In this lesson, we'll learn how to add two fractions: five-sixths and two-thirds. First, let's understand what these fractions represent. Five-sixths means we divide something into six equal parts and take five of them. Two-thirds means we divide something into three equal parts and take two of them. To add fractions, we need pieces of the same size.
To add fractions, they must have the same denominator. Our fractions five-sixths and two-thirds have different denominators: six and three. We cannot directly add them because their pieces are different sizes. We need to find a common denominator. The least common multiple of six and three will work. Let's list multiples of six: six, twelve, eighteen, twenty-four, and so on. Now multiples of three: three, six, nine, twelve, fifteen, and so on. The smallest number that appears in both lists is six, so the least common multiple is six.
Now we convert both fractions to have the common denominator of six. Five-sixths already has denominator six, so it stays as five-sixths. For two-thirds, we multiply both numerator and denominator by two to get four-sixths. This is because multiplying by two over two is the same as multiplying by one, which doesn't change the value. Visually, we can see that two-thirds and four-sixths represent the same amount - just divided into different sized pieces.
Now that both fractions have the same denominator, we can add them. We add five-sixths plus four-sixths. When denominators are the same, we simply add the numerators and keep the denominator. So we have five plus four over six, which equals nine-sixths. Visually, we're combining five parts and four parts to get nine parts out of six total parts. Notice that we have more parts than make up one whole.
Nine-sixths is an improper fraction because the numerator is larger than the denominator. We can convert it to a mixed number. Nine divided by six equals one with a remainder of three, so nine-sixths equals one and three-sixths. Now we can simplify three-sixths by finding the greatest common divisor of three and six, which is three. Dividing both numerator and denominator by three gives us one-half. Therefore, our final simplified answer is one and one-half.
Let's verify our answer. We started with five-sixths plus two-thirds. We converted two-thirds to four-sixths. Then we added five-sixths plus four-sixths to get nine-sixths. Finally, we simplified nine-sixths to one and one-half. To verify, we can convert one and one-half back to an improper fraction: one times two plus one over two equals three-halves. Three-halves is equivalent to nine-sixths, confirming our answer is correct. The complete solution shows that five-sixths plus two-thirds equals one and one-half.