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#5 Compulsory Number System & Arithmetic Math 5

Rational Numbersपरिमेय संख्याएँ: गुणधर्म, दशमलव रूप व संक्रियाएं

Rational numbers form the foundational bedrock of arithmetic in the SCERT UP and NCERT Class 6-8 curriculum, covering closure, commutativity, associativity, distributive properties, and operations on decimal forms. For the Super TET Junior Exam, expect high-yield questions on multiplicative inverse, standard forms, non-terminating recurring decimals, and density property.

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Syllabus Focus & High-Yield Strategy: Rational numbers form the foundational bedrock of arithmetic in the SCERT UP and NCERT Class 6-8 curriculum, covering closure, commutativity, associativity, distributive properties, and operations on decimal forms. For the Super TET Junior Exam, expect high-yield questions on multiplicative inverse, standard forms, non-terminating recurring decimals, and density property.

1. Fundamental Axioms, Definitions & Foundational Properties

Definition of Rational Number: Any number that can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are integers, and $q \neq 0$. The set of all rational numbers is denoted by $\mathbb{Q}$.
Closure Property: The set of rational numbers is closed under addition ($a+b \in \mathbb{Q}$), subtraction ($a-b \in \mathbb{Q}$), and multiplication ($a \times b \in \mathbb{Q}$). However, division is closed only when excluding the divisor $0$ ($a \div 0$ is undefined).
Commutative Property: Addition and multiplication are commutative for rational numbers: $a + b = b + a$ and $a \times b = b \times a$. Subtraction and division are strictly non-commutative ($a - b \neq b - a$).
Associative Property: Addition and multiplication satisfy associativity: $(a + b) + c = a + (b + c)$ and $(a \times b) \times c = a \times (b \times c)$. Subtraction and division lack associative validity.
Exam Pro-Tip: Always remember that zero ($0$) is the additive identity and one ($1$) is the multiplicative identity for rational numbers.
Memory Mnemonic: C-CAD — Closure, Commutative, Associative, Distributive properties apply to Addition & Multiplication!

2. Formulas, Standard Theorems & Algorithmic Procedures

Category / CaseFormula / Mathematical RuleTopper Calculation Shortcut
Additive InverseFor $\frac{p}{q}$, additive inverse is $-\frac{p}{q}$ such that $\frac{p}{q} + \left(-\frac{p}{q}\right) = 0$Simply flip the sign of the numerator/denominator.
Multiplicative Inverse (Reciprocal)For $\frac{p}{q}$ ($p \neq 0$), reciprocal is $\frac{q}{p}$ such that $\frac{p}{q} \times \frac{q}{p} = 1$Flip the fraction upside down without changing its sign.
Distributive Law over Addition: For any rational numbers $a, b, c$, the distributive law states $a \times (b + c) = (a \times b) + (a \times c)$.
Decimal Expansion Theorem: A rational number $\frac{p}{q}$ (in simplest form) has a terminating decimal expansion if and only if the prime factorization of $q$ is of the form $2^n 5^m$, where $n, m$ are non-negative integers.
Recurring Decimal Conversion Algorithm: To convert pure recurring decimal $0.\bar{a}$ to a fraction, write $\frac{a}{9}$. For mixed recurring decimal $0.a\bar{b}$, use $\frac{ab - a}{90}$.
15-Second Speed-Math Trick: To convert $0.2\bar{3}$ to a fraction rapidly: Subtract the non-recurring part from the full number ($23 - 2 = 21$), put one $9$ for the recurring digit and one $0$ for the non-recurring decimal place to get $\frac{21}{90} = \frac{7}{30}$.

3. Word Problem Modeling & Practical Applications

Standard Form Reduction: To reduce $\frac{-45}{72}$ to standard form, compute $\text{HCF}(|-45|, 72) = 9$. Divide numerator and denominator by $9$ to get $\frac{-5}{8}$.
Rational Numbers Between Two Numbers: To find $n$ rational numbers between $a$ and $b$, equalize denominators to a common multiple $m$ or use the mean formula $\frac{a+b}{2}$ recursively.
Word Problem (Temperature/Elevation): If a submarine is at $-\frac{35}{4}$ meters below sea level and ascends by $\frac{15}{2}$ meters, final position is $-\frac{35}{4} + \frac{15}{2} = \frac{-35 + 30}{4} = -\frac{5}{4}$ meters.
Proportion & Scale Application: Dividing a total quantity $Q$ into rational ratios $\frac{a}{b} : \frac{c}{d}$ requires clearing denominators first by multiplying by $\text{LCM}(b, d)$.
UP Exam High-Yield Tip: Super TET Junior exams frequently test the density property: there exist infinitely many rational numbers between any two distinct rational numbers.

4. Mathematical Edge Cases, Sign Conventions & Common Traps

Critical Distinction: The negative sign in a rational number like $\frac{-p}{q}$ or $-\frac{p}{q}$ applies to the entire number; never treat the numerator and denominator negative signs as cancelling unless both are negative (yielding a positive number).
Edge Case / Exception: Division by zero is undefined ($\frac{a}{0} = \text{Undefined}$). However, $0$ divided by any non-zero rational number is zero ($0 \div \frac{p}{q} = 0$).
Common Calculation Trap: Confusing additive inverse with multiplicative inverse. Additive inverse changes the sign ($-a$), whereas multiplicative inverse flips the value ($a^{-1}$).
Trap Prevention Trick: When simplifying complex rational expressions involving nested brackets, always resolve innermost parentheses first and check sign distributions carefully.
Memory Mnemonic: FLIP — For Multiplicative Inverse, Leave sign, Invert parts!

Comprehensive Mathematical Matrix & Method Comparison

Parameter / DimensionMethod / Concept AMethod / Concept BExam Speed & Accuracy Takeaway
Decimal ConversionTerminates ($q = 2^n 5^m$)Non-terminating RecurringCheck prime factors of denominator immediately.
Inverse OperationsAdditive Inverse ($-\frac{p}{q}$)Multiplicative Inverse ($\frac{q}{p}$)Never confuse sign change with reciprocal.
Density PropertyFinite IntegersInfinite Rational NumbersAlways infinite rationals between two numbers.
Standard FormDenominators must be positiveHCF of $|p|$ and $q$ must be $1$Reduce fractions to lowest terms before comparing.
Practice Progress: 0 of 10 Answered
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Question 1 of 10 Super TET Junior Mathematics

1. What is the multiplicative inverse of $-\frac{3}{5} \times \left(-\frac{15}{9} ight)?$

Correct Answer: Option A

First, evaluate the expression: $-\frac{3}{5} \times \left(-\frac{15}{9}\right) = \frac{3 \times 15}{5 \times 9} = \frac{45}{45} = 1$. The multiplicative inverse of $1$ is $1$, since $1 \times 1 = 1$. Distractors like $-1$ arise from sign multiplication errors.

Question 2 of 10 Super TET Junior Mathematics

2. Which of the following rational numbers has a terminating decimal expansion?

Correct Answer: Option A

A rational number $\frac{p}{q}$ in simplest form terminates if prime factors of $q$ are only $2$ and $5$. For $\frac{7}{40}$, $40 = 2^3 \times 5^1$, which terminates. Denominators $30$ ($2 \times 3 \times 5$), $45$ ($3^2 \times 5$), and $21$ ($3 \times 7$) contain prime factors other than $2$ and $5$, resulting in non-terminating recurring decimals.

Question 3 of 10 Super TET Junior Mathematics

3. What is the decimal representation of the pure recurring decimal $0.\bar{54}$ expressed in simplest fractional form?

Correct Answer: Option B

Let $x = 0.545454...$. Multiplying by $100$: $100x = 54.5454...$. Subtracting $x$: $99x = 54 \implies x = \frac{54}{99}$. Dividing numerator and denominator by their HCF ($9$) gives $\frac{6}{11}$. Option 2 is correct.

Question 4 of 10 Super TET Junior Mathematics

4. If $x = -\frac{2}{3}$ and $y = \frac{4}{5}$, what is the value of $\frac{x+y}{x-y}?

Correct Answer: Option A

Calculate numerator $x+y = -\frac{2}{3} + \frac{4}{5} = \frac{-10 + 12}{15} = \frac{2}{15}$. Calculate denominator $x-y = -\frac{2}{3} - \frac{4}{5} = \frac{-10 - 12}{15} = \frac{-22}{15}$. Then $\frac{x+y}{x-y} = \frac{2/15}{-22/15} = \frac{2}{-22} = -\frac{1}{11}$. Option 0 is correct.

Question 5 of 10 Super TET Junior Mathematics

5. How many rational numbers exist between any two distinct rational numbers?

Correct Answer: Option D

The set of rational numbers is dense. Between any two distinct rational numbers, there exist infinitely many rational numbers, which can be generated using the mean method or by common denominator expansion.

Question 6 of 10 Super TET Junior Mathematics

6. What is the standard form of the rational number $\frac{-84}{-144}?

Correct Answer: Option B

First, negative signs in both numerator and denominator cancel: $\frac{-84}{-144} = \frac{84}{144}$. Find HCF of $84$ and $144$, which is $12$. Dividing both by $12$ yields $\frac{84 \div 12}{144 \div 12} = \frac{7}{12}$.

Question 7 of 10 Super TET Junior Mathematics

7. Which of the following properties does NOT hold true for subtraction of rational numbers?

Correct Answer: Option B

Subtraction of rational numbers is not commutative because $a - b \neq b - a$ in general (e.g., $\frac{1}{2} - \frac{1}{3} = \frac{1}{6}$, while $\frac{1}{3} - \frac{1}{2} = -\frac{1}{6}$). Thus, commutativity fails for subtraction.

Question 8 of 10 Super TET Junior Mathematics

8. If $\frac{3}{4}$ of a number is $\frac{15}{28}$, what is the number?

Correct Answer: Option A

Let the number be $x$. Given $\frac{3}{4}x = \frac{15}{28}$. Solving for $x$: $x = \frac{15}{28} \times \frac{4}{3} = \frac{15 \times 4}{28 \times 3} = \frac{5}{7}$. Option 0 is correct.

Question 9 of 10 Super TET Junior Mathematics

9. What is the value of $0.3\bar{6} + 0.\bar{23}?

Correct Answer: Option D

Convert the recurring decimals: 0.3666...=11/30 and 0.2323...=23/99. With common denominator 990, their sum is 363/990+230/990=593/990. The answer is Option D.

Question 10 of 10 Super TET Junior Mathematics

10. What is the additive inverse of $\frac{-5}{-9}?

Correct Answer: Option B

First simplify the given number: $\frac{-5}{-9} = \frac{5}{9}$. The additive inverse of $\frac{5}{9}$ is $-\frac{5}{9}$, since $\frac{5}{9} + \left(-\frac{5}{9}\right) = 0$.

5-Minute Exam Simulation Mode

Experience the real pressure of Super TET Junior (Part 2C Mathematics). You will face 10 randomized speed-math questions with negative marking (-1 for incorrect answers) and a strict 5-minute countdown.

Time Limit:

5 Minutes (300s)

Marking Scheme:

+3 Marks | -1 Negative

Passing Benchmark:

21+ Marks (70%)

High-Yield Rapid Recall Formulas & Laws (Must-Remember Pointers)

#1
A rational number is any number expressible as $\frac{p}{q}$ with $q \neq 0$ and $p, q \in \mathbb{Z}$.
#2
Closure, commutativity, and associativity hold for addition and multiplication of rational numbers.
#3
Additive inverse of $\frac{p}{q}$ is $-\frac{p}{q}$; multiplicative inverse is $\frac{q}{p}$.
#4
Division by zero is undefined, whereas zero divided by any non-zero rational number is zero.
#5
Terminates in decimal form if and only if denominator prime factors are exclusively powers of 2 and 5.
#6
Pure recurring decimal $0.\bar{a}$ converts to fraction as $\frac{a}{9}$.
#7
Mixed recurring decimal $0.a\bar{b}$ converts to fraction as $\frac{ab - a}{90}$.
#8
The set of rational numbers possesses the density property, meaning infinitely many rationals exist between any two.

Speed Math & Memory Mnemonics Vault (Exam Retention Anchors)

Rational Operations Rule

Memory Key: COIN

C - Closure, O - Operations, I - Inverse, N - Negative rule.

Decimal Conversion Shortcut

Memory Key: PQR

P - Prime factors 2 & 5, Q - Quotient rules, R - Recurring fraction bars.

Negative Marking Traps & Common Blunders (-1 Penalty Avoidance)

Negative Marking Alert #1:

Negative Marking Trap 1: Forgetting that a negative sign in both numerator and denominator makes the rational number positive.

Negative Marking Alert #2:

Negative Marking Trap 2: Confusing additive identity ($0$) with multiplicative identity ($1$).

Negative Marking Alert #3:

Negative Marking Trap 3: Applying commutativity or associativity to subtraction and division.

Negative Marking Alert #4:

Negative Marking Trap 4: Miscalculating recurring decimal conversions by ignoring the non-recurring prefix digits.

Self-Assessment Checklist

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