Arranging of Fractions
Pointwise AISSEE Class 6 Mathematics guide for Arranging of Fractions (Topic 25). Learn cross-multiplication comparison, LCM denominator equalization, decimal ranking, and ascending vs descending orders.
Overview & Fundamental Concept
Pointwise AISSEE Class 6 Mathematics guide for Arranging of Fractions (Topic 25). Learn cross-multiplication comparison, LCM denominator equalization, decimal ranking, and ascending vs descending orders.
Official Syllabus Coverage Breakdown
| Syllabus Item | Prescribed Scope | Exam Relevance & Pedagogical Objective |
|---|---|---|
| Item 25 | Comparison of fractions, like and unlike fractions, arranging fractions in ascending order, arranging in descending order, cross-multiplication technique. | Master fraction comparison axioms, cross-multiplication shortcuts, LCM like fraction conversion, and ascending/descending ordering algorithms for AISSEE Class VI. |
1. The Three Fundamental Fraction Comparison Axioms
- Case 1 (Like Fractions — Same Denominator): If two or more fractions have the same denominator, the fraction with the larger numerator is the greater fraction: $$\frac{7}{12} > \frac{5}{12} > \frac{2}{12}$$
- Case 2 (Same Numerator — Different Denominators): If two or more fractions have the same numerator, the fraction with the smaller denominator is the greater fraction (because dividing the whole into fewer parts yields larger pieces!): $$\frac{4}{5} > \frac{4}{7} > \frac{4}{11}$$
- Case 3 (Unlike Fractions — Different Numerators and Denominators): Must be converted to equivalent like fractions using the $\text{LCM}$ of their denominators, or compared using pairwise cross-multiplication.
| Fractions Condition | Comparison Rule | Example Comparison |
|---|---|---|
| Same Denominators | Larger Numerator is Greater | $\frac{8}{15} > \frac{4}{15}$ |
| Same Numerators | Smaller Denominator is Greater | $\frac{5}{8} > \frac{5}{12}$ |
| Unlike Fractions | Convert via LCM of Denominators | $\frac{2}{3} (=\frac{8}{12}) > \frac{3}{4} (=\frac{9}{12}) \to \frac{3}{4} > \frac{2}{3}$ |
Cadet Mnemonic • Denominator Inverse Rule
When numerators are equal:
- Bigger bottom = Smaller slice
- Smaller bottom = Bigger slice
2. Pairwise Cross-Multiplication Comparison Shortcut
- The Cross-Multiplication Method: To compare any two fractions $\frac{a}{b}$ and $\frac{c}{d}$ in two seconds:
- 1. Multiply the numerator of the first by the denominator of the second ($a \times d$).
- 2. Multiply the denominator of the first by the numerator of the second ($b \times c$).
- 3. Compare the cross products:
- • If $a \times d > b \times c \implies \frac{a}{b} > \frac{c}{d}$
- • If $a \times d < b \times c \implies \frac{a}{b} < \frac{c}{d}$
- • If $a \times d = b \times c \implies \frac{a}{b} = \frac{c}{d}$ (Equivalent fractions).
- Example: Compare $\frac{5}{8}$ and $\frac{7}{11}$: $5 \times 11 = 55$ and $8 \times 7 = 56$. Since $55 < 56$, $\frac{5}{8} < \frac{7}{11}$!
Exam Trick • Upward Arrow Cross-Multiplication
Always multiply upwards from bottom to top! Write the product above the numerator: 5 × 11 = 55 above 5; 8 × 7 = 56 above 7. Since 56 > 55, 7/11 is larger!
3. Ordering Protocols: Ascending vs. Descending Order
- Ascending Order: Arranging numbers from the smallest to the greatest (increasing order: $x_1 < x_2 < x_3 < \dots$).
- Descending Order: Arranging numbers from the greatest to the smallest (decreasing order: $x_1 > x_2 > x_3 > \dots$).
- The 3-Step LCM Standardization Algorithm:
• Step 1: Find the $\text{LCM}$ of all the denominators.
• Step 2: Convert each fraction into an equivalent fraction with the $\text{LCM}$ as its common denominator.
• Step 3: Compare the numerators and arrange in the required order. - Decimal Value Method (Fastest for 3+ Fractions): Convert each fraction into decimal form up to 2 decimal places and rank directly (e.g., $\frac{1}{2} = 0.50, \frac{2}{3} = 0.67, \frac{3}{5} = 0.60$). Ascending: $\frac{1}{2} < \frac{3}{5} < \frac{2}{3}$.
AISSEE Trap Alert • Ascending vs Descending Inversion
Make sure you read the question stem carefully! Arranging in descending order when asked for ascending order is a heartbreaking 3-mark loss.
Visual Concept Blueprint • Vector Architecture
Speed Shortcuts & Mental Calculation Strategies
Master these high-yield time-saving heuristics curated by top AISSEE scorers to eliminate calculation bottlenecks and answer questions in under 45 seconds.
Shortcut Technique 1: Same Gap Shortcut
If the difference between numerator and denominator is the same in all proper fractions (e.g., 2/3, 3/4, 4/5, all have gap 1), the fraction with the **largest numerator is the greatest**: $2/3 < 3/4 < 4/5$!
Shortcut Technique 2: Benchmark 1/2 Elimination
Compare fractions against 1/2: If numerator is less than half the denominator, fraction is < 1/2. If greater than half, fraction is > 1/2.
Shortcut Technique 3: Tournament Bracket for Greatest/Smallest
To find only the largest among 4 fractions, compare pairs with cross-multiplication, then compare the winners.
Interactive 3D Revision Flashcards Deck
Click any card to flip and reveal the key exam rule or topper secret.
REVISION FLASHCARD
What is the core rule & definition for Arranging of Fractions?
CADET MEMORY KEY
Cross-Product Rule: \frac{a}{b} > \frac{c}{d} \iff a \times d > b \times c
REVISION FLASHCARD
Cadet Shortcut: How to solve Arranging of Fractions in under 30 seconds?
CADET MEMORY KEY
Fast MCQ Elimination: Check the last digit (unit digit) and estimate the order of magnitude to eliminate 2 distractors in under 10 seconds.
REVISION FLASHCARD
Exam Trap Alert: What pitfall do most candidates fall into?
CADET MEMORY KEY
Ascending vs Descending: Ascending: smallest to largest (↑). Descending: largest to smallest (↓).
REVISION FLASHCARD
Rapid Memory Anchor for AISSEE Hall Recall:
CADET MEMORY KEY
Equal Gap Rule: For proper fractions with equal (d - n) gap, fraction with larger numbers is larger (3/4 > 2/3).
REVISION FLASHCARD
Which is greater: 4/7 or 5/9?
CADET MEMORY KEY
Cross-multiply: 4 × 9 = 36 and 7 × 5 = 35. Since 36 > 35, 4/7 is greater than 5/9.
Common Traps & Examination Pitfalls
Avoid the Speed-Guess Trap: Under AISSEE pressure, candidates frequently misread units (e.g., mixing metres and centimetres, or per hour and per second) and confuse complementary ($90^\circ$) with supplementary ($180^\circ$) angles. Read every stem carefully and check units before selecting your OMR option.
Key Identities & Formulas
Cross-Product Rule
$$\frac{a}{b} > \frac{c}{d} \iff a \times d > b \times c$$
Pairwise fraction inequality criterion.
Equal Gap Law
$$\frac{n}{n + k} < \frac{n + 1}{(n + 1) + k}$$
For proper fractions with equal (d - n) difference.
LCM Equivalence
$$\frac{a_i}{b_i} = \frac{a_i \times (\text{LCM} / b_i)}{\text{LCM}}$$
Standardization to common denominator.
Step-by-Step Solved Examination Models
Solved Example 1
Model Problem
Arrange the fractions $\frac{2}{3}, \frac{3}{5}, \frac{7}{10}$ in ascending order.
Detailed Solution:
LCM of denominators (3, 5, 10) is 30.
$\frac{2}{3} = \frac{20}{30} = 0.666...$
$\frac{3}{5} = \frac{18}{30} = 0.60$
$\frac{7}{10} = \frac{21}{30} = 0.70$
Ascending order: $\frac{18}{30} < \frac{20}{30} < \frac{21}{30} \implies \frac{3}{5} < \frac{2}{3} < \frac{7}{10}$.
Solved Example 2
Model Problem
Which is greater: $\frac{5}{8}$ or $\frac{7}{12}$?
Detailed Solution:
Cross multiplication method:
$5 \times 12 = 60$
$7 \times 8 = 56$
Since $60 > 56$, $\frac{5}{8} > \frac{7}{12}$.
Solved Example 3
Model Problem
Arrange the following fractions in ASCENDING order: $\frac{3}{4}, \frac{5}{6}, \frac{7}{12}, \frac{2}{3}$.
Detailed Solution:
1. Denominators are 4, 6, 12, 3. Find their $\text{LCM}$: $\text{LCM}(4, 6, 12, 3) = 12$.
2. Convert each fraction to have denominator 12:
• $\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}$
• $\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}$
• $\frac{7}{12} = \frac{7}{12}$
• $\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}$
3. Compare numerators: $7 < 8 < 9 < 10$.
4. In ascending order: $\frac{7}{12} < \frac{2}{3} < \frac{3}{4} < \frac{5}{6}$. Correct Option is (A).
2. Convert each fraction to have denominator 12:
• $\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}$
• $\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}$
• $\frac{7}{12} = \frac{7}{12}$
• $\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}$
3. Compare numerators: $7 < 8 < 9 < 10$.
4. In ascending order: $\frac{7}{12} < \frac{2}{3} < \frac{3}{4} < \frac{5}{6}$. Correct Option is (A).
AISSEE High-Yield Practice Drills (10 Questions)
Simulated exam drill matching NTA question style, difficulty, and phrasing. Click an option for instant feedback and pedagogical explanation.
0
Score (Points)
0%
Accuracy
0
Cadet Streak 🔥
Question 1 of 10
Exam Standard
Which of the following is the largest fraction?
Correct Answer: Option (D)
Notice each fraction has difference $(d - n) = 1$. In proper fractions with equal differences, the fraction with the largest numbers is greatest: $11/12 = 0.916 > 7/8 = 0.875 > 5/6 = 0.833 > 3/4 = 0.75$.
Question 2 of 10
Exam Standard
Arrange in descending order: $\frac{4}{5}, \frac{7}{10}, \frac{3}{4}$.
Correct Answer: Option (A)
Convert to decimals: $4/5 = 0.80$, $3/4 = 0.75$, $7/10 = 0.70$. Descending: $4/5 > 3/4 > 7/10$.
Question 3 of 10
Exam Standard
Which of the following is the smallest fraction?
Correct Answer: Option (A)
$2/7 \approx 0.285$, $3/8 = 0.375$, $4/9 \approx 0.444$, $5/11 \approx 0.454$. The smallest is $2/7$.
Question 4 of 10
Exam Standard
Compare: $\frac{4}{9}$ and $\frac{5}{11}$. Which symbol correctly fits $\frac{4}{9} \_\_\_ \frac{5}{11}$?
Correct Answer: Option (A)
$4 \times 11 = 44$ and $9 \times 5 = 45$. Since $44 < 45$, $\frac{4}{9} < \frac{5}{11}$.
Question 5 of 10
Exam Standard
If denominators are identical, which fraction is larger?
Correct Answer: Option (A)
For like fractions, the value increases directly with the numerator (e.g. $7/10 > 3/10$).
Question 6 of 10
Exam Standard
If numerators are identical, which fraction is smaller?
Correct Answer: Option (A)
Dividing the same quantity into more pieces yields smaller pieces: $1/8 < 1/4$.
Question 7 of 10
Exam Standard
Arrange in ascending order: $\frac{1}{2}, \frac{2}{5}, \frac{3}{10}$.
Correct Answer: Option (A)
In decimals: $3/10 = 0.3$, $2/5 = 0.4$, $1/2 = 0.5$. Ascending: $3/10 < 2/5 < 1/2$.
Question 8 of 10
Exam Standard
Which fraction lies strictly between $\frac{1}{3}$ and $\frac{1}{2}$?
Correct Answer: Option (A)
$\frac{1}{3} = \frac{4}{12}$ and $\frac{1}{2} = \frac{6}{12}$. Exactly between them is $\frac{5}{12}$.
Question 9 of 10
Exam Standard
What is the difference between the greatest and smallest of $\frac{1}{2}, \frac{1}{3}, \frac{1}{4}$?
Correct Answer: Option (A)
Greatest = $1/2$, Smallest = $1/4$. Difference = $\frac{1}{2} - \frac{1}{4} = \frac{1}{4}$.
Question 10 of 10
Exam Standard
Arrange in descending order: $\frac{5}{6}, \frac{5}{8}, \frac{5}{12}, \frac{5}{7}$.
Correct Answer: Option (A)
When numerators are equal, smaller denominator implies larger fraction: $6 < 7 < 8 < 12 \implies \frac{5}{6} > \frac{5}{7} > \frac{5}{8} > \frac{5}{12}$.
AISSEE Timed Mini Mock Simulator
Simulate real exam pressure with a 5-minute countdown clock and instant OMR scorecard analysis.
NTA OMR Simulator
AISSEE 5-Minute Timed Mini Mock Test
5 Exam-Calibrated MCQs • 15 Marks Total • 05:00 Minutes Countdown Clock
05:00
Real AISSEE Pattern
+3 Marks per question, 0 Negative
+3 Marks per question, 0 Negative
Strict 5-Min Timer
Auto-submits when clock expires
Auto-submits when clock expires
Instant Report
Score, accuracy & full review
Score, accuracy & full review
Question 1 of 5
05:00
Cadet Mock Performance Report
Here is your official evaluation breakdown for this topic simulator.
Score
0 / 15 Marks
Accuracy
0%
Time Spent
00:00
Last-Minute Cadet Revision Matrix
Rapid recall checkpoints to review 24 hours before stepping into the examination hall.
| Concept Checkpoint | High-Yield Exam Rule & Memory Anchor |
|---|---|
| Same Denominator | Larger numerator gives the greater fraction. |
| Same Numerator | Smaller denominator gives the greater fraction. |
| Cross-Multiplication | a/b > c/d if and only if ad > bc. |
| Ascending vs Descending | Ascending = Smallest to Largest; Descending = Largest to Smallest. |
Frequently Asked Questions
How many questions from Arranging of Fractions appear in the AISSEE Class VI exam?
Typically, 2 to 3 questions directly test Arranging of Fractions in the 50-question Mathematics paper, contributing 6 to 9 marks towards the 150-mark section total.
Is there any negative marking for incorrect answers in AISSEE Class VI?
No. The National Testing Agency (NTA) does not apply negative marking in the AISSEE Class VI entrance examination. Candidates should attempt every single question.
What is the best way to revise Arranging of Fractions before the exam?
Review the high-yield formulas, practice the 10 interactive drill MCQs on this page, and memorize the revision checkpoints matrix.