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The World of Numbers - Filling the Spaces: Fractions and Rational Numbers

Grade 9CBSE

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

🔑Concepts

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A rational number is a number that can be expressed in the form pq\frac{p}{q}, where pp and qq are integers and q≠0q \neq 0.

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Fractions are a subset of rational numbers where pp and qq are typically whole numbers and q>0q > 0. Rational numbers include negative values like −23-\frac{2}{3}.

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A rational number pq\frac{p}{q} is in its standard form (or simplest form) if pp and qq have no common factor other than 11, and the denominator qq is a positive integer.

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Every integer nn is a rational number because it can be written as n1\frac{n}{1}.

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Equivalent Rational Numbers: Multiplying or dividing the numerator and denominator of a rational number by the same non-zero integer gives an equivalent rational number, i.e., pq=p×mq×m\frac{p}{q} = \frac{p \times m}{q \times m}.

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Density Property: Between any two distinct rational numbers, there are infinitely many rational numbers.

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Rational numbers have a decimal expansion that is either terminating (e.g., 0.250.25) or non-terminating repeating (e.g., 0.333...0.333... or 0.3‾0.\overline{3}).

📐Formulae

Rational Number Form: pq, where p,q∈Z,q≠0\text{Rational Number Form: } \frac{p}{q}, \text{ where } p, q \in \mathbb{Z}, q \neq 0

Additive Inverse: a+(−a)=0\text{Additive Inverse: } a + (-a) = 0

Multiplicative Inverse (Reciprocal): ab×ba=1\text{Multiplicative Inverse (Reciprocal): } \frac{a}{b} \times \frac{b}{a} = 1

Commutative Property of Addition: a+b=b+a\text{Commutative Property of Addition: } a + b = b + a

Distributive Property: a(b+c)=ab+ac\text{Distributive Property: } a(b + c) = ab + ac

Finding a number between x and y:x+y2\text{Finding a number between } x \text{ and } y: \frac{x + y}{2}

💡Examples

Problem 1:

Find the sum of −35-\frac{3}{5} and 23\frac{2}{3}.

Solution:

−35+23=−3×315+2×515=−9+1015=115\begin{array}{r} \frac{-3}{5} + \frac{2}{3} = \frac{-3 \times 3}{15} + \frac{2 \times 5}{15} \\ = \frac{-9 + 10}{15} \\ = \frac{1}{15} \end{array}

Explanation:

To add rational numbers with different denominators, we find the Least Common Multiple (LCM) of the denominators (55 and 33), which is 1515, and then convert each fraction accordingly.

Problem 2:

Find two rational numbers between 13\frac{1}{3} and 12\frac{1}{2}.

Solution:

Method: Make denominators the same and larger. Multiply 13\frac{1}{3} by 1010\frac{10}{10} to get 1030\frac{10}{30}. Multiply 12\frac{1}{2} by 1515\frac{15}{15} to get 1530\frac{15}{30}. Two numbers between them are 1130\frac{11}{30} and 1230\frac{12}{30}.

Explanation:

By expanding the range using equivalent fractions with a common denominator, we can easily identify multiple rational numbers between two values.

Problem 3:

Simplify −4560\frac{-45}{60} to its standard form.

Solution:

HCF of 45 and 60 is 15.\text{HCF of } 45 \text{ and } 60 \text{ is } 15. −45÷1560÷15=−34\frac{-45 \div 15}{60 \div 15} = \frac{-3}{4}

Explanation:

To convert a rational number to its standard form, divide both the numerator and the denominator by their Highest Common Factor (HCF).

Problem 4:

Verify the distributive property a(b+c)=ab+aca(b + c) = ab + ac for a=12a = \frac{1}{2}, b=23b = \frac{2}{3}, and c=13c = \frac{1}{3}.

Solution:

LHS: 12×(23+13)=12×33=12×1=12\frac{1}{2} \times (\frac{2}{3} + \frac{1}{3}) = \frac{1}{2} \times \frac{3}{3} = \frac{1}{2} \times 1 = \frac{1}{2}

RHS: (12×23)+(12×13)=26+16=36=12(\frac{1}{2} \times \frac{2}{3}) + (\frac{1}{2} \times \frac{1}{3}) = \frac{2}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2}

LHS = RHS.

Explanation:

The distributive property allows us to multiply a sum by a number or multiply each addend separately and then add the results.