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Number System - Fractions: Types, Comparison, Operations (+, -, ×, ÷)

Grade 6ICSE

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

🔑Concepts

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A fraction ab\frac{a}{b} represents 'a' equal parts out of 'b' total parts of a whole. Visually, if a circular pizza is cut into 8 equal slices and you take 3, the fraction is 38\frac{3}{8}. The top number is the Numerator and the bottom is the Denominator.

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Fractions are classified into three main types: Proper fractions (numerator < denominator, e.g., 23\frac{2}{3}), Improper fractions (numerator ≥\ge denominator, e.g., 54\frac{5}{4}), and Mixed fractions (a whole number plus a proper fraction, e.g., 1141\frac{1}{4}). An improper fraction like 32\frac{3}{2} can be visualized as one full object and half of another.

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Equivalent fractions represent the same value even though they look different. For example, 12\frac{1}{2}, 24\frac{2}{4}, and 48\frac{4}{8} all represent the same half of a shape. You can find them by multiplying or dividing both the numerator and denominator by the same non-zero number.

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Like fractions have the same denominator (e.g., 15,35\frac{1}{5}, \frac{3}{5}), while Unlike fractions have different denominators (e.g., 12,13\frac{1}{2}, \frac{1}{3}). Like fractions can be compared directly by their numerators; for unlike fractions, you must first find a common denominator.

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Comparing fractions can be done using the Cross-Multiplication method. To compare ab\frac{a}{b} and cd\frac{c}{d}, we compare the products a×da \times d and b×cb \times c. If ad>bcad > bc, then ab>cd\frac{a}{b} > \frac{c}{d}. On a number line, a larger fraction is always to the right of a smaller one.

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Addition and Subtraction of unlike fractions requires converting them into equivalent fractions with a common denominator, usually the Least Common Multiple (LCM) of the denominators. Once the denominators are the same, you simply add or subtract the numerators while keeping the denominator constant.

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Multiplication of fractions is the process of finding a 'part of a part'. Visually, 12×12\frac{1}{2} \times \frac{1}{2} means half of a half, which is 14\frac{1}{4}. Calculation-wise, you multiply the numerators together and the denominators together.

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Division of fractions involves the 'Reciprocal'. To divide by a fraction, you multiply by its reciprocal (the fraction flipped upside down). For example, 23÷57\frac{2}{3} \div \frac{5}{7} becomes 23×75\frac{2}{3} \times \frac{7}{5}.

📐Formulae

General Form: NumeratorDenominator\text{General Form: } \frac{\text{Numerator}}{\text{Denominator}}

Equivalent Fractions: ab=a×kb×k (where k≠0)\text{Equivalent Fractions: } \frac{a}{b} = \frac{a \times k}{b \times k} \text{ (where } k \neq 0)

Addition (Like): ac+bc=a+bc\text{Addition (Like): } \frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}

Addition (Unlike): ab+cd=(a×LCM/b)+(c×LCM/d)LCM\text{Addition (Unlike): } \frac{a}{b} + \frac{c}{d} = \frac{(a \times \text{LCM}/b) + (c \times \text{LCM}/d)}{\text{LCM}}

Multiplication: ab×cd=a×cb×d\text{Multiplication: } \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}

Division: ab÷cd=ab×dc\text{Division: } \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}

Reciprocal of ab is ba\text{Reciprocal of } \frac{a}{b} \text{ is } \frac{b}{a}

💡Examples

Problem 1:

Solve: 34+16−12\frac{3}{4} + \frac{1}{6} - \frac{1}{2}

Solution:

  1. Find the LCM of denominators 4,6,4, 6, and 22. The LCM is 1212.
  2. Convert each fraction to an equivalent fraction with denominator 1212: 3×34×3=912\frac{3 \times 3}{4 \times 3} = \frac{9}{12} 1×26×2=212\frac{1 \times 2}{6 \times 2} = \frac{2}{12} 1×62×6=612\frac{1 \times 6}{2 \times 6} = \frac{6}{12}
  3. Perform the operations: 912+212−612=9+2−612=512\frac{9}{12} + \frac{2}{12} - \frac{6}{12} = \frac{9 + 2 - 6}{12} = \frac{5}{12}.

Explanation:

To add or subtract unlike fractions, we first find a common denominator (LCM), convert all fractions to that denominator, and then combine the numerators.

Problem 2:

Divide 2132\frac{1}{3} by 79\frac{7}{9}

Solution:

  1. Convert the mixed fraction to an improper fraction: 213=(2×3)+13=732\frac{1}{3} = \frac{(2 \times 3) + 1}{3} = \frac{7}{3}.
  2. Set up the division: 73÷79\frac{7}{3} \div \frac{7}{9}.
  3. Multiply by the reciprocal of the divisor: 73×97\frac{7}{3} \times \frac{9}{7}.
  4. Simplify: 7×93×7=6321\frac{7 \times 9}{3 \times 7} = \frac{63}{21}.
  5. Reduce to lowest terms: 63÷2121÷21=3\frac{63 \div 21}{21 \div 21} = 3.

Explanation:

First, convert mixed numbers to improper fractions. Then, change the division sign to multiplication and flip the second fraction to its reciprocal before simplifying.