Working Problems With Fractions

To add two fractions together with different denominators, remember, we need to get a common denominator. If we want to add one fifth plus two thirds, we need to rewrite these fractions so that they have the same denominator. The easiest way to get a common denominator is to take our two denominators and multiply them together. And so if we do that three times five gives us 15. And so what we're doing here when we want to add one fifth plus two thirds. We're going to multiply that three times five to get 15. 15 is the least common multiple of three and five, to take this first fraction and get that common denominator of 15, what we're doing is multiplying the five by three, if we multiply the denominator by three, then we also have to multiply the numerator by three. Because then we're really just multiplying that one fifth by 3/3 or one, so we're not changing the value of the fraction. Then for the second fraction to get the common denominator, we have to multiply that three by five. If we multiply the denominator by five, then we also have to multiply the numerator by five. Doing this gives us 3/15 plus 10/15. Now we have two fractions being added together with the same denominator, so we can just add the numerators together to get 13/15. If we want to add one third plus two, at first, this looks different than the last example. But remember, two is the same thing as 2/1. Here again, we're adding two fractions with different denominators. Now, in this case, the common denominator would be three. When we multiply three times one, we just get three. The first fraction already has a denominator of three, so we don't have to change anything there. But in the second fraction, we can multiply that denominator by three to get the common denominator of three, which means we also have to multiply the numerator by three as well. This gives us one third plus six thirds. Now we have two fractions with the same denominator, so we can just add the numerators together. To get seven thirds. Subtraction works the same way. Here we have three fourths minus one third. Before we can do the subtraction, we need to get a common denominator. We have three fourths minus one third. If we multiply the three and the four together, we'll get 12 12 is going to be our common denominator. For the first fraction, I'm going to multiply by 3/3, and for the second fraction, we'll multiply by 4/4. Which will give us 9/12 -4/12. We've rewritten both terms as equivalent fractions with the same denominator. Now we can do the subtraction just by subtracting those two terms in the numerator, nine minus four, will give us five, and this is over 12. In this example, we have a word problem. It says, Emily baked a batch of cookies and divided them into two different containers. In the first container, she placed two thirds of the cookies. In the second container, she placed one fourth of the cookies. Later, she decided to combine the cookies from both containers into a single large container, and we want to know what fraction of the total batch of cookies is now in the large container. In that first container, she had two thirds of her batch of cookies. Then in the second container, she had one fourth of her batch of cookies. To figure out what fraction of the total batch of cookies would be in the large container when she combines them, we're going to add these two fractions together. Here again, we have two fractions with different denominators, so we're going to need to get a common denominator. So if we multiply three and four, that'll give us 12. 12 will be our common denominator. This first fraction will multiply by 4/4. The second fraction will multiply by 3/3, which gives us 8/12 plus 3/12. Now we have two fractions with the same denominator, so we can add these together by adding the numerators, eight plus three will give us 11. Over 12. That means in the large container, she has 11/12 of her original batch of cookies. For another example, this says a store is having a buy one get one, half off sale. The original price of an item is X dollars. What would the total price be before taxes for both items? So it's a buy one get one, half off sale, which means we're going to buy one item at the original price, and the second item is half off. If the first item costs X dollars, then the second item would be half of that. X divided by two. To find the total price before taxes, we're going to add X plus X over two. Here again, you can think of X really as being X over one. Even though we have an X here, we're still just adding two fractions with different denominators. We need to get a common denominator, which in this case would be two, to get that common denominator in that first fraction, we could multiply by 2/2. The second fraction already has a two in the denominator so we don't have to change anything. Which gives us two X over two plus X over two. Now we have two fractions with the same denominator being added together. We can add the numerators together. Two X plus one X would give us three X, and this is over two. Multiplying two fractions together is fairly straightforward. When you want to multiply two fractions together, remember that you can do this by multiplying the numerators together and the denominators together. To multiply one third and two fifths, we can do this by multiplying the one times the two over the three times the five, which gives us 2/15. T multiply two fractions together, you can just think about multiplying straight across. Sarah is planning a 300 mile road trip. She drove three fifths of the total distance on the first day. How many miles did she drive on the first day? We know she's planning 300 miles. She's already driven three fifths of that total on the first day. We want to know how many miles that is. We can do this by taking our 300 miles and multiplying by three fifths. Now remember, 300 is the same as 300/1, we really are just multiplying two fractions together. We can multiply the numerators together to get 900. And the denominators together five times one just gives us five. Then we could simplify this just a little bit more. 900/5 is 180. On that first day, she drove 180 miles. To divide two fractions, remember, the rule is that we're going to multiply by the reciprocal of that second fraction. For something like one third, divided by two fifths, this is going to be the same thing as one third times the reciprocal, five halves. Again, when we want to multiply two fractions together, we can do this by multiplying straight across. This would be one times 5/2 times three or five sixth. Emma has two thirds of a cake and she wants to share it equally among her four friends. How much cake will each friend receive? Here Emma has two thirds of a cake. And she wants to divide this cake equally among her four friends among four people. We're going to take this two thirds and we're going to divide by four. But remember, four is the same thing as 4/1, when we're dividing two fractions, we're going to flip that second fraction and multiply. This is really the same thing as two thirds times one fourth, now we are multiplying two fractions together, so we can multiply straight across. This will give us two times one, which is 2/3 times four, which is 12. Now notice here the numerator and denominator both divisible by two. We could reduce this fraction a little bit. To goes into 21 time and two goes into 126 times. This fraction will simplify to one sixth, which means that each of her friends gets one sixth of the cake.