We are tasked with adding two fractions: 5/6 and 3/8. To understand this, let's first visualize what these fractions represent. A fraction like 5/6 means we have 5 parts out of 6 equal parts of a whole. Similarly, 3/8 means 3 parts out of 8 equal parts. To add fractions, we need to work with pieces of the same size, which requires finding a common denominator.
To add fractions, we need pieces of the same size. This means we must find a common denominator. We can't directly add 5/6 and 3/8 because their pieces are different sizes. We find the Least Common Multiple of 6 and 8 by listing their multiples. The multiples of 6 are 6, 12, 18, 24, 30, and so on. The multiples of 8 are 8, 16, 24, 32, 40, and so on. The smallest number that appears in both lists is 24, so 24 is our least common multiple and will be our common denominator.
Now we convert both fractions to equivalent fractions with denominator 24. For 5/6, we multiply both numerator and denominator by 4, giving us 20/24. For 3/8, we multiply both numerator and denominator by 3, giving us 9/24. These equivalent fractions represent the same amounts as the originals, but now they have the same denominator, making them ready for addition.
Now that both fractions have the same denominator, we can add them. We simply add the numerators while keeping the denominator the same. So 20/24 plus 9/24 equals 20 plus 9 over 24, which is 29/24. Visually, we can see this by combining the colored parts from both fraction bars.
Our result is 29/24, which is an improper fraction because the numerator is larger than the denominator. We can convert this to a mixed number. We divide 29 by 24, which gives us 1 with a remainder of 5. This means 29/24 equals 1 and 5/24. Finally, we check if 5/24 can be simplified by finding the greatest common divisor of 5 and 24. Since the GCD is 1, the fraction is already in its lowest terms. Therefore, our final answer is 1 and 5/24.
Let's verify our answer by converting to decimals. 5/6 is approximately 0.833, and 3/8 is exactly 0.375. Adding these gives us 1.208. Now let's convert our answer 1 and 5/24 to decimals. 5/24 is approximately 0.208, so 1 and 5/24 equals about 1.208. This confirms our answer is correct. To summarize the process: First, we found the common denominator by calculating the least common multiple. Second, we converted both fractions to equivalent fractions with that denominator. Third, we added the numerators while keeping the denominator the same. Fourth, we simplified the result by converting to a mixed number and checking if it could be reduced further. The final answer is 1 and 5/24.