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Algebra 1 Fundamentals: Simplifying, Adding, Subtracting, and Dividing Integer Fractions
Transcript of Algebra 1 Fundamentals: Simplifying, Adding, Subtracting, and Dividing Integer Fractions
Algebra 1 Fundamentals: Fractions
Fractions are a way of expressing numbers in terms of wholes and parts of wholes.
For example, a pie can be cut in half to create 2 equal pie pieces called halves. The pie can then be cut in half again in the other direction to create 4 equal pieces of pie: these are called fourths.
You can also see from the chart that one half pie piece is made up of 2 fourths pie pieces:
Mixed Numbers and
Mixed Numbers and Improper Fractions are ways we can express numbers greater than 1.
Adding & Subtracting Fractions
If we took a pie that was already cut in half and multiplied the number of pieces by cutting each of them in half, it would create 4 total pieces, each being equal to one fourth of the pie:
by Patrick Bennett, Saint Mary's High School
For some students mastering fractions seems out of reach.
Some students resort to their calculator to turn fractions into decimals. Other students ignore or avoid fractions altogether.
However, in many cases working with fractions in an algebra problem is actually much easier than working with decimals, and is a skill almost any algebra student can get comfortable with after some practice!
Let's review the fundamentals of working with fractions - hopefully you'll discover they're well within your grasp!
One half =
This pie is cut up into ten pieces ("tenths"). It is easy to see that one half of the pie is equal to 5 tenths:
The first pie is divided into 3 pieces - "thirds". The second pie is divided into nine pieces - "ninths".
We can see from these pies that six ninths is equal to two thirds:
In math we like to state numeric values in the simplest way possible!
So, given a fraction-number like "two fourths", we'd rather say "one half". We'd rather say "two thirds" instead of "six ninths".
How can we figure out if the fraction we're dealing with is in it's simplest form? We can do it by factoring!
The mixed number three and one half is the same as the improper fraction seven halves.
In many cases in algebra, using an improper fraction for our calculations will be easier and more reliable. We can use a simple technique to change mixed numbers into improper fractions.
If we need to convert the other way around, from an improper fraction to a mixed number, we simply divide, then put the remainder over the denominator:
As a matter of fact, the fraction bar is also called the "division bar".
To add or subtract fractions, the denominators (bottoms) must be the same. When they are the same, simply add across the top, but leave the bottom alone. After doing any operations with fractions, always simplify!
To add fractions when there are mixed numbers, simply convert the mixed numbers to improper fractions first, then add the fractions as above, and simplify.
What about adding or subtracting fractions when the denominators are *not* the same?
This shows us how we can multiply fractions: multiply across the top, and also across the bottom:
One third pieces multiplied into four pieces each:
To add or subtract fractions with unlike denominators, we have to transform them into fractions that have a common denominator. We can do this if we "unsimplify" them by using a multiplier.
Well we can do it the other way around too!
So, if we had to add fractions such as these:
we first must "unsimplify" so that both are in fourths:
Step 1: transform one half into fourths by multiplying both the top and bottom by 2:
Step 2: Rewrite the problem with the common denominator, and add the fractions:
Least Common Denominators
This trick of changing the denominators before adding or subtracting them is called using the "Least Common Denominator" or "LCD". This is just a fancy term for the smallest number on the bottom of the fractions that can be the same for all of them. Also, you may recall that the LCD is just the Least Common Multiple ("LCM") of the denominators of the fractions.
A lot of the time it's easy to see what the LCD is. Other times it isn't so easy, and we'll need a reliable method for finding it - this is where factoring again comes to the rescue!
Finding the LCD:
Factor each denominator to primes;
Borrow each base factor, along with it's greatest power;
Multiply out to find the LCD.
We can tell from using a pie chart that the answer is 5 fourths!
Dividing a fraction by another fraction is like asking "how many small pie pieces will fit into this big piece?"
This division problem is asking "how many one-sixth pieces of pie will fit into a one half piece of pie?"
This gives us a clue about solving this sort of division mathematically: notice if we flip the second fraction over and multiply instead of dividing, we will get the correct result:
This "flipped" fraction is called the "reciprocal": when we need to divide by a fraction, we can use the fraction's reciprocal and multiply instead! So, to divide fractions just remember "
flip and multiply
Fractions are a way to express parts of whole numbers. Using pies and slices of pie to represent fractions and math problems with fractions can be very helpful.
Simplifying fractions is just a way of expressing the same quantity with smaller numbers. To simplify, use factoring and cancel the common factors on the top and bottom of the fraction.
As a rule of thumb, fractions should always be fully simplified in the final answer.
Mixed Numbers and Improper Fractions are two different ways of expressing numbers bigger than 1. Generally, it is better to use Improper Fractions in math calculations. However, if a problem is *asked* as a Mixed Number, it should normally be *answered* as a Mixed Number too: remember in this case to convert your Improper Fractions back to Mixed Numbers.
Adding and subtracting fractions requires a *common denominator*. If your fractions don't already have one, use factoring to find the Least Common Denominator, then continue. Add and subtract the numbers across the top, but leave the denominator on the bottom alone.
Multiply fractions by multiplying across the top, then across the bottom. Don't forget to simplify!
Divide fractions by inserting just one step: first flip the second fraction to get its reciprocal, then continue with multiplication. Remember, for division: "flip and multiply"!