Question

Find each sum or difference, showing each step of your work. Give your answers in lowest terms. If an answer is greater than 1 , write it as a mixed number. $$3 \frac{1}{4}+1 \frac{1}{3}$$

Short answer

4 \frac{7}{12}

Step-by-step solution

Convert Mixed Numbers to Improper Fractions

Convert each mixed number to an improper fraction. For \(3 \frac{1}{4}\), multiply 3 by 4 and add 1:\[3 \frac{1}{4} = \frac{3 \times 4 + 1}{4} = \frac{13}{4}\]For \(1 \frac{1}{3}\), multiply 1 by 3 and add 1:\[1 \frac{1}{3} = \frac{1 \times 3 + 1}{3} = \frac{4}{3}\]

Find a Common Denominator

The denominators of the fractions are 4 and 3. The least common multiple of 4 and 3 is 12. Rewrite both fractions with this common denominator:\[\frac{13}{4} = \frac{13 \times 3}{4 \times 3} = \frac{39}{12}\]\[\frac{4}{3} = \frac{4 \times 4}{3 \times 4} = \frac{16}{12}\]

Add the Fractions

Now that the fractions have the same denominator, add the numerators:\[\frac{39}{12} + \frac{16}{12} = \frac{39 + 16}{12} = \frac{55}{12}\]

Simplify the Fraction

\(\frac{55}{12}\) is an improper fraction. Divide the numerator by the denominator to convert it into a mixed number:\[\frac{55}{12} = 4 \frac{7}{12}\]

Key concepts

convert mixed numbers to improper fractions

To solve the problem, we first need to convert mixed numbers to improper fractions. This conversion makes it easier to perform arithmetic operations like addition and subtraction. A mixed number consists of a whole number and a fraction. To convert it, follow these steps:

Create an improper fraction by multiplying the whole number by the fraction's denominator.

• For example, if you have the mixed number \( 3 \frac{1}{4} \) (where 3 is the whole number and 1/4 is the fractional part), multiply the whole number (3) by the denominator (4).

Add the resulting product to the numerator of the fraction.

• In our example, this would be \( 3 \times 4 + 1 = 12 + 1 = 13 \).

Write the resulting sum over the original denominator, forming an improper fraction.

• So, \( 3 \frac{1}{4} \) becomes \( \frac{13}{4} \).

Repeat the process for any other mixed numbers.

Following this method, \( 1 \frac{1}{3} \) can be converted to \( \frac{4}{3} \). The conversion step simplifies working with the mixed numbers in the upcoming calculations.

common denominator

After converting the mixed numbers to improper fractions, it's important to find a common denominator for the fractions we are working with. Here’s why and how:

Fractions need to have the same denominator to be added or subtracted directly. The denominators we have are 4 and 3.

Identify the least common multiple (LCM) of the denominators to make the math simpler. The LCM of 4 and 3 is 12.

• To express \( \frac{13}{4} \) with a denominator of 12, multiply both the numerator and the denominator by 3: \( \frac{13 \times 3}{4 \times 3} = \frac{39}{12} \).
• For \( \frac{4}{3} \), multiply both the numerator and the denominator by 4: \( \frac{4 \times 4}{3 \times 4} = \frac{16}{12} \).

Now, both fractions have a common denominator of 12.

This step makes adding or subtracting the fractions straightforward, as we can simply add their numerators while keeping the common denominator.

simplifying fractions

Once we have the fractions with a common denominator, the process of adding them can begin. This step might yield an improper fraction, which we will then need to simplify.

To add \( \frac{39}{12} \) and \( \frac{16}{12} \), add the numerators: \( 39 + 16 = 55 \).

This results in \( \frac{55}{12} \). Since this is an improper fraction (the numerator is larger than the denominator), we should convert it back to a mixed number.

• Divide the numerator by the denominator: \( 55 \div 12 = 4 \) with a remainder of 7.
• The whole number part is 4, and the fractional part is \( \frac{7}{12} \).
• So, \( \frac{55}{12} \) becomes \( 4 \frac{7}{12} \).

Check if the resulting fraction part can be simplified further. In this case, 7 and 12 share no common factors other than 1, so \( \frac{7}{12} \) is already in its simplest form.

Performing these simplifications makes the final answer more understandable and is expected in most math assignments.