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Fractions and Decimals - Multiplication of Fractions

Grade 7CBSE

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

🔑Concepts

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Multiplication of a Fraction by a Whole Number: To multiply a whole number by a fraction, we multiply the whole number with the numerator of the fraction while keeping the denominator the same. Visually, if we have 3 circles and 14\frac{1}{4} of each circle is shaded, the total shaded area is represented as 3×14=343 \times \frac{1}{4} = \frac{3}{4}, which looks like three-quarters of a single circle.

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The 'Of' Operator: In mathematics, the word 'of' represents multiplication. For example, 12\frac{1}{2} of 1010 is calculated as 12×10=5\frac{1}{2} \times 10 = 5. Visually, this is like taking a collection of 10 items and splitting them into two equal groups; 'half' of the collection refers to the count of items in one of those groups.

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Multiplication of a Fraction by another Fraction: This is calculated by multiplying the numerators together to get the new numerator and multiplying the denominators together to get the new denominator. Visually, imagine a unit square; if we divide the width into 12\frac{1}{2} and the height into 13\frac{1}{3}, the area of the overlapping rectangle where these two sections meet is 12×13=16\frac{1}{2} \times \frac{1}{3} = \frac{1}{6} of the total square.

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Product of Proper Fractions: When we multiply two proper fractions (where the numerator is less than the denominator), the product is always smaller than each of the fractions being multiplied. For instance, 23×45=815\frac{2}{3} \times \frac{4}{5} = \frac{8}{15}. Here, 815\frac{8}{15} is less than 23\frac{2}{3} and also less than 45\frac{4}{5}.

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Product of Improper Fractions: When we multiply two improper fractions, the product is always greater than each of the two fractions. For example, 32×54=158\frac{3}{2} \times \frac{5}{4} = \frac{15}{8}. In this case, 158\frac{15}{8} (which is 1.8751.875) is larger than both 1.51.5 and 1.251.25.

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Multiplying Mixed Fractions: To multiply mixed fractions, we first convert them into improper fractions and then multiply the numerators and denominators. Visually, 1121\frac{1}{2} is treated as 32\frac{3}{2} equal parts before performing the operation.

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Simplification before Multiplication: It is often easier to simplify or 'cancel' common factors in the numerator and denominator before performing the final multiplication. This keeps the numbers smaller and reduces the need for simplification at the end.

📐Formulae

a×bc=a×bca \times \frac{b}{c} = \frac{a \times b}{c}

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

Product of Fractions=Product of NumeratorsProduct of Denominators\text{Product of Fractions} = \frac{\text{Product of Numerators}}{\text{Product of Denominators}}

💡Examples

Problem 1:

Find the value of 312×473\frac{1}{2} \times \frac{4}{7}.

Solution:

Step 1: Convert the mixed fraction 3123\frac{1}{2} into an improper fraction: 312=(3×2)+12=723\frac{1}{2} = \frac{(3 \times 2) + 1}{2} = \frac{7}{2} Step 2: Multiply the improper fraction by 47\frac{4}{7}: 72×47\frac{7}{2} \times \frac{4}{7} Step 3: Multiply numerators and denominators: 7×42×7=2814\frac{7 \times 4}{2 \times 7} = \frac{28}{14} Step 4: Simplify the resulting fraction: 2814=2\frac{28}{14} = 2

Explanation:

We first change the mixed number into a format that allows for easy multiplication (improper fraction). Then we apply the rule of multiplying numerators and denominators. Finally, we simplify the result to its lowest form.

Problem 2:

Sushant reads 13\frac{1}{3} part of a book in 1 hour. How much part of the book will he read in 2152\frac{1}{5} hours?

Solution:

Step 1: Identify the rate of reading: 13\frac{1}{3} part per hour. Step 2: Identify the total time: 2152\frac{1}{5} hours. Step 3: Convert the mixed fraction time to an improper fraction: 215=(2×5)+15=115 hours2\frac{1}{5} = \frac{(2 \times 5) + 1}{5} = \frac{11}{5} \text{ hours} Step 4: Multiply the rate by the total time: 13×115=1×113×5=1115\frac{1}{3} \times \frac{11}{5} = \frac{1 \times 11}{3 \times 5} = \frac{11}{15}

Explanation:

To find the total work done, we multiply the work done per unit time (the fraction of the book read in one hour) by the total time spent. The result 1115\frac{11}{15} represents the portion of the book completed.