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This is a short reminder on Fractions for a grade 10 learner who just forgot how fractions are operated and how they can handle complex fractions in mathematical problems.

Simply put, this a fraction...

Mind the line between 1 and 4. I call it a fraction line. Numbers above it make up our numerator, and numbers below it make up our denominator. In this case 1 is our numerator and 4 is our denominator. But, look at this one for example.

Remember, what's above our fraction line makes up our numerator. Therefore, the sum of 1 and 2 is our numerator. The difference of 3 and 2 makes our denominator.

A fraction cannot have a zero as its denominator. This is one of the things we tend to forget when we do calculations mentally.

A fraction is a division of two numbers. In illu-1, we can say 1 is divided by 4. But often times, this is not how you'll hear professionals call it out... you will most likely hear a fourth or a quarter. This is common with other numbers, but they have their names (not necessarily names) too. Look at these few ones.

• 1

  3

  third

• 1

  4

  forth / quarter

• 1

  5

  fifth

• 1

  6

  sixth

• 1

  7

  seventh

• 1

  8

  eighth

• 1

  9

  nineth

• 1

  10

  tenth

Now? What will you call 2/3 or 8/5? Keep thinking about this questions...

Even though I will only talk about basic fractions, it is important to know that fractions can be classified into atleast 3 types. We have Proper Fractions, Improper Fractions and Mixed Fractions.

A proper fraction is a fraction whose denominator is greater than its numerator. For example 1/5.On the other hand, an Improper Fraction is a fraction whose denominator is less than its numerator, take 5/2 for example. A mixed fraction is a proper or improper fraction mixed with a whole number like 3(4/5).

Fractions can be negative if its numerator or denominator is negative. But, if the numerator and the denominator are both negative, our fraction will be positive. Sounds a bit confusing, let's look into it.

take this four fifths for example.

This fraction is positive, and its numerator and denominator are both positive. Let's turn its denominator into a negative 5.

Remember I said this fraction was negative, but now, our negative denominator will make our fraction negative. Like this

Now, if we have a negative numerator value, what will it do?

This will also change our fraction to a negative fraction.

...but, what if our numerator and denominator are both negative?

We usually say, these negative signs will cancel each other out. But, what we simply mean is, 1 negative will make our fraction negative, and then the other negative will make our negative fraction positive. Hence we say they cancel out.

...Tricky question. What if our negative fraction has a negative denominator?

I'll let you think about that for a second, as I continue. Signs of a fraction is also one of the few things we tend to forget about when we do calculations, this is why it is important for me to start with them. Also, it lays an important foundation of simplifying your fractions to a accepted standard. A 4/-5 is not accepted as a simplified answer.

Now that I mentioned standards, one of the important ones is that your fractions needs to be in their most simplified versions. Take this fraction for instance...

Five fifteenth sounds a bit rough, but besides that, it can be simplified. We simply say what is the highest common factor between 5 and 15? We know it is 5. Then we say, how many times does 5 go into 5? 1 time. But in 15? 3 times. Therefore our simplified version of 5/15 is...

Which is a third! You wont need to say 5 fifteenth anymore... because its simpliest equivalent is a third! Now, mind that word, "equivalent".

Before we jump to our next chapter, what was your answer on simplifying a negative fraction whose denominator is a negative?

A negative denominator changes the sign of a fraction, regardless of what it is. In this case, it is a negative, so it will change it into a positive.

First of all, fractions can be tricky sometimes. So, take it easy with yourself. Once you good, let's roll.

First of all, this doesnt matter whether you add 3/4 into 5 or 5 into 3/4. Remember that. This method works for both of the cases. Here we first multiply 5 and the denominator, and then add that product to the numerator.

Then we can just calculate and simplify our fraction if needed.

Let's try one more example.

Remember, it doesn't matter whether we have 2 + 5/3 or 5/3 + 2. This method is the same.

First step : I multiply my 2 with my denominator

Second step : I add the product I got from my first step to the numerator.

Last step : I then calculate and simplify if needed

It's important to notice that my denominator didn't change.

There is a slight difference between addition and subtraction I need to mention. But you will pick it up as you go through this section and take your quizzes.

Remember when we were adding a fraction and a number, order didnt matter much. But when subtracting numbers, order matter. 3-2 is not the same as 2-3. One gives us a positive 1 and the other one gives us a negative 1.

If we are subtracting a number from a fraction like we have here, we need to always subtract the product we get from multiplying the denominator and the number from the numerator. This order matter.

Then we can simply follow with our calculations and simplify if needed.

The only thing we need to keep in mind is this order. Let's try one more example. Now we are subtracting a fraction from a number. Unlike subtracting a number from a fraction.

So, we gonna subtract the numerator from the product we get from multiplying our denominator and the number

First step : I multiply 2 with my denominator

Second step : I subtract the numerator from the product I get from first step.

Last step : I then calculate and simplify if needed

As you can see, the order matter. The reason why this order matter is because, proper fractions are normally smaller numbers so if we subtract a big number from a small number, this difference will be different from subtracting a small number from a bigger number.

This is the most easiest of these operations. So, you can relax a bit. Let's look at how we can multiply a fraction and a number or vice versa...

Whenever you have a problem similar to this one, always remember that we just multiply the numerator with our number.

Then we can just calculate and simplify...

You always multiply the number with the numerator. It is seriously that simple. And then calculate, lastly simplify if need be. Let's look at it again...

Remember that 3 x 2 = 2 x 3... so, it doesn't matter if we have a number first or the fraction. Our method is the same

First step : I multiply 2 and my numerator.

Last step : I then calculate and simplify if needed

We can conclude that order doesnt matter when we multiply two terms.

We gonna perform some black magic on this section. But I need you to pay close attention to what I will be talking about and the order I do things. Dividing a number is not as intuitive as other operations we've looked into in our previous chapters.

Before we continue, I want to mention this. Every number has an invisible denominator of 1. We assume this because first, it doesn't change the value of the number, and second, it is going to make my method easy to understand... so you can assume that 5 is 5/1 like...

Let's look at our first example...

Let's remember that 5 is just 5/1...

There are 2 things you will now do... change the division operator to a multiplication operator and then swap the the left side fraction's numerator and denominator...

That was the black magic I was telling you about. You can actually do this everytime you are dividing with a number. This also means 10/2 is 10 * 1/2. The last step is multiplying the numerator with the numerator and denominator with the denominator.

Then calculate and simplify if you need to.

After understanding the Fraction and Number type of problems and you did you enough quizzes on them, I believe these coming sections will be a bit easy for you than they were for me many years ago.

Let's look at how we can sum two fractions firs. I need to mention that there are a couple of ways you can approach two fractions, but I chose this method because I find it to be the easiest.

My approach is, I will first multiply the denominators, which are 4 and 6. Then I cross multiply, 6 with 3 and 4 with 5... then sum the 6 x 3 + 4 x 5 on my numerator. It sounds like witchcraft, but lemme show you...

In my denominator, I simply multiplied my denominators which are 4 and 6. Then in my numerator, I summed the cross products of 6 x 3 and 4 x 5. I find it easy this way. But let's finish our calculations and simplify if needed.

On this example, I'll show you another method which is popular in schools.

We will first look at multiples of our denominators. We have 3 and 5, multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, 27, 30... etc, and of 5 are 5, 10, 15, 20, 25, 30, 35...etc.

Between these set of multiples, some numbers are common, such as 15 and 30... that is why they are called as common multiples. Between these two, 15 is the lowest common multiple, and it is called LCM.

Since 15 is our LCM, we need a number that we can multiply 3 with to make it 15 and multiply 5 with to make it 15. Those numbers are 5 and 3 respectively. And this is not coincidental, but we will get to that later.

If we multiply 3 with 5, or 5 with 3, this will change the value of our expression. Which we dont want. So, we need to come up with a way that doesnt change the value of our expression but ends up changing our denominators. If we multiply 2/3 with 5/5 and 4/5 with 3/3, it magically changes our denominators to 15 but the value of the expression remains the same.

...and that looks kind of suspicious, I know. But, stick with me. Remember how we multiplied 2 fractions in our previous chapter? Let's do the same...

Now that our denominators are the same, we can just add the numerators and divide with one of our denominator.

Let's go over another example...

First step : Get our multiples. For 2 are 2, 4, 6, 8, 10, 12...etc and for 6 are 6, 12, 18, 24...etc. 6 is our LCM. So, we need to multiply 2 with 3 for it to be 6 and 6 with 1... we will talk more about this LCM thing later. For now, I'm trying to simplify my explanations.

Second step : Multiply 1/2 with 3/3 and 5/6 with 1/1... 1/1 is basically 1. But just write it, you will ommit it when you are getting good with this.

Third step : Calculate your fraction multiplications..

Last step : Now denominators are the same, add the numerators and divide with the denominator.

Try both methods and choose one that resonates with you.

There is a slight difference between addition and subtraction I need to mention. But you will pick it up as you go through this section and take your quizzes.

Remember when we were adding a fraction and a number, order didnt matter much. But when subtracting numbers, order matter. 3-2 is not the same as 2-3. One gives us a positive 1 and the other one gives us a negative 1.

If we are subtracting a number from a fraction like we have here, we need to always subtract the product we get from multiplying the denominator and the number from the numerator. This order matter.

Then we can simply follow with our calculations and simplify if needed.

The only thing we need to keep in mind is this order. Let's try one more example. Now we are subtracting a fraction from a number. Unlike subtracting a number from a fraction.

So, we gonna subtract the numerator from the product we get from multiplying our denominator and the number

First step : I multiply 2 with my denominator

Second step : I subtract the numerator from the product I get from first step.

Last step : I then calculate and simplify if needed

As you can see, the order matter. The reason why this order matter is because, proper fractions are normally smaller numbers so if we subtract a big number from a small number, this difference will be different from subtracting a small number from a bigger number.

There is a slight difference between addition and subtraction I need to mention. But you will pick it up as you go through this section and take your quizzes.

Remember when we were adding a fraction and a number, order didnt matter much. But when subtracting numbers, order matter. 3-2 is not the same as 2-3. One gives us a positive 1 and the other one gives us a negative 1.

If we are subtracting a number from a fraction like we have here, we need to always subtract the product we get from multiplying the denominator and the number from the numerator. This order matter.

Then we can simply follow with our calculations and simplify if needed.

The only thing we need to keep in mind is this order. Let's try one more example. Now we are subtracting a fraction from a number. Unlike subtracting a number from a fraction.

So, we gonna subtract the numerator from the product we get from multiplying our denominator and the number

First step : I multiply 2 with my denominator

Second step : I subtract the numerator from the product I get from first step.

Last step : I then calculate and simplify if needed

As you can see, the order matter. The reason why this order matter is because, proper fractions are normally smaller numbers so if we subtract a big number from a small number, this difference will be different from subtracting a small number from a bigger number.

There is a slight difference between addition and subtraction I need to mention. But you will pick it up as you go through this section and take your quizzes.

Remember when we were adding a fraction and a number, order didnt matter much. But when subtracting numbers, order matter. 3-2 is not the same as 2-3. One gives us a positive 1 and the other one gives us a negative 1.

If we are subtracting a number from a fraction like we have here, we need to always subtract the product we get from multiplying the denominator and the number from the numerator. This order matter.

Then we can simply follow with our calculations and simplify if needed.

The only thing we need to keep in mind is this order. Let's try one more example. Now we are subtracting a fraction from a number. Unlike subtracting a number from a fraction.

So, we gonna subtract the numerator from the product we get from multiplying our denominator and the number

First step : I multiply 2 with my denominator

Second step : I subtract the numerator from the product I get from first step.

Last step : I then calculate and simplify if needed

As you can see, the order matter. The reason why this order matter is because, proper fractions are normally smaller numbers so if we subtract a big number from a small number, this difference will be different from subtracting a small number from a bigger number.