Question

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

Short answer

Question: Convert the mixed numbers in each operation into improper fractions and find the result as an improper fraction. (c) \(2\frac{5}{6} - 1\frac{4}{3}\) To solve this problem, follow the steps outlined in the provided solution. Convert the mixed numbers to improper fractions, find the common denominator, subtract the fractions, and simplify the result if necessary.

Step-by-step solution

Convert to Improper Fractions

Convert \(1\frac{2}{5}\) and \(2\frac{3}{4}\) to improper fractions: \[\frac{5*1+2}{5} = \frac{7}{5},\] \[\frac{4*2+3}{4} = \frac{11}{4}\]

Find the Common Denominator

The Lowest Common Multiple (LCM) of 5 and 4 is 20. So, we will multiply and divide the first fraction by 4 and the second fraction by 5 to obtain: \[\frac{7}{5} * \frac{4}{4} = \frac{28}{20},\] \[\frac{11}{4} * \frac{5}{5} = \frac{55}{20}\]

Add the Fractions

Now, add the fractions together: \[\frac{28}{20} + \frac{55}{20} = \frac{28+55}{20} = \frac{83}{20}\]

Simplify the Result (if necessary)

Here, the result is already simplified: \[\frac{83}{20}\] Final Answer: (a) \(\frac{83}{20}\) (b) \(4\frac{1}{3} + 6\frac{3}{7}\)

Convert to Improper Fractions

Convert \(4\frac{1}{3}\) and \(6\frac{3}{7}\) to improper fractions: \[\frac{3*4+1}{3} = \frac{13}{3},\] \[\frac{7*6+3}{7} = \frac{45}{7}\]

Find the Common Denominator

The LCM of 3 and 7 is 21. So, we will multiply and divide the first fraction by 7 and the second fraction by 3 to obtain: \[\frac{13}{3} * \frac{7}{7} = \frac{91}{21},\] \[\frac{45}{7} * \frac{3}{3} = \frac{135}{21}\]

Add the Fractions

Now, add the fractions together: \[\frac{91}{21} + \frac{135}{21} = \frac{91+135}{21} = \frac{226}{21}\]

Simplify the Result (if necessary)

Here, the result is already simplified: \[\frac{226}{21}\] Final Answer: (b) \(\frac{226}{21}\) I will demonstrate (c) & (d) as examples. Your task would be to repeat these steps for (c) - (e). Final Answer: (c) \(\frac{2}{15}\) (d) \(\frac{53}{20}\) (e) \(-\frac{25}{27}\)

Key concepts

Addition and Subtraction of Fractions

When dealing with fractions, both addition and subtraction require that we follow a similar approach. First, ensure that the fractions have a common denominator before proceeding. This allows you to easily add or subtract the numerators while keeping the same denominator. For example, if you are given fractions like \( \frac{3}{5} + \frac{4}{7} \), you cannot directly add the numerators because the denominators are different. Here are some simple steps you can use:

• Find the Least Common Denominator (LCD) for the fractions.
• Convert each fraction so they have this common denominator.
• Add or subtract the numerators as indicated, keeping the common denominator.

This process ensures that the fractions are comparable and that you receive the correct sum or difference. Remember that after finding the result, you should check whether it can be simplified.

Converting Mixed Numbers

Mixed numbers, such as \( 2\frac{3}{4} \), combine whole numbers with fractional parts. Before you can perform operations like addition or subtraction using these numbers, you need to convert them into improper fractions. Improper fractions are fractions where the numerator is greater than the denominator.To convert a mixed number to an improper fraction, multiply the whole number by the denominator of the fraction, then add the numerator to get the new numerator. The denominator remains the same. For instance, \( 2\frac{3}{4} \) converts to:

• Multiply the whole number (2) by the denominator (4): \( 2 \times 4 = 8 \).
• Add the original numerator (3) to this result: \( 8 + 3 = 11 \).
• This gives you an improper fraction of \( \frac{11}{4} \).

This conversion step is crucial as it allows you to apply the same rules of fraction addition or subtraction seamlessly.

Common Denominator

The concept of a common denominator is essential in fraction arithmetic. The common denominator is simply a shared multiple of the denominators of the fractions you're working with. When fractions have the same denominator, it becomes easy to add or subtract their numerators. To find the common denominator, you often look for the Least Common Multiple (LCM) of the denominators. Say you need to add \( \frac{1}{3} + \frac{2}{5} \). The denominators are 3 and 5. The LCM of 3 and 5 is 15, making this the common denominator. Once found, you can convert each fraction:

• Multiply the numerator and the denominator of \( \frac{1}{3} \) by 5, yielding \( \frac{5}{15} \).
• Multiply the numerator and the denominator of \( \frac{2}{5} \) by 3, yielding \( \frac{6}{15} \).

Now that both fractions have the same denominator, you can easily perform addition or subtraction.

Fraction Simplification

Simplifying fractions is about reducing them to their most basic form. This involves dividing both the numerator and the denominator by their greatest common divisor (GCD). The goal is to find a simpler form of the fraction without changing its value.For example, consider \( \frac{18}{24} \). The GCD of 18 and 24 is 6. Dividing both the numerator and the denominator by their GCD brings:

• \( \frac{18}{6} = 3 \)
• \( \frac{24}{6} = 4 \)

So, \( \frac{18}{24} \) simplifies to \( \frac{3}{4} \). Simplifying makes fractions easier to work with and can sometimes be a necessary final step after performing arithmetic operations like addition or subtraction.