Official Syllabus Item: 4, 27, 36, 37 Arithmetic & Number Theory

Fractions: Types, Operations & Comparisons

Comprehensive masterclass on fractions for RIMC Stage-I: proper, improper, mixed, number line representation, comparing, arranging, and descriptive word problems.

Overview & Fundamental Concept

Fractions represent parts of a whole or collections of equal parts. In RIMC descriptive mathematics, candidates are tested on rigorous classifications (proper, improper, mixed, unit, like/unlike), geometric representation on the real number line, equivalent forms, irreducible reductions via HCF, ascending/descending ordering using the LCM common denominator method, and multi-step word problems involving sequential remainders.

Key Theoretical Principles

In proper fractions, Numerator < Denominator (value between 0 and 1).
In improper fractions, Numerator >= Denominator (value >= 1); convertible to mixed numerals.
To compare or arrange unlike fractions, convert them to equivalent fractions with common LCM denominator.
Two fractions are equivalent if their cross-products are equal (ad = bc).
A fraction is in simplest form when gcd(Numerator, Denominator) = 1.
In remainder word problems, sequential expenditures must be evaluated on intermediate remainders.

Illustrated Conceptual Diagrams & Visual Schematics

Fraction Classification Models & Cross-Multiplication Scheme

Fraction Types: Proper Fractions ($<1$, numerator $<$ denominator), Improper Fractions ($ge 1$, numerator $ge$ denominator), Mixed Numbers (whole number $+$ proper fraction). To compare fractions $rac{a}{b}$ and $rac{c}{d}$ instantly, cross-multiply: $a cdot d gtrless b cdot c$.

Topper Strategy & Descriptive Step Breakdown

Mastering RIMC Mathematics Standard

State the LCM of all denominators explicitly before ordering fractions. Show each equivalent fraction calculation. In remainder word problems, define the total variable x and show each stage's balance clearly.

Step 1: Given Information & Variable Declaration (1-2 Marks)

Clearly write down the data provided in the question. State any assumption (e.g., 'Let the cost price be ₹ x').

Step 2: Formula & Conceptual Foundation (1-2 Marks)

State the exact mathematical theorem or formula in standard algebraic notation before entering numeric values.

Step 3: Systematic Calculation & Reduction (2-4 Marks)

Carry out each step on a fresh line. Avoid combining three operations in one step so examiners can award partial credit.

Step 4: Final Statement with Proper Units (1 Mark)

End with a concluding statement highlighting the answer in a neat box or underline with explicit units (e.g., '₹ 1,500', '45 km/h').

Key Formulas & Conversion Cheat Sheet

Fraction Division Reciprocal

$$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}$$

Cross-Product Equivalence Test

$$\frac{a}{b} = \frac{c}{d} \iff a \cdot d = b \cdot c$$

Mixed Fraction Conversion

$$q\frac{r}{d} = \frac{qd + r}{d} \quad \text{and} \quad \frac{N}{D} = q\frac{r}{D}$$

Fraction of a Quantity

$$\text{Value} = \frac{a}{b} \text{ of } Q = \frac{a \times Q}{b}$$

Solved Model Examples (Descriptive Format with Step Mark Distribution)

Solved Model 1 Model Solution

Arrange the fractions 5/6, 7/9, 11/12, and 13/18 in ascending and descending order.

Step-by-Step Model Solution:

Step 1: LCM(6, 9, 12, 18) = 36.\nStep 2: Equivalent fractions: 5/6 = 30/36, 7/9 = 28/36, 11/12 = 33/36, 13/18 = 26/36.\nStep 3: Comparing numerators: 26 < 28 < 30 < 33.\nAscending Order: 13/18 < 7/9 < 5/6 < 11/12.\nDescending Order: 11/12 > 5/6 > 7/9 > 13/18.

Solved Model 2 Model Solution

A bamboo pole has 1/3 in mud, 1/4 in water, and 5 m above water. Find the total length of the pole.

Step-by-Step Model Solution:

Step 1: Submerged fraction = 1/3 + 1/4 = 7/12.\nStep 2: Fraction above water = 1 - 7/12 = 5/12.\nStep 3: (5/12) × L = 5 m => L = (5 × 12) / 5 = 12 m.\nFinal Answer: Total length is 12 metres.

Solved Model 3 Model Solution

A man spent 2/5 of his salary on rent, 1/4 of remainder on education, and 1/3 of what was left on food. If he saves ₹ 9,000, find his salary.

Step-by-Step Model Solution:

Step 1: Remainder 1 = x - 2/5 x = 3/5 x.\nStep 2: Education = 1/4 × 3/5 x = 3/20 x. Remainder 2 = 3/5 x - 3/20 x = 9/20 x.\nStep 3: Food = 1/3 × 9/20 x = 3/20 x. Final savings = 9/20 x - 3/20 x = 6/20 x = 3/10 x.\nStep 4: 3/10 x = 9000 => x = ₹ 30,000.\nFinal Answer: Monthly salary is ₹ 30,000.

RIMC Practice & Drill Exercises

Attempt these questions on paper following proper step-wise conventions, then click Reveal Solution to verify your answers.

Question 1 5 Marks

Arrange in descending order: 2/3, 3/5, 7/10, 8/15.