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Fractions - Multiplication and Division of Fractions

Grade 5ICSE

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

🔑Concepts

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Multiplying a Fraction by a Whole Number: To multiply a fraction by a whole number, we multiply the numerator of the fraction by the whole number and keep the denominator the same. For example, if you have three containers each filled to 15\frac{1}{5} of their capacity, you can visualize them as three separate bars with one-fifth shaded. Combining them results in 35\frac{3}{5} of a bar being shaded.

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Multiplying a Fraction by a Fraction: To find the product of two fractions, multiply the numerators together and the denominators together. Visually, this is like finding a part of a part. Imagine a square divided into 3 vertical columns (representing 13\frac{1}{3}) and then dividing those columns into 2 horizontal rows. The overlapping area representing 12\frac{1}{2} of 13\frac{1}{3} is one small block out of 6 total blocks, or 16\frac{1}{6}.

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The 'Of' Operator: In fraction problems, the word 'of' acts as a multiplication sign. For example, '12\frac{1}{2} of 2020' means 12×20\frac{1}{2} \times 20. If you have a collection of 20 marbles and you take half of them, you are performing multiplication to find that you have 10 marbles.

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Reciprocal (Multiplicative Inverse): Two numbers are reciprocals if their product is 11. To find the reciprocal of a fraction, simply flip it so the numerator becomes the denominator and vice versa. For example, the reciprocal of 34\frac{3}{4} is 43\frac{4}{3}. Visually, if 34\frac{3}{4} represents three-quarters of a whole, its reciprocal represents the amount needed to scale that part back to a full units-based ratio.

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Dividing a Fraction by another Fraction: Division is the process of finding how many times the divisor fits into the dividend. To divide, we 'Keep' the first fraction, 'Change' the division sign to multiplication, and 'Flip' the second fraction to its reciprocal. For example, 12÷14\frac{1}{2} \div \frac{1}{4} asks 'How many quarters fit into one-half?'. Visualizing a half-circle, you can see that two quarter-circles fit perfectly inside it, so the answer is 22.

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Simplification (Cancellation Method): Before multiplying fractions, it is often easier to simplify them by dividing any numerator and any denominator by their Greatest Common Divisor (GCD). This 'diagonal' or 'vertical' cancellation makes the numbers smaller and the final multiplication much simpler.

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Division involving Mixed Numbers: Before multiplying or dividing, always convert mixed numbers into improper fractions. For instance, 1121 \frac{1}{2} should be converted to 32\frac{3}{2}. Once converted, you can apply the standard rules for multiplication or division.

📐Formulae

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

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

Whole Number×Fraction=n×ab=n×ab\text{Whole Number} \times \text{Fraction} = n \times \frac{a}{b} = \frac{n \times a}{b}

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

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

Mixed Number to Improper Fraction: wab=(w×b)+ab\text{Mixed Number to Improper Fraction: } w \frac{a}{b} = \frac{(w \times b) + a}{b}

💡Examples

Problem 1:

Multiply 512\frac{5}{12} by 415\frac{4}{15} and give the answer in the simplest form.

Solution:

Step 1: Write the multiplication expression: 512×415\frac{5}{12} \times \frac{4}{15}. \ Step 2: Use the cancellation method to simplify before multiplying. Divide 55 and 1515 by their GCD (55): 112×43\frac{1}{12} \times \frac{4}{3}. \ Step 3: Divide 44 and 1212 by their GCD (44): 13×13\frac{1}{3} \times \frac{1}{3}. \ Step 4: Multiply the remaining numerators and denominators: 1×13×3=19\frac{1 \times 1}{3 \times 3} = \frac{1}{9}.

Explanation:

The problem uses the product of fractions rule. By simplifying (canceling) common factors in the numerators and denominators first, we avoid working with large numbers like 20180\frac{20}{180} and go straight to the simplest form.

Problem 2:

Divide 3133 \frac{1}{3} by 59\frac{5}{9}.

Solution:

Step 1: Convert the mixed number 3133 \frac{1}{3} to an improper fraction: 313=(3×3)+13=1033 \frac{1}{3} = \frac{(3 \times 3) + 1}{3} = \frac{10}{3}. \ Step 2: Write the division expression: 103÷59\frac{10}{3} \div \frac{5}{9}. \ Step 3: Apply the Keep-Change-Flip rule (multiply by the reciprocal of the divisor): 103×95\frac{10}{3} \times \frac{9}{5}. \ Step 4: Simplify by canceling common factors: 1010 and 55 divide by 55 (leaving 22 and 11); 99 and 33 divide by 33 (leaving 33 and 11). \ Step 5: Multiply the simplified fractions: 21×31=61=6\frac{2}{1} \times \frac{3}{1} = \frac{6}{1} = 6.

Explanation:

This solution demonstrates converting a mixed number to an improper fraction first, which is a required step before any division. Then, the division is converted to multiplication by the reciprocal.