by the phrase "Please Excuse My Dear Aunt Sally". What it means in the Order of Operations is "Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction". When using this you must remember that multiplication and division are together, multiplication doesn't come before division. The same rule applies to addition and subtraction. 

Linear Equations - Slope Forms

This page assumes you have some basic knowledge of linear equations and slope. In the linear equations basics section we discussed the standard form of a linear equation where Ax + By = C. 

There are other ways that linear equations can be written that can help provide useful information for graphing. They are called slope forms. There is the slope-intercept form and the point-slope form. 

Slope-Intercept Form 

The slope intercept form uses the following equation: 

y = mx + b

In this equation, x and y are still the variables. The coefficients are m and b. These are numbers. 

The advantage of putting a linear equation in this form is that the number for m equals the slope and the number for b equals the y-intercept. This makes the line the equation represents simple to graph. 

m = slope
b = intercept 

slope = (change in y) divided by the (change in x) = (y2 - y1)/(x2 - x1) 

intercept = the point where the line crosses (or intercepts) the y-axis 

Example Problems: 

1) Graph the equation y = 1/2x + 1 

From the equation y = mx + b we know that: 

m = slope = ½
b = intercept = 1 

1) Graph the equation y = 3x - 3 

From the equation y = mx + b we know that: 

m = slope = 3
b = intercept = -3 



Point-Slope Form 

The point-slope form of linear equation is used when you know the coordinates of one point on the line and the slope. The equation looks like this: 

y - y1 = m(x - x1)

y1, x1 = the coordinates of the point you know
m = the slope, which you know
x, y = variables 

Example Problems: 

Graph a line that passes through the coordinate (2,2) and has a slope of 3/2. Write the equation in the slope-intercept form. 

See the graph below. First we plotted the point (2,2) on the graph. Then we found another point using a rise of 3 and a run of 2. We drew a line between these two points. 






To write this equation in slope-intercept form we use the equation: 

y = mx + b 

We already know that the slope (m) = 3/2 from the question. The y-intercept (b) we can see is at -1 from the graph. We can fill in m and b to get the answer: 

y = 3/2x -1 

Things to Remember

• Slope-intercept form is y = mx + b.
• Point-slope form is y - y1 = m(x - x1).
• We can write a linear equation in three different ways: standard form, slope-intercept form, and point-slope form.