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Rational Numbers - Operations on Rational Numbers (Addition, Subtraction, Multiplication, Division)

Grade 7CBSE

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

🔑Concepts

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Rational Numbers are numbers that can be expressed in the form pq\frac{p}{q}, where pp and qq are integers and q≠0q \neq 0. Visually, they represent points on a number line that can exist between integers.

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To add rational numbers with the same denominator, simply add the numerators and keep the common denominator. For example, 27+37\frac{2}{7} + \frac{3}{7} means taking 2 parts and 3 parts of a whole divided into 7 equal sections, resulting in 57\frac{5}{7}.

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To add or subtract rational numbers with different denominators, first find the Least Common Multiple (LCM) of the denominators to convert them into like fractions. This is like resizing the sub-divisions on a number line so that the jumps are of equal length.

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The Additive Inverse of a rational number ab\frac{a}{b} is −ab-\frac{a}{b}, such that their sum is 0. On a number line, this is the point at the same distance from zero but in the opposite direction.

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Subtraction of rational numbers is defined as adding the additive inverse of the number being subtracted. For example, 23−14\frac{2}{3} - \frac{1}{4} is the same as 23+(−14)\frac{2}{3} + (-\frac{1}{4}).

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Multiplication involves multiplying the numerators together and the denominators together. If you visualize a rectangle with side lengths ab\frac{a}{b} and cd\frac{c}{d}, the area represents the product acbd\frac{ac}{bd}.

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The Reciprocal (Multiplicative Inverse) of a non-zero rational number ab\frac{a}{b} is ba\frac{b}{a}. Multiplying a number by its reciprocal always results in 1.

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Division of one rational number by another is performed by multiplying the first number by the reciprocal of the second. This is expressed as ab÷cd=ab×dc\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}.

📐Formulae

Addition (Same Denominator): ab+cb=a+cb\frac{a}{b} + \frac{c}{b} = \frac{a + c}{b}

Addition (Different Denominator): ab+cd=ad+bcbd\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}

Additive Inverse: ab+(−ab)=0\frac{a}{b} + (-\frac{a}{b}) = 0

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

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

Multiplicative Identity: ab×1=ab\frac{a}{b} \times 1 = \frac{a}{b}

Additive Identity: ab+0=ab\frac{a}{b} + 0 = \frac{a}{b}

💡Examples

Problem 1:

Simplify: −37+23\frac{-3}{7} + \frac{2}{3}

Solution:

Step 1: Find the LCM of the denominators 7 and 3. LCM(7, 3) = 21. Step 2: Convert to like fractions: −3×37×3=−921\frac{-3 \times 3}{7 \times 3} = \frac{-9}{21} and 2×73×7=1421\frac{2 \times 7}{3 \times 7} = \frac{14}{21}. Step 3: Add the numerators: −9+1421=521\frac{-9 + 14}{21} = \frac{5}{21}. Final Answer: 521\frac{5}{21}

Explanation:

To add fractions with different denominators, we find a common ground (LCM) so the parts are of the same size before combining them.

Problem 2:

Divide −45\frac{-4}{5} by 815\frac{8}{15}

Solution:

Step 1: Write the division expression: −45÷815\frac{-4}{5} \div \frac{8}{15}. Step 2: Multiply by the reciprocal of the divisor: −45×158\frac{-4}{5} \times \frac{15}{8}. Step 3: Simplify by canceling common factors: −4×155×8=−1×31×2\frac{-4 \times 15}{5 \times 8} = \frac{-1 \times 3}{1 \times 2}. Step 4: Multiply the remaining terms: −32\frac{-3}{2}. Final Answer: −32\frac{-3}{2} or −112-1\frac{1}{2}

Explanation:

Division is the inverse of multiplication. We multiply the dividend by the reciprocal of the divisor and simplify the resulting fraction to its lowest terms.