Comparing Fractions

In this lesson, you will learn about comparing fractions. Before you start this lesson, I recommend that you study or review my lesson about fractions. There are many ways to compare fraction.

Comparing Fractions

Equivalent Fractions
Equivalent fractions are fractions that represent the same quantity.

example:
1/2 = 2/4 = 4/8

Fundamental Law of Fractions: the value of a fraction does not change when its numerator and denominator are both multiplied by the same number (not zero).
[Remembering that multiplication by ½ can also be considered division by 2, this law also extends to "division".]


Simplifying Fractions
Reducing Fractions

When working with fractions, it is a common practice to list fractional answers in their "simplest form". The simplest form usually gives the reader a better idea of what the fraction represents.

To simplify (or reduce) a fraction, look for a number that will divide exactly into BOTH the numerator and the denominator. Try to find the "largest" such value (the greatest common divisor).

Example:
3/6 = 1/2

If you don't choose the LARGEST common factor (the GCD), you can still reduce (or simplify) the fraction, but you will have to do the division process more than one time.

Less Than, Greater Than, Equal To
Comparing fractions as to their size.
A. Simple Cut-up Visual Comparisons
These diagrams show a pictorial representation of what is occurring in relation to the size of the fractions. Each of the whole bars are divided into 5, 3, or 6 equal pieces and compared. Unfortunately, it can be difficult to compare fractions in this simple cut-up visual manner. If we did not know that 2/3 = 4/6 in the last comparison, it would be difficult to tell from the diagram whether the fractions were equal or just really close to being equal.

B. Compare using a Common Denominator
1. Comparing fractions that already have the same denominator is easy. The fraction with the larger numerator will be the larger fraction. Since the denominators are the same, you are comparing "pieces" (parts) of the same size.

2. Comparing fractions that have different denominators is harder. The solution requires that we create equivalent fractions that have the same denominators and then compare them. When looking for this same denominator (called the common denominator), look for the smallest possible number (called the least common multiple or the least common denominator) to keep the calculations as simple as possible.

If we re-examine our visual models from above, we can see that when the whole bar is divided equally for BOTH fractions, using the common denominator, we have a more reliable and accurate portrayal of the fraction sizes.

3. Compare using the Cross Multipy Rule
We saw in Ratios and Proportions that the Cross Multiplication Algorithm (Rule) was the result of rewriting fractions to have the same denominator and then comparing the numerators.
Here's how this process works:
To compare two fractions,

• multiply the numerator from the fraction on the left by the denominator of the fraction on the right.
• Place this number above the fraction on the left.
• Now, multiply the denominator from the fraction on the left by the numerator of the fraction on the right. 
• Place this number above the fraction on the right.
• Place the appropriate inequalitiy symbol (< , >, or even =)

As with the Cross Multiplication Algorithm (Rule), this cross multiply application to comparing fractions is actually the result of rewriting fractions to have the same denominators (which we do not write down) and then comparing the numerators.

4. Ordering Fractions
Place in ascending order (from smallest on the left to largest on the right)




This article was originally published on The mathbits notebook. Read the original article.

Source:
https://mathbitsnotebook.com/JuniorMath/FractionsDecimals/FDcomparing.html

Operation of Algebraic Expression

Before we see how to add and subtract integers, we define terms and factors.

Terms and Factors

A term in an algebraic expression is an expression involving letters and/or numbers (called factors), multiplied together.

Example
The algebraic expression

5x

is an example of one single term. It has factors 5 and x.

The 5 is called the coefficient of the term and the x is a variable.

Like Terms

"Like terms" are terms that contain the same variables raised to the same power.

Example
3x and 7x are like terms.

Adding and Subtracting Terms

Important: We can only add or subtract like terms.

Why? Think of it like this. On a table we have 4 pencils and 2 books. We cannot add the 4 pencils to the 2 books - they are not the same kind of object.

We go get another 3 pencils and 6 books. Altogether we now have 7 pencils and 8 books. We can't combine these quantities, since they are different types of objects.

Next, our sister comes in and grabs 5 pencils. We are left with 2 pencils and we still have the 8 books.

Similarly with algebra, we can only add (or subtract) similar "objects", or those with the same letter raised to the same power.

Example
Simplify 13x + 7y − 2x + 6a

13x + 7y − 2x + 6a

The only like terms in this expression are 13x and −2x. We cannot do anything with the 7y or 6a.

So we group together the terms we can subtract, and just leave the rest:

(13x − 2x) + 6a + 7y

= 6a + 11x + 7y

We usually present our variables in alphabetical order, but it is not essential.

Multiplication of Algebraic Expressions

When we multiply algebraic expressions, we need to remember the Index Laws from the Numbers chapter.

Let's see how algebra multiplication works with a series of examples.

Example
Multiply (x + 5)(a − 6)

We multiply this out as follows. We take each term of the first bracket and multiply them by the second bracket. Then we expand out the result.

(x + 5)(a − 6)

= x(a − 6) + 5(a − 6)

= ax − 6x + 5a − 30

We cannot do any more with this answer. There are no like terms, so we cannot simplify it in any way.

Dividing by a Fraction

Recall the following when dividing algebraic expressions.

The reciprocal of a number x, is 1/x

For example, the reciprocal of 5 is 1/5 and the reciprocal of 5/3 is 3/5

To divide by a fraction, you multiply by the reciprocal of the fraction.




This article was originally published on The Interactive Mathematics. Read the original article.

Source:
https://www.intmath.com/basic-algebra/1-addition-subtraction-algebra.php

https://www.intmath.com/basic-algebra/2-multiplication-algebra.php

https://www.intmath.com/basic-algebra/3-division-algebra.php