We want to add two fractions: 3/4 plus 2/3. First, let's understand what these fractions represent. A fraction like 3/4 means we take 3 parts out of 4 equal parts of a whole. Similarly, 2/3 means 2 parts out of 3 equal parts. To add them, we need a common way to measure both parts.
To add fractions with different denominators, we need a common denominator. The denominators here are 4 and 3. We cannot directly add fractions with different denominators. We need to find a common multiple of 4 and 3. Let's list the multiples: for 4 we have 4, 8, 12, 16, 20, and so on. For 3 we have 3, 6, 9, 12, 15, and so on. The smallest number that appears in both lists is 12, which is the least common multiple.
Now we convert both fractions to have the common denominator of 12. For 3/4, we multiply both numerator and denominator by 3, giving us 9/12. For 2/3, we multiply both numerator and denominator by 4, giving us 8/12. Visually, we can see this as dividing each pie into 12 equal parts and shading the appropriate number of parts.
Now that both fractions have the same denominator, we can add them by adding the numerators. 9/12 plus 8/12 equals 17/12. Visually, this means we have 17 parts out of 12 equal parts. This is more than one whole, so we'll need to convert it to a mixed number.
The fraction 17/12 is an improper fraction because the numerator is larger than the denominator. To convert it to a mixed number, we divide 17 by 12. This gives us 1 with a remainder of 5. So 17/12 equals 1 and 5/12. Visually, this is one complete pie plus 5 parts out of 12 in another pie.
Let's verify our answer and summarize the steps. First, we found the least common multiple of 4 and 3, which is 12. Then we converted both fractions to have denominator 12: 3/4 became 9/12 and 2/3 became 8/12. We added the numerators to get 17/12. Finally, we converted 17/12 to the mixed number 1 and 5/12. To verify, we can convert to decimals: 3/4 is 0.75, 2/3 is approximately 0.667, and their sum is approximately 1.417. Our answer 1 and 5/12 is also approximately 1.417, confirming our solution.