Decimal Numbers
Pointwise AISSEE Class 6 Mathematics guide for Decimal Numbers (Topic 21). Master place values tenths/hundredths/thousandths, fraction-decimal conversion, column alignment, and multiplication/division decimal shift rules.
Overview & Fundamental Concept
Pointwise AISSEE Class 6 Mathematics guide for Decimal Numbers (Topic 21). Master place values tenths/hundredths/thousandths, fraction-decimal conversion, column alignment, and multiplication/division decimal shift rules.
Official Syllabus Coverage Breakdown
| Syllabus Item | Prescribed Scope | Exam Relevance & Pedagogical Objective |
|---|---|---|
| Item 21 | Decimal fractions, tenths, hundredths, thousandths, comparing decimals, conversion between fractions and decimals, addition, subtraction, multiplication, and division. | Master decimal positional notation, tenths/hundredths/thousandths values, like decimal conversions, column alignment for addition/subtraction, and shift rules for multiplication/division for AISSEE Class VI. |
1. Decimal Place Value Architecture & Taxonomy
- Anatomy of a Decimal Number: A decimal number consists of two distinct components separated by a dot called the decimal point: The Whole Number part (to the left of the decimal) and the Decimal part (to the right of the decimal).
- Decimal Place Values to the Right of Decimal:
- • Tenths Place: First place after decimal, representing $\frac{1}{10} = 0.1$.
- • Hundredths Place: Second place after decimal, representing $\frac{1}{100} = 0.01$.
- • Thousandths Place: Third place after decimal, representing $\frac{1}{1,000} = 0.001$.
- • Ten-thousandths Place: Fourth place, representing $\frac{1}{10,000} = 0.0001$.
- Expanded Form of Decimals: Expressing a decimal as the sum of the place values of all its digits: $$53.478 = 50 + 3 + \frac{4}{10} + \frac{7}{100} + \frac{8}{1,000}$$
- Trailing Zeros Invariance Property: Adding any number of zeros to the extreme right of the decimal part does not alter its numerical value: $0.4 = 0.40 = 0.400$.
| Tens ($10$) | Units ($1$) | Decimal Point | Tenths ($\frac{1}{10}$) | Hundredths ($\frac{1}{100}$) | Thousandths ($\frac{1}{1000}$) |
|---|---|---|---|---|---|
| 3 | 8 | . | 5 | 0 | 0 |
| 0 | 7 | . | 0 | 4 | 0 |
| 4 | 2 | . | 6 | 8 | 9 |
Cadet Mnemonic • Place Value Direction Anchor
Remember place values away from decimal:
- Left of point grows by ×10: Units, Tens, Hundreds...
- Right of point shrinks by ÷10: Tenths, Hundredths, Thousandths...
2. Like vs. Unlike Decimals & Comparing Rules
- Like Decimals: Decimal numbers having the exact same number of decimal places (e.g., $3.45$ and $12.89$ both have 2 decimal places).
- Unlike Decimals: Decimal numbers having different counts of decimal places (e.g., $4.2$ has 1 place, $6.385$ has 3 places).
- Converting Unlike to Like Decimals: Append trailing zeros to the decimal part until all numbers have the same number of decimal digits (e.g., $4.2 \to 4.200$ and $6.385$).
- Comparison Protocol:
• Step 1: Compare the whole number parts. The number with the larger whole number part is greater ($15.2 > 9.875$).
• Step 2: If whole number parts are equal, compare the digits at the tenths place ($4.71 > 4.69$).
• Step 3: If tenths are equal, compare hundredths, then thousandths.
Exam Trick • Equalize Places Before Comparing
Which is greater: 0.6 or 0.58? Equalize decimal places: 0.60 vs 0.58. Since 60 > 58, 0.6 is greater! Never judge size by the number of digits.
3. Arithmetic Operations on Decimals
- Addition & Subtraction Rule: Always write the numbers in vertical columns such that the decimal points are strictly aligned in a vertical line. Fill missing decimal places with zeros, then add or subtract normally.
- Multiplication of Decimals: 1. Multiply the numbers as if they were whole numbers (ignoring decimal points). 2. Count the total number of decimal places in both factors combined. 3. Place the decimal point in the product counting from right to left by that total count. Example: $1.2 \times 0.04 = 0.048$ (1 place + 2 places = 3 decimal places).
- Division by 10, 100, 1000: Shift the decimal point to the LEFT by as many places as there are zeros: $45.6 \div 10 = 4.56$; $45.6 \div 100 = 0.456$.
- Division by a Decimal Divisor: Multiply both dividend and divisor by $10, 100, \dots$ to make the divisor a whole number, then divide normally: $$\frac{7.2}{0.08} = \frac{7.2 \times 100}{0.08 \times 100} = \frac{720}{8} = 90$$
AISSEE Trap Alert • The Subtraction Column Trap
When subtracting 0.75 from 4: Write 4 as 4.00! Then compute 4.00 − 0.75 = 3.25. Writing 4 − 0.75 = 4.75 is a catastrophic error.
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: Fraction to Decimal Denominator Powers
Convert denominators to powers of 10 (10, 100, 1000) for instant conversion: $\frac{3}{4} = \frac{75}{100} = 0.75$; $\frac{7}{25} = \frac{28}{100} = 0.28$; $\frac{3}{8} = \frac{375}{1000} = 0.375$.
Shortcut Technique 2: Decimal Multiplication Shift Trick
Multiplying by 10, 100, 1000 shifts decimal point to the **RIGHT** by the count of zeros: $0.054 \times 100 = 5.4$.
Shortcut Technique 3: Divisor Purification
Never divide by a decimal! Always shift the decimal point in the divisor all the way to the right to make it an integer, shifting the dividend by the same count.
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 Decimal Numbers?
CADET MEMORY KEY
Multiply by 10^k: x \times 10^k \implies \text{Shift point } k \text{ places RIGHT}
REVISION FLASHCARD
Cadet Shortcut: How to solve Decimal Numbers 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
Trailing Zeros: Adding trailing zeros after the decimal point does NOT change value (3.5 = 3.50 = 3.500).
REVISION FLASHCARD
Rapid Memory Anchor for AISSEE Hall Recall:
CADET MEMORY KEY
Point Alignment: Always line up decimal points vertically when adding or subtracting.
REVISION FLASHCARD
What is 0.024 ÷ 0.006?
CADET MEMORY KEY
Multiply both by 1,000 to make divisor a whole number: 24 ÷ 6 = 4.
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
Decimal Multiplication Rule
$$\text{Places in Product} = \text{Places in } A + \text{Places in } B$$
Sum of decimal digits in factors.
Division Decimal Purification
$$\frac{A}{B} = \frac{A \times 10^k}{B \times 10^k}$$
Making divisor a whole integer.
Decimal Expanded Identity
$$0.abc = \frac{a}{10} + \frac{b}{100} + \frac{c}{1000}$$
Positional base-10 fractional expansion.
Step-by-Step Solved Examination Models
Solved Example 1
Model Problem
Evaluate: $45.6 + 7.85 - 12.045$.
Detailed Solution:
Line up decimal places:
$45.600 + 7.850 = 53.450$.
$53.450 - 12.045 = 41.405$.
Solved Example 2
Model Problem
Evaluate: $1.25 \times 0.08$.
Detailed Solution:
Multiply without decimals: $125 \times 8 = 1000$.
Total decimal places = $2 + 2 = 4$.
Place decimal: $0.1000 = 0.1$.
Solved Example 3
Model Problem
Evaluate: $14.25 \times 0.04 \div 0.003$.
Detailed Solution:
1. Follow BODMAS/order of operations. First multiply: $14.25 \times 0.04$.
2. $1425 \times 4 = 5700$. Total decimal places: 2 (in 14.25) + 2 (in 0.04) = 4 places. Product $= 0.5700 = 0.57$.
3. Now divide by $0.003$: $\frac{0.57}{0.003}$.
4. Multiply numerator and denominator by 1,000 to eliminate the decimal in the divisor: $\frac{0.57 \times 1,000}{0.003 \times 1,000} = \frac{570}{3} = 190$. Correct Option is (A).
2. $1425 \times 4 = 5700$. Total decimal places: 2 (in 14.25) + 2 (in 0.04) = 4 places. Product $= 0.5700 = 0.57$.
3. Now divide by $0.003$: $\frac{0.57}{0.003}$.
4. Multiply numerator and denominator by 1,000 to eliminate the decimal in the divisor: $\frac{0.57 \times 1,000}{0.003 \times 1,000} = \frac{570}{3} = 190$. Correct Option is (A).
AISSEE High-Yield Practice Drills (10 Questions)
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Question 1 of 10
Exam Standard
Convert $\frac{3}{8}$ into a decimal number.
Correct Answer: Option (A)
$\frac{3}{8} = 3 \div 8 = 0.375$.
Question 2 of 10
Exam Standard
What is the place value of digit 6 in the decimal 23.468?
Correct Answer: Option (A)
The second digit after the decimal point represents hundredths: $\frac{6}{100} = 0.06$.
Question 3 of 10
Exam Standard
Evaluate: $14.4 \div 1.2$.
Correct Answer: Option (A)
Multiply numerator and denominator by 10: $144 / 12 = 12$.
Question 4 of 10
Exam Standard
Express 45 paise in rupees as a decimal.
Correct Answer: Option (A)
1 rupee = 100 paise. 45 paise = $45 / 100 = \text{Rs } 0.45$.
Question 5 of 10
Exam Standard
Which of the following decimals is the greatest?
Correct Answer: Option (A)
Compare tenths digits: 0.700 has 7 in tenths place, which is greater than 6 or 0.
Question 6 of 10
Exam Standard
Evaluate: $0.4 \times 0.4 \times 0.4$.
Correct Answer: Option (A)
$4 \times 4 \times 4 = 64$. Total 3 decimal digits $\implies 0.064$.
Question 7 of 10
Exam Standard
Subtract 3.85 from 10.
Correct Answer: Option (A)
$10.00 - 3.85 = 6.15$.
Question 8 of 10
Exam Standard
Divide $2.75$ by 100.
Correct Answer: Option (A)
Dividing by 100 shifts the decimal point 2 places to the left: $0.0275$.
Question 9 of 10
Exam Standard
Convert 2.5 into an improper fraction in simplest form.
Correct Answer: Option (A)
$2.5 = 25/10 = 5/2$.
Question 10 of 10
Exam Standard
Evaluate: $2.5 \times 4 + 3.6 \div 3$.
Correct Answer: Option (A)
$2.5 \times 4 = 10$. $3.6 \div 3 = 1.2$. Sum = $10 + 1.2 = 11.2$.
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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 |
|---|---|
| Tenths & Hundredths | Tenths is 1/10 (0.1); Hundredths is 1/100 (0.01); Thousandths is 1/1000 (0.001). |
| Addition Alignment | Decimal points must be aligned vertically in a straight column. |
| Product Places | Total decimal places in product = sum of decimal places of factors. |
| Trailing Zeros | Trailing zeros at the end of a decimal part do not change its value (0.5 = 0.50). |
Frequently Asked Questions
How many questions from Decimal Numbers appear in the AISSEE Class VI exam?
Typically, 2 to 3 questions directly test Decimal Numbers 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 Decimal Numbers before the exam?
Review the high-yield formulas, practice the 10 interactive drill MCQs on this page, and memorize the revision checkpoints matrix.