Comparing, Adding, and Subtracting Fractions

Theory: When two fractions have the same denominator, compare their numerators. The greater numerator represents the greater fraction.

Example:
Question: Which is greater: \(\frac{4}{7}\) or \(\frac{5}{7}\) ​?
Solution: Compare 4 and 5 ⇒ 5>4
So, \(\frac{5}{7}\) > \(\frac{4}{7}\)​

Theory: To compare fractions with different denominators, first convert them to have a common denominator using LCM, then compare the numerators.

Example:
Question: Compare \(\frac{2}{3}\) and \(\frac{3}{5}\)​
Solution: LCM of 3 and 5 = 15
Convert: \(\frac{2}{3}\) = \(\frac{10}{15}\), \(\frac{3}{5}\) = \(\frac{9}{15}\)​
Compare: \(\frac{10}{15}\) > \(\frac{9}{15}\)​ ⇒ \(\frac{2}{3}\) > \(\frac{3}{5}\)

Theory: To perform operations on unlike fractions, find the LCM of denominators and rewrite both fractions using that LCM.

Example:
Question: Convert \(\frac{3}{4}\)​ and \(\frac{2}{6}\)​ to like denominators.
Solution: LCM of 4 and 6 = 12
\(\frac{3}{4}\) = \(\frac{9}{12}\), \(\frac{2}{6}\) = \(\frac{4}{12}\)​

Theory: When denominators are same, simply add numerators and keep the denominator unchanged.

Example:
Question: \(\frac{5}{9} + \frac{2}{9}\) = ?
Solution: Add numerators: 5+2 = 7
Answer: \(\frac{7}{9}\)

Theory: Find the LCM of denominators, convert to equivalent like fractions, and add.

Example:
Question: \(\frac{1}{2}\) + \(\frac{1}{3}\) = ?
Solution: LCM = 6
\(\frac{1}{2} = \frac{3}{6}, \frac{1}{3} = \frac{2}{6}\)
Add: \(\frac{3 + 2}{6}\) = \(\frac{5}{6}\)

Theory: Subtract numerators directly when denominators are equal.

Example:
Question: \(\frac{5}{8} – \frac{3}{8}\) = ?
Solution: \(\frac{5 – 3}{8} = \frac{2}{8} = \frac{1}{4}\)

Theory: Convert to like denominators using LCM, then subtract numerators.

Example:
Question: \(\frac{5}{6} – \frac{1}{4}\) = ?
Solution: LCM = 12
\(\frac{5}{6} = \frac{10}{12}, \frac{1}{4} = \frac{3}{12}\)
Subtract: \(\frac{10 – 3}{12} = \frac{7}{12}\)

Theory: LCM (Least Common Multiple) helps in finding a common base for converting unlike fractions.

Example:
Question: Find LCM of 3 and 5, and convert \(\frac{2}{3}\)​ and \(\frac{1}{5}\)​ to like fractions.
Solution: LCM = 15
\(\frac{2}{3} = \frac{10}{15}, \frac{1}{5} = \frac{3}{15}\)

Theory: Fractions can be shown using shaded portions of equal-sized shapes to help visualize sum.

Example:
Question: Use rectangles to show \(\frac{1}{4} + \frac{2}{4}\)
Solution: Draw a rectangle in 4 parts, shade 1 part and 2 parts separately. Total shaded: 3 parts ⇒ \(\frac{3}{4}\)

Theory: Use shaded diagrams to show how subtraction works by removing parts from a whole.

Example:
Question: Subtract \(\frac{1}{3}\)​ from \(\frac{2}{3}\) using a bar diagram.
Solution: Shade 2 parts of a bar with 3 equal parts, remove 1 part ⇒ remaining = \(\frac{1}{3}\)

Theory: Add fractions to solve real-life scenarios like combining lengths, volumes, or time.

Example:
Question: Rina walked \(\frac{1}{5}\) km in the morning and \(\frac{2}{5}\) km in the evening. How far did she walk in total?
Solution: Add: \(\frac{1 + 2}{5}\) = \(\frac{3}{5}\) km

Theory: Subtract fractions to find remaining quantity or difference in real-life cases.

Example:
Question: A jug holds \(\frac{7}{8}\) L water. \(\frac{3}{8}\) L is poured out. How much remains?
Solution: \(\frac{7 – 3}{8} = \frac{4}{8} = \frac{1}{2}\) L

Theory: Always simplify final answers to lowest terms for clarity.

Example:
Question: Simplify \(\frac{6}{9}\)
Solution: GCD of 6 and 9 = 3
⇒ \(\frac{6}{9} = \frac{2}{3}\)​

Theory: Used in cooking, budgeting, shopping, and time management. Fractions combine or separate quantities in various contexts.

Example:
Question: A recipe needs \(\frac{2}{3}\)​ cup of flour, and you’ve already added \(\frac{1}{3}\)​. How much more do you need?
Solution: \(\frac{2}{3} – \frac{1}{3} = \frac{1}{3}\)​ cup

Theory: Mistakes may occur in LCM, numerator conversions, or improper subtraction. Always verify with diagrams or steps.

Example:
Question: A student finds \(\frac{1}{2} + \frac{1}{3} = \frac{2}{5}\). Is it correct?
Solution: No. Correct LCM = 6 ⇒ \(\frac{1}{2} = \frac{3}{6}, \frac{1}{3} = \frac{2}{6}\)
So sum = \(\frac{5}{6}\) not \(\frac{2}{5}\)​