In this lesson, we're going to focus on adding and subtracting fractions. Let's say if we have 3 divided by 5 plus 4 divided by 7. How can we add these two fractions? Well here's a simple technique. First multiply 5 and 7. This will give you a common denominator of 35. Next, multiply 3 and 7. 3 times 7 is 21. And also, there's a plus in between.
Multiply 5 times 4, which is 20. 21 plus 20 is 41. So the answer is 41 divided by 35. Let's try another example. Let's say if we want to subtract 7 over 8 minus 2 over 9. Let's use the same technique. Let's multiply the two denominators, 8 and 9, which... is 72 and then the next one is going to be 7 times 9 which is 63 minus 8 times 2 which is 16 now let's subtract what is 63 minus 16 this is going to be 47 Now 47 is not divisible by 2, nor is it divisible by 3. So this is it. That's the final answer.
So now you know how to add or subtract two fractions. Now what if we wanted to add or subtract, let's say, three fractions instead of two what should we do in this case so let's say we wish to combine 3 over 4 plus 5 over 3 minus 7 7 over 2. Whenever you wish to add or subtract fractions, the denominator has to be the same. The denominator is the bottom part of the fraction. And right now they're all different. So how can we make them the same?
get the common denominator. If you want to find the least common denominator, make a list. All of the multiples of 2 are 2, 4, 6, 8, 10, 12, 14, and so forth.
Multiples of 3 are 3, 6, 9, 12, 15, 18, and so forth. And multiples of 4 are 4, 8, 12, 16, 20. What is the least common multiple? We're looking for a multiple that is common to all three numbers, but it's the lowest. The least common multiple is 12. 12 is common to 2, 3, and 4, and it's the lowest of such numbers.
Now granted, 24 is also a common multiple, and if you use 24, you can get the right answer, you just gotta simplify it at the end. So if you're ever unsure about how to find the least common denominator, you can find any common denominator. One simple technique is simply to multiply these three.
4 times 3 times 2 is 24, and you could use 24 and still get the right answer. So now that we know the least common denominator is 12, let's multiply each fraction in such a way to get 12. The first fraction, let's multiply the top and the bottom by 3, because 3 times 4 is 12. The second one, let's use 4, and for the last one, let's use 6. So looking at the first one, 3 times 3 is 9, 3 times 4 is 12. 4 times 5 is 20, 4 times 3 is 12. 7 times 6 is 42, 2 times 6 is 12. Now that we have the same... denominator we can combine the numerators. 9 plus 20 is 29 and 29 minus 42 is negative 13. So this is the final answer it's negative 13 divided by 12. So now it's your turn. Go ahead and try this example.
8 over 5 minus 2 over 3 plus 9 divided by 4. So go ahead and add these three fractions. So this time, to find the least common multiple, we're just going to multiply 5, 3, and 4. It may not be the least common multiple, but it is a common denominator, just so you know. If we multiply 5 times 4 times 3, this will give us 5 times 4 is 20, 20 times 3 is 60. So 60 is going to be the common denominator that we're going to try to get.
So we're going to multiply the first fraction by 12, because 12 times 5 is 60, and the second one by 20, because 20 times 3 is 60. By the way, if you want to find out the number, divide it. 60 divided by 5 will give you the 12. 60 divided by 3 will give you the 20. And 60 divided by 4 will give us the number that we need to multiply this fraction by, which is 15. Now, 12 times 8, that's 96. 5 times 12, we know it's 60. 2 times 20 is 40. 3 times 20 is 60. And 9 times 15. 15 times 10 is 150. So if you take away 15 from that, you'll get 15 times 9, so that's 135. And 4 times 15 is 60. Now, 96 minus 40, that's positive 56. And 56 plus 135. Let's go ahead and add those two numbers the old-fashioned way. 5 plus 6 is 11, carry over the 1. 1 plus 3 plus 5 is 9, plus 1. So the final answer is 191 divided by 60. In this lesson, we're going to focus on multiplying two fractions. Whenever you need to multiply, multiply the numbers across the fractions. 3 times 7 is equal to 21. and 5 times 2 is equal to 10 and so this is it.
The answer is 21 over 10. That's all you need to do when multiplying fractions. But sometimes the numbers may not be that small. Let's say if we have larger numbers. What should we do in this case? Now we can multiply.
cross we can multiply 24 and 45 which will give us a big number but do we really want to do that when multiplying fractions with large numbers it's in your best interest to break down the large number into small numbers for instance 24 is basically 6 times 4 27 is 9 times 3 45 is 9 times 5 and 30 is 6 times 5 you want to break it in such a way that you can cancel some numbers here we can cancel a 5 because we have one on top and the other on the bottom the same is true for the 9 and we can cancel a 6 so the So therefore, the final answer is 4 over 3. So we were able to get the final answer without multiplying 24 by 45. That step was necessary. Plus it's going to take some time and you need a calculator. Doing it this way requires no use of a calculator. Try this one. Multiply 56 divided by 77 by 35 over 40. Now 56 is 8 times 7. 77 is 11 times 7, 35 is 7 times 5, and 40 is 8 times 5. So we can cancel an 8, we can cancel a 7, and we can cancel a 5, leaving the final answer of 7 over 11. So now you know how to multiply two fractions.
In this lesson, we're going to focus on dividing two fractions. Let's use 8 over 5 as an example, and let's divide it by 12 over 7. Now, perhaps you heard of the expression keep, change, flip. It's useful when dividing fractions.
Keep the first fraction the same way, change, division to multiplication, and flip the second fraction. And now you can do it. So, 8 times 7 is 56, but we can simplify it before we multiply. 8 is basically 4 times 2, and 12 is 4 times 3. So we can cancel a 4. And now we can multiply.
2 times 7 is 14. And 5 times 3 is 15. So the final answer is 14 over 15. Try this one. What's 4 divided by 3? Divided by... 9 over 5. So using the expression keep change flip, let's keep the first fraction the same.
Let's change division to multiplication, and let's flip the second fraction. Now there's nothing to cancel, so let's multiply across. 4 times 5 is 20, 3 times 9 is 27. And so we can't reduce this fraction, that's the answer.
now what if you see a problem that looks like this 36 over 54 divided by 64 over 48 If you have a fraction written this way, what should you do? This expression is equivalent to saying 36 over 54 divided by 64 over 48. And then we can use the keep, change, flip principle. Let's keep the first fraction the same. Let's change division to multiplication, and then let's flip the second fraction. And now let's simplify.
So 36 is basically 9 times 4. 54 is 9 times 6. 48 is 16 times 3. And 64 is 16 times 4. So right now we can cancel a 9, we can cancel a 16, and we can cancel a 4. So what we have left over is 3 over 6. Now, 3 over 6 can be reduced. We can divide both numbers by 3. 3 divided by 3 is equal to 1, 6 divided by 3 is 2. So the final answer is 1 over 2.