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Fractions and Decimals - Division of Fractions

Grade 7CBSE

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

🔑Concepts

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Reciprocal of a Fraction: Two non-zero numbers are called reciprocals of each other if their product is 11. To find the reciprocal of a fraction, we interchange the numerator and the denominator. For example, the reciprocal of 37\frac{3}{7} is 73\frac{7}{3}. Visually, this is like flipping the fraction upside down.

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Division of a Whole Number by a Fraction: To divide a whole number by a fraction, we multiply the whole number by the reciprocal of that fraction. For example, 3÷123 \div \frac{1}{2} is solved as 3×2=63 \times 2 = 6. Visually, this means finding how many 'half-sized' pieces fit into three whole units.

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Division of a Fraction by a Whole Number: To divide a fraction by a whole number, we multiply the fraction by the reciprocal of the whole number. If we have 12÷3\frac{1}{2} \div 3, it becomes 12×13=16\frac{1}{2} \times \frac{1}{3} = \frac{1}{6}. Visually, this represents taking half of an object and splitting that half into three equal smaller parts.

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Division of a Fraction by another Fraction: When dividing one fraction by another, we multiply the first fraction (the dividend) by the reciprocal of the second fraction (the divisor). This is commonly known as the 'Keep-Change-Flip' rule: Keep the first fraction, Change the division sign to multiplication, and Flip the second fraction.

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Division involving Mixed Fractions: When dealing with mixed fractions, they must first be converted into improper fractions before performing division. For instance, to divide 2132\frac{1}{3} by 25\frac{2}{5}, first convert 2132\frac{1}{3} to 73\frac{7}{3} and then proceed with the multiplication by the reciprocal.

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Division by 1 and Identity: Any fraction divided by 11 results in the same fraction (e.g., 34÷1=34\frac{3}{4} \div 1 = \frac{3}{4}). Conversely, any non-zero fraction divided by itself results in 11 (e.g., 58÷58=1\frac{5}{8} \div \frac{5}{8} = 1). On a number line, this shows that the entire length of a fraction fits into itself exactly one time.

📐Formulae

Reciprocal of ab=ba\frac{a}{b} = \frac{b}{a} (where a,b≠0a, b \neq 0)

Whole Number ÷\div Fraction: a÷bc=a×cba \div \frac{b}{c} = a \times \frac{c}{b}

Fraction ÷\div Whole Number: ab÷c=ab×1c\frac{a}{b} \div c = \frac{a}{b} \times \frac{1}{c}

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

Mixed Fraction conversion: wab=(w×b)+abw\frac{a}{b} = \frac{(w \times b) + a}{b}

💡Examples

Problem 1:

Find the result of 49÷23\frac{4}{9} \div \frac{2}{3}.

Solution:

Step 1: Keep the first fraction: 49\frac{4}{9}. Step 2: Change the division sign to multiplication: ×\times. Step 3: Flip the second fraction to find its reciprocal: The reciprocal of 23\frac{2}{3} is 32\frac{3}{2}. Step 4: Multiply the fractions: $$ \frac{4}{9} \times \frac{3}{2} = \frac{4 \times 3}{9 \times 2} = \frac{12}{18}

Step 5: Simplify the result by dividing both numerator and denominator by their greatest common divisor (6): $$ \frac{12 \div 6}{18 \div 6} = \frac{2}{3}

Explanation:

To divide two fractions, we multiply the dividend by the reciprocal of the divisor and then simplify the resulting fraction to its lowest terms.

Problem 2:

Divide 3123\frac{1}{2} by 77.

Solution:

Step 1: Convert the mixed fraction to an improper fraction: 312=(3×2)+12=723\frac{1}{2} = \frac{(3 \times 2) + 1}{2} = \frac{7}{2} Step 2: Rewrite the division problem: 72÷7\frac{7}{2} \div 7. Step 3: Multiply by the reciprocal of the divisor (77): The reciprocal of 77 is 17\frac{1}{7}. Step 4: Perform multiplication: $$ \frac{7}{2} \times \frac{1}{7} = \frac{7 \times 1}{2 \times 7} = \frac{7}{14}

Step 5: Simplify the fraction: $$ \frac{7 \div 7}{14 \div 7} = \frac{1}{2}

Explanation:

Mixed fractions must be converted to improper fractions first. Dividing by a whole number is the same as multiplying by a unit fraction where the whole number is the denominator.