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Fractions - Addition and Subtraction of Fractions

Grade 6CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Like Fractions: Fractions that have the same denominator are called like fractions. Visually, this is equivalent to having slices of the exact same size from identical wholes (for example, two pizzas both cut into 8 equal slices). To add or subtract them, you simply add or subtract the numerators while keeping the denominator the same.

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Unlike Fractions: Fractions with different denominators are unlike fractions. Visually, these represent parts of different sizes, such as comparing a large slice (half) to a small slice (one-eighth). These cannot be added or subtracted directly; they must first be converted into 'Like Fractions' so that the slice sizes match.

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Equivalent Fractions: To make denominators the same, we use equivalent fractions. Visually, this is like taking a slice of pizza and cutting it into smaller, equal pieces without changing the total amount of pizza you have. This is done by multiplying both the numerator and denominator by the same non-zero number.

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Common Denominator using LCM: The most efficient way to convert unlike fractions is to find the Least Common Multiple (LCM) of the denominators. This value becomes the new 'Common Denominator,' ensuring that all fractions are expressed in terms of the same-sized parts.

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Addition and Subtraction Process: For like fractions, the rule is ac±bc=a±bc\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}. For unlike fractions, find the LCM, convert to equivalent fractions, then perform the addition or subtraction on the new numerators.

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Mixed Fractions Conversion: Before adding or subtracting mixed numbers like 2132\frac{1}{3}, it is often easiest to convert them into improper fractions. Visually, a mixed number represents whole units plus a fraction; converting it to an improper fraction represents the entire value as a total number of equal-sized pieces.

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Simplification: The final answer should always be expressed in its simplest form. This means dividing both the numerator and the denominator by their Highest Common Factor (HCF) until they have no common factors other than 1. Visually, this is grouping smaller slices back into the largest possible equal slices.

📐Formulae

ac+bc=a+bc\frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}

ac−bc=a−bc\frac{a}{c} - \frac{b}{c} = \frac{a-b}{c}

wab=(w×b)+abw\frac{a}{b} = \frac{(w \times b) + a}{b}

Common Denominator=LCM of denominators\text{Common Denominator} = \text{LCM of denominators}

💡Examples

Problem 1:

Find the sum of 35\frac{3}{5} and 27\frac{2}{7}

Solution:

  1. Find the LCM of denominators 55 and 77. Since they are prime numbers, LCM=5×7=35\text{LCM} = 5 \times 7 = 35. \n2. Convert to equivalent fractions with denominator 3535: \n 35=3×75×7=2135\frac{3}{5} = \frac{3 \times 7}{5 \times 7} = \frac{21}{35} \n 27=2×57×5=1035\frac{2}{7} = \frac{2 \times 5}{7 \times 5} = \frac{10}{35} \n3. Add the numerators: \n 2135+1035=21+1035=3135\frac{21}{35} + \frac{10}{35} = \frac{21+10}{35} = \frac{31}{35}

Explanation:

To add unlike fractions, we find a common denominator (35), convert both fractions, and then add the numerators.

Problem 2:

Subtract 312−1343\frac{1}{2} - 1\frac{3}{4}

Solution:

  1. Convert mixed fractions to improper fractions: \n 312=(3×2)+12=723\frac{1}{2} = \frac{(3 \times 2) + 1}{2} = \frac{7}{2} \n 134=(1×4)+34=741\frac{3}{4} = \frac{(1 \times 4) + 3}{4} = \frac{7}{4} \n2. Find the LCM of 22 and 44, which is 44. \n3. Convert 72\frac{7}{2} to an equivalent fraction with denominator 44: \n 72=7×22×2=144\frac{7}{2} = \frac{7 \times 2}{2 \times 2} = \frac{14}{4} \n4. Subtract the numerators: \n 144−74=14−74=74\frac{14}{4} - \frac{7}{4} = \frac{14-7}{4} = \frac{7}{4} \n5. Convert back to a mixed fraction: \n 74=134\frac{7}{4} = 1\frac{3}{4}

Explanation:

Mixed fractions are first turned into improper fractions. After finding a common denominator, we subtract and then simplify the result back into a mixed fraction.