Basic Math Adding and Subtracting Fractions with Different Denominators

Welcome back to Educator.com.0000

This lesson, we are going to add and subtract fractions with different denominators.0002

Before we begin with that, let's review over the lesson on least common multiple, the LCM.0013

This I believe was a few lessons ago.0022

If you want, you can go back to that lesson.0025

We are just going to do a brief example here.0028

To find the LCM of 6 and 4, I am going to take the two numbers.0031

I am going to do the factor tree method on each of them.0041

For 6, a factor pair of 6 is going to be 2 and 3.0046

They are both prime; I am going to circle them both.0053

For 4, it is going to be 2 and 2; I am going to circle them.0056

Here, to find the LCM, I am going to look at what they have in common.0072

I know that I have a 2 here; I also have a 2 here.0082

I am going to write that 2 by itself.0087

This pair cancels out one of them; I am going to write a 2.0093

The other numbers, 3 and 2, the other remaining numbers are going to go tag along with it.0098

Again to find the LCM, I just find what they have in common.0112

Since there is a 2 here and a 2 here, one of those 2s get cancelled.0119

It is going to be 2 times 3 times 2.0124

2 times 3 is 6; 6 times 2 is 12; my LCM is 12.0130

The LCM of 6 and 4 is going to be 12.0139

LCM is the same thing as LCD.0147

LCM stands for least common multiple; LCD is least common denominator.0154

When I am using those two numbers as my denominators, then it is going to be called LCD.0162

But I am still going to find the LCM between those two denominators.0170

The reason for this, whenever I add fractions or subtract fractions,0174

I have to make sure that these denominators are the same.0179

In order to make them the same, I need to find the LCM or the LCD between the two numbers.0184

Here 6 and 4, just like what we did, the example, we know that the LCM is 12.0193

The LCM, 1/6, what I am going to do is I am going to make 1/6 the same fraction with the denominator becoming 12.0204

Same thing here, 3/4, the denominator is going to change to 12.0221

I want to figure out what these top numbers are going to be, my numerators.0229

How do I go from a 6 to a 12?--what do I multiply it by?0235

I multiplied this by 2; or I can do 12 divided by 6; I get 2.0240

Since I multiplied the 6 by 2 to get 12, I need to also multiply the top number by 2.0250

1 times 2 is 2; the fraction 1/6 became 2/12.0258

These are the same fractions; 1/6 is the same thing as 2/12.0267

Same thing here, 3/4; to go from 4 to 12, I have to multiply it by a 3.0274

Then I have to multiply the top number by 3; 3 times 3 is 9.0285

3/4, because I multiplied the top and the bottom by the same number, these fractions become the same.0294

They are the same fraction; 3/4 is the same thing as 9/12.0304

Now since I know that 2/12 is the same thing as 1/6 and 9/12 is the same thing as 3/4,0310

I can add these two fractions, this fraction and this fraction.0318

If I add these two, then my answer will be the same as if I add these two.0330

I have to do that because these denominators are different.0337

I have to make them the same by converting these fractions,0341

by changing these fractions so that the denominators will be the same.0344

Now that they are, I am going to take my numerators and add them together.0351

2 plus 9 is 11; here the denominator is 12; 12.0357

Then my denominator here has to stay the same as a 12.0365

2/12 plus 9/12 is 11/12; or I can say that 1/6 plus 3/4 is 11/12.0369

Let's do another example; here I am going to subtract.0382

But before I do that, I have to check my denominators.0388

This denominator is an 8; this one is a 4; they are different.0391

I have to find the LCD or the LCM between 8 and 4 so that I can make the denominators the same.0397

I am going to take 8 and 4; you could do the factor tree.0407

4 and 2; circle that one; that is a prime.0419

2 and 2; this is 2 and 2; look at this one.0424

There is a common one here; I am going to cancel out one of them.0433

I have another common between these two so I am going to cancel out one of them.0438

Then I just multiply all the remaining circled numbers.0442

It is 2 times 2 times 2 which is 8.0447

If I look at these, I can just look at them and figure out what the LCM is by looking at the multiples.0455

Multiples of 8 would be 8, 16, 24, and so on.0464

For 4, it would be 4, 8, 12, 16, and so on.0469

You are going to find the smallest common multiple between them which is 8.0473

Here I am going to change this fraction and this fraction so that their denominators will be the same.0480

For this fraction, 7/8, my LCM is already 8.0489

My LCM or my LCD, it is already 8.0494

For that one, I can just keep it the way it is.0498

For this one however, 1/4, I have to convert it; I have to change it.0504

I need a top number; 4; to get 8, I multiply it by 2.0515

Again I have to multiply the same number to the top which is 2.0523

Whenever you are converting fractions, as long as you multiply the top0530

and the bottom by the same number, then your fraction will stay the same.0533

Even if you change the numbers, it is still the same fraction; 1/4 became 2/8.0538

Now I am going to rewrite my problem, 7/8 minus 2/8.0548

Make sure the denominators are the same.0556

If they are not the same, then you did something wrong.0558

Go back and check your work.0561

Since they are the same, I can go ahead and subtract them.0565

7 minus 2 which is 5; then my denominator, 8.0568

8 here; it stays an 8 there; 7/8 minus 1/4 is going to equal 5/8.0576

Let's add this next problem, 9/10 plus 3/15.0591

Again I have to check my denominators; they are not the same.0599

I have to find the least common denominator with them.0602

I am going to take 10; do the factor tree which is 5 and 2.0606

Circle them if they are prime; only circle them if they are prime.0612

Then 15, this becomes 5 and 3.0616

If you are confused about how to find the LCD or LCM,0625

then you can go back and look at the lesson on that one before continuing.0630

My LCM or I am just going to call it the LCD since they are my denominators.0636

I look for any common numbers between them; they have a 5; 5 is common.0642

Whenever they have something in common, just cancel one of them out.0650

That is all they have in common.0654

Then for my LCD, I am just going to write out the remaining circled numbers.0656

Remember they can only be circled; 5 times 2 times 3.0662

5 times 2 is 10; 10 times 3 is 30; my LCD is going to be 30.0667

I have to change this fraction so that my denominator will become 30.0676

Same thing here, change this fraction so my denominator will be 30.0681

9/10, going to convert it; I can take 30 divided by 10; that is 3.0688

I know that I did 10 times 3 to get 30.0702

Again you have to do it to both the top and the bottom, the same number.0707

That is the only way you are going to have the same fraction because you don't want to change your fraction.0711

Even if you are changing the numbers, it is still the same fraction.0715

9 times 3 is 27.0720

I am going to do the same thing for the other fraction.0727

15 times 2 was 30; 3 times 2... again multiply it by the same number.0733

It is going to be 6.0742

Since 9/10 is the same thing as 27/30 and 3/15 is the same as 6/30, I need to add my new fractions.0747

Again double check your denominators; make sure they are the same.0764

It is going to be 27 plus 6.0769

27 plus 6 is 33 over... your denominator will stay the same.0772

It is 33/30; let's look at this fraction.0785

This is your answer; this is a solution to this problem.0788

But I have an improper fraction because the top number, the numerator, is bigger than the denominator.0793

You can either leave it like this; this is still the correct answer; or I can simplify it.0800

I know that a 3 goes into 33 and a 3 goes into 30.0811

I can take that number, the common number, the common factor between 33 and 30,0819

divide it to both the top and the bottom.0828

Remember as long as you are doing the same thing to the top and to the bottom of the fraction,0831

you are not changing it; you are just simplifying it.0835

33 divided by 3 is 11; 30 divided by 3 is 10.0839

This is your new improper fraction, 11/10.0850

Since it is an improper fraction, we can change it to a mixed number.0854

Or we can just leave it like that; that is fine too.0858

But if I do want to change it to a mixed number,0861

then this 10 fits into the top number 11 only one time.0864

10 fits into 11 only one time.0873

How many left over do I have?--only one.0877

My denominator always has to stay the same.0881

11/10 is the same thing as 1 and 1/10.0884

Another example, we are going to take 11/20 and subtract it to 11/30.0896

My denominators are different; I have to find the common denominator.0903

I can take 20; 5 is a prime number; I am going to circle it.0910

4, 2, and 2; I circle those; and then 30.0918

For this one, I can either do 3 and 10 or I can do 15 and 2, any factor pair.0927

Let's do 3 and 10; here 3 is a prime number; I am going to circle it.0933

10 is 5 and 2; they are both prime.0939

I am going to look for any numbers they have in common.0945

Here; I have a 2 here; and I have a 2 here.0949

I am going to cancel one of them out.0954

Here I have a 5; and I have a 5 here.0957

I am going to cancel just one of them out.0960

Any others?--nope, that is it.0963

My LCD or my LCM is going to be 2 times 2 times 5 times 3.0967

This is going to be 4 times 5 which is 20, times 3 which is 60; my LCD is 60.0981

Then my next step is going to be to change each fraction so that their denominator will become 60.0993

20, to figure out what you have to multiply to 20 to get 60,1004

I can just take 60 and divide it by 20.1009

This is going to be 3; 20 times 3 was 60.1014

Again you have to multiply the top number by the same number.1019

11 times 3 is 33.1022

For the second fraction, 11/30, 30 times 2 is 60.1028

Multiply the top number by that number; 22.1040

11/20 is the same thing as 33/60; I am going to subtract.1047

Then 11/30 is the same thing as 22/60.1055

Again double check your denominators; make sure that they are the same.1061

Since they are, now I can subtract; 33 minus 22 which is 11 over...1068

Keep your denominator the same; do not add or subtract your denominators.1078

11/20 minus 11/30 became 11/60.1085

Let's do another example; this example, 23/95 plus 4/5.1093

In order for me to add these two fractions, I have to make sure they have a common denominator.1104

In this case, they don't; 95 is this denominator; 5 is the other one.1108

I have to look for the common denominator.1115

For 95, I can either look for the LCM, the least common denominator or least common multiple, between 95 and 5.1122

Or I can list all the multiples out and see the smallest common multiple.1134

I know that 95 is divisible by 5 because any number that ends in a 5 or 0 is divisible by 5.1143

In this case, a 5, if this number is divisible by this number,1156

then this becomes the new common denominator, the least common denominator.1162

Or if you want to just do the factor tree to find the least common denominator, then you can do that too.1167

95 is going to be 5 times 19; these are both prime numbers.1175

I am going to circle them; 5 is just 5 and 1.1189

To find the LCD, I am going to look for any factors they have in common.1197

Here, there is a 5 here and a 5 here.1207

I am going to cancel only one of them out.1210

Whenever they have something in common, just cancel only one of them out.1212

Then I am going to write all the circled numbers again; 5 times 19.1217

This is just a 1 so I don't have to write that.1225

5 times 19 I know is 95.1227

My LCD, my least common denominator, is going to be 95.1232

For this fraction here, since the denominator is already 95, I don't have to change it.1239

This one can stay as it is.1246

This one however, I have to change that 5 to make it a 95 so they will have a common denominator1250

because that is the only way I can add these fractions, if their denominators are the same.1256

For this fraction right here, I need to change it so that the denominator will become 95.1260

I am going to take this 95, divide it by 5 to see what I have to multiply this by.1271

That is 19; here I am going to take this and multiply it by 19.1281

This will become 76; 4/5 became 76/95.1296

Make sure you multiply it by the same number.1311

You have to multiply the top and the bottom number by the same number.1314

That way you are not changing the fraction.1318

You are just changing the numbers; but they are still equal fractions.1321

Now I am going to do 23/95 plus 76/95.1326

Again I have to make sure the denominators are the same.1340

If they are not the same at this point, then there is something wrong.1344

Go back and check your work.1347

But since they are the same, I can go ahead and add the fractions.1350

23 plus 76, I am going to add the numerators together.1354

If I add them, it is going to be 99.1358

Here denominator stays the same; it is 95 here; 95 here.1365

My denominator is going to become 95; 23/95 plus 4/5 is 99/95.1372

That is it for this lesson; thank you for watching Educator.com.1385