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Number System - Fractions and Decimals: Multiplication and Division

Grade 7ICSE

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

🔑Concepts

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Multiplying Fractions: To multiply two or more fractions, multiply the numerators together to get the new numerator and the denominators together to get the new denominator. For example, 23×57=1021\frac{2}{3} \times \frac{5}{7} = \frac{10}{21}. Visually, this can be understood as taking a 'part of a part'. Imagine a rectangle representing a whole; if you shade 57\frac{5}{7} of it and then find 23\frac{2}{3} of that shaded area, the resulting portion is 1021\frac{10}{21} of the original rectangle.

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Reciprocal of a Fraction: The reciprocal (or multiplicative inverse) of a non-zero fraction ab\frac{a}{b} is ba\frac{b}{a}. When a fraction is multiplied by its reciprocal, the product is always 11. For instance, the reciprocal of 49\frac{4}{9} is 94\frac{9}{4} because 49×94=1\frac{4}{9} \times \frac{9}{4} = 1. Visually, a reciprocal 'flips' the ratio between the numerator and denominator.

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Dividing Fractions: Dividing a fraction by another fraction is equivalent to multiplying the first fraction by the reciprocal of the second. To calculate 34÷12\frac{3}{4} \div \frac{1}{2}, you change it to 34×21=64=32\frac{3}{4} \times \frac{2}{1} = \frac{6}{4} = \frac{3}{2}. This operation determines how many times the divisor fits into the dividend.

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Multiplying Decimals: To multiply decimals, ignore the decimal points and multiply them as whole numbers. After finding the product, count the total number of decimal places in both original numbers and place the decimal point in the product so it has the same total number of decimal places. For example, in 0.4×0.020.4 \times 0.02, there are 1+2=31 + 2 = 3 decimal places, so the result is 0.0080.008. Visually, multiplying 0.1×0.10.1 \times 0.1 is like taking 11 small square out of a 10×1010 \times 10 grid, which equals 0.010.01.

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Decimal Multiplication/Division by Powers of 10: Multiplying a decimal by 10,100,10, 100, or 10001000 shifts the decimal point to the right by as many places as there are zeros. Dividing a decimal by 10,100,10, 100, or 10001000 shifts the decimal point to the left. For example, 5.67×100=5675.67 \times 100 = 567 and 5.67÷10=0.5675.67 \div 10 = 0.567. This is a visual shift of digits across the place value columns.

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Division of Decimals: To divide a decimal by another decimal, multiply both the dividend and the divisor by the same power of 1010 (like 10,100,…10, 100, \dots) to make the divisor a whole number. For instance, to solve 0.25÷0.50.25 \div 0.5, multiply both by 1010 to get 2.5÷5=0.52.5 \div 5 = 0.5. The decimal point in the quotient is placed directly above the decimal point in the dividend during long division.

📐Formulae

Product of Fractions = Product of NumeratorsProduct of Denominators\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}

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

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

Decimal ×10n=Move decimal n places right\text{Decimal } \times 10^n = \text{Move decimal } n \text{ places right}

Decimal ÷10n=Move decimal n places left\text{Decimal } \div 10^n = \text{Move decimal } n \text{ places left}

💡Examples

Problem 1:

Evaluate: 214×23÷1122 \frac{1}{4} \times \frac{2}{3} \div 1 \frac{1}{2}

Solution:

Step 1: Convert mixed fractions to improper fractions: 214=942 \frac{1}{4} = \frac{9}{4} and 112=321 \frac{1}{2} = \frac{3}{2}. \ Step 2: Rewrite the expression: 94×23÷32\frac{9}{4} \times \frac{2}{3} \div \frac{3}{2}. \ Step 3: Change division to multiplication by the reciprocal: 94×23×23\frac{9}{4} \times \frac{2}{3} \times \frac{2}{3}. \ Step 4: Multiply the numerators and denominators: 9×2×24×3×3=3636=1\frac{9 \times 2 \times 2}{4 \times 3 \times 3} = \frac{36}{36} = 1.

Explanation:

We first standardise the fractions by converting them to improper forms, then apply the BODMAS rule (specifically the division-to-multiplication rule) and simplify the resulting expression.

Problem 2:

Divide 30.9430.94 by 0.70.7.

Solution:

Step 1: Write the division as a fraction: 30.940.7\frac{30.94}{0.7}. \ Step 2: Multiply both numerator and denominator by 1010 to make the divisor a whole number: 30.94×100.7×10=309.47\frac{30.94 \times 10}{0.7 \times 10} = \frac{309.4}{7}. \ Step 3: Perform long division: 309.4÷7309.4 \div 7. \ 30÷7=430 \div 7 = 4 remainder 22. \ 29÷7=429 \div 7 = 4 remainder 11. \ Place decimal point: 14÷7=214 \div 7 = 2. \ Step 4: The result is 44.244.2.

Explanation:

To divide by a decimal, we convert the divisor into a whole number by shifting the decimal point in both the divisor and dividend. Then, we perform normal division while keeping the decimal point in its new aligned position.