Question

Calculate the following, expressing your answer as a mixed fraction: (a) \(2 \frac{1}{4}+3 \frac{1}{3}\) (b) \(2 \frac{4}{5}-1 \frac{2}{3}\) (c) \(5 \frac{2}{3}-1 \frac{1}{2}+2 \frac{1}{5}\) (d) \(5-4 \frac{2}{7}+\frac{1}{3}\) (e) \(\frac{9}{10}+\frac{6}{7}-1 \frac{2}{5}\)

Short answer

Question: Perform the following mixed-fraction operations and write your answer in the simplest form as a mixed fraction: (a) \(2 \frac{1}{4}+3 \frac{1}{3}\) (b) \(2 \frac{4}{5}-1 \frac{2}{3}\) (c) \(5 \frac{2}{3}-1 \frac{1}{2}+2 \frac{1}{5}\) (d) \(5-4 \frac{2}{7}+\frac{1}{3}\) (e) \(\frac{9}{10}+\frac{6}{7}-1 \frac{2}{5}\)

Step-by-step solution

Convert mixed fractions to improper fractions

To do this, multiply the whole number by the denominator and then add the numerator. Place the result over the original denominator: \((2 * 4) + 1 = 9\), so \(2 \frac{1}{4}=\frac{9}{4}\). Similarly, \((3 * 3) + 1 = 10\), so \(3 \frac{1}{3}=\frac{10}{3}\).

Find a common denominator

To add or subtract fractions with different denominators, we must first find a common denominator. The least common multiple (LCM) of 4 and 3 is 12. Multiply both the numerator and the denominator for each fraction by the necessary factor to make their denominators equal to 12: \(\frac{9}{4} * \frac{3}{3}=\frac{27}{12}\). \(\frac{10}{3} * \frac{4}{4}=\frac{40}{12}\).

Perform the arithmetic

Add the two fractions: \(\frac{27}{12}+\frac{40}{12}=\frac{67}{12}\).

Convert the improper fraction to a mixed fraction

Divide the numerator by the denominator: \(67\div 12=5\) remainder \(7\). So the mixed fraction is \(5 \frac{7}{12}\). The final answer is \(2 \frac{1}{4}+3 \frac{1}{3}=5 \frac{7}{12}\). (b) \(2 \frac{4}{5}-1 \frac{2}{3}\) Follow the same steps as above, but with subtraction to obtain the answer: \(2 \frac{4}{5}-1 \frac{2}{3}=1 \frac{1}{15}\). (c) \(5 \frac{2}{3}-1 \frac{1}{2}+2 \frac{1}{5}\) Follow the same steps as above but with multiple operations to obtain the answer: \(5 \frac{2}{3}-1 \frac{1}{2}+2 \frac{1}{5}=6 \frac{17}{30}\). (d) \(5-4 \frac{2}{7}+\frac{1}{3}\) Follow similar steps as above with the whole number included to obtain the answer: \(5-4 \frac{2}{7}+\frac{1}{3}= \frac{16}{21}\). (e) \(\frac{9}{10}+\frac{6}{7}-1 \frac{2}{5}\) Follow the same steps as above but with different operations to obtain the answer: \(\frac{9}{10}+\frac{6}{7}-1 \frac{2}{5} = \frac{7}{70}\).

Key concepts

Mixed Fractions

A mixed fraction is a combination of a whole number and a proper fraction. It's like saying "three and a half oranges," where "three" is the whole part and "one-half" is the fraction part. This kind of fraction is often more intuitive when comparing or calculating with everyday numbers.

To work with mixed fractions in mathematical operations, we typically convert them to improper fractions. This helps because improper fractions can be more easily manipulated during calculations like addition or subtraction. A mixed fraction such as 2 \( \frac{3}{4} \) is turned into an improper fraction by multiplying the whole number by the fraction's denominator and then adding the numerator (e.g., \( (2\times4) + 3 = 11/4 \)). This gives us the improper fraction \( \frac{11}{4} \), which can now be used in calculations like any other fractions.

Improper Fractions

Improper fractions have numerators that are larger than their denominators, such as \( \frac{9}{4} \). These fractions represent a quantity greater than one whole, which is sometimes easier to work with in operations compared to mixed numbers.

For example, in the exercise, we converted mixed fractions to improper fractions to streamline the addition and subtraction processes. An improper fraction is calculated by multiplying the whole number by the denominator and adding the numerator (e.g., \( 3\times3 + 1 = 10 \), so \( 3 \frac{1}{3} = \frac{10}{3} \)). Using improper fractions allows us to directly apply the rules of arithmetic without having to juggle the whole and fractional parts separately during calculations.

Least Common Multiple

The least common multiple (LCM) is crucial when dealing with fractions that have different denominators. To add or subtract fractions, they must share the same denominator, and the LCM helps us find the smallest shared multiple of two or more numbers.

In the exercise, when adding \( \frac{9}{4} \) and \( \frac{10}{3} \), we found that their common denominator was 12. We computed this by finding the LCM of 4 and 3. The LCM helps standardize denominators, making it straightforward to add or subtract fractions without altering their original values.

• Step: Find LCM by listing multiples of the numbers.
• Use LCM to adjust fractions to the same base (denominator).
• Perform arithmetic operations on these adjusted fractions.

Fraction Addition

Adding fractions is only possible if they share the same denominator. When dealing with more complex fractions, like mixed or improper fractions, we often convert them to a uniform denominator using the LCM, allowing straightforward addition.

After converting fractions in the given exercise, we proceed with addition by adding the numerators while keeping the denominator constant. For instance, adding \( \frac{27}{12} \) and \( \frac{40}{12} \) results in \( \frac{67}{12} \). The process looks like this:

• Ensure common denominators (achieved by LCM).
• Add numerators directly.
• Simplify if necessary or convert back to mixed fraction.

Fraction Subtraction

Just like with addition, subtracting fractions requires a common denominator. This is why the LCM is frequently used to equalize denominators before subtracting.

Let's consider subtracting two fractions. The process is similar to addition, where we first align the denominators and then subtract the numerators. For instance, from the step-by-step solution, \( \frac{48}{15} - \frac{25}{15} = \frac{23}{15} \). This improper fraction can further be transformed back into a mixed fraction if desired:

• Convert fractions to improper if necessary.
• Find LCM to adjust denominators.
• Subtract numerators and resolve the fraction into its simplest form where applicable.