Question

Perform the indicated operation. \(12 \frac{3}{4}+6 \frac{2}{5}\)

Short answer

Answer: The sum is \(19 \frac{3}{20}\).

Step-by-step solution

Converting mixed numbers to improper fractions

In order to add the mixed numbers, we first need to convert them into improper fractions. To do this, we multiply the whole number by the denominator of the fraction, then add the numerator. For instance, the first mixed number \(12 \frac{3}{4}\) would be calculated as: \((12 * 4) + 3 = 51\). So, the improper fraction is \(\frac{51}{4}\). Applying the same method for the second mixed number \(6 \frac{2}{5}\), we get: \((6 * 5) + 2 = 32\). So, the improper fraction is \(\frac{32}{5}\).

Finding a common denominator

Now, we need to find a common denominator for the two improper fractions \(\frac{51}{4}\) and \(\frac{32}{5}\). In this case, the lowest common multiple (LCM) between 4 and 5 is 20. Hence, the common denominator is 20.

Performing the addition

To perform the addition of these fractions, we need to adjust the numerators according to the common denominator. So, we will multiply the numerator and denominator of the first fraction by 5 and the numerator and the denominator of the second fraction by 4. \(\frac{51}{4} \cdot \frac{5}{5} = \frac{255}{20}\) and \(\frac{32}{5} \cdot \frac{4}{4} = \frac{128}{20}\) Now, we can perform the addition: \(\frac{255}{20} + \frac{128}{20} = \frac{255 + 128}{20}\) This simplifies to: \(\frac{383}{20}\)

Simplifying the result

Finally, we will simplify the result and convert it back into a mixed number if necessary. To do this, we can ask the following question: "How many times does the denominator (20) fit into the numerator (383)?" It turns out that \(20 \cdot 19 = 380\), which is very close to 383, so the whole number part of our mixed number is 19. We can subtract these 380 parts from our original fraction to find out what's left: \(383 - 380 = 3\) So, we have 3 parts left. We will put them above the original denominator(20), and the resulting mixed number is: \(19 \frac{3}{20}\)