Official Syllabus Item: 21, 38
Arithmetic & Number Theory
Decimal Numbers & Everyday Applications
Master decimal place values, converting decimals to fractions, comparing, addition, subtraction, multiplication, division, and measurement conversions for RIMC.
Overview & Fundamental Concept
Decimals extend the positional place-value system to fractional parts using powers of 10. In RIMC Stage-I descriptive mathematics, candidates are tested on place-value expanded forms, converting unlike decimals to like decimals, vertical column addition and subtraction, decimal point shift rules for products and quotients, and real-world multi-step word problems involving money, metric lengths, masses, and capacities.
Key Theoretical Principles
- Appending trailing zeroes to the decimal part does not alter numerical value (0.5 = 0.50 = 0.500).
- To compare decimals, compare whole number parts first; if equal, compare tenths, hundredths, etc.
- In column addition and subtraction, align decimal points vertically and pad with trailing zeroes.
- In division by a decimal, multiply numerator and denominator by 10^k to make the divisor a whole number.
- A simplified fraction p/q terminates into a finite decimal if and only if q = 2^m × 5^n.
Illustrated Conceptual Diagrams & Visual Schematics
Decimal Place Value Chart & Scale Shifting Dynamics
Decimal Mechanics: Moving left from the decimal point multiplies place values by $10$ (Units, Tens, Hundreds). Moving right divides place values by $10$ (Tenths, Hundredths, Thousandths). Multiplying by $10^n$ shifts the decimal point $n$ places to the RIGHT; dividing shifts $n$ places to the LEFT.
Topper Strategy & Descriptive Step Breakdown
Mastering RIMC Mathematics Standard
Always set up vertical column layouts with aligned decimal points. When converting mixed units (e.g. ₹ and paise, kg and grams), convert all items to a uniform decimal unit before executing arithmetic.
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
Expanded Decimal Notation
$$a.bcd = a + \frac{b}{10} + \frac{c}{100} + \frac{d}{1000}$$
Multiplication Decimal Places
$$\text{Places in Product} = d_1 + d_2$$
Division Integer Transformation
$$\frac{x}{y} = \frac{x \times 10^k}{y \times 10^k} \quad (y \times 10^k \in \mathbb{N})$$
Money Conversion
$$1\text{ paisa} = ₹ 0.01 \implies p\text{ paise} = ₹ \frac{p}{100}$$
Mass & Length Conversions
$$1\text{ g} = 0.001\text{ kg}; \quad 1\text{ cm} = 0.01\text{ m}; \quad 1\text{ mm} = 0.001\text{ m}$$
Solved Model Examples (Descriptive Format with Step Mark Distribution)
Solved Model 1
Model Solution
Arrange in ascending and descending order: 3.05, 3.50, 3.005, 3.55, 3.505.
Step-by-Step Model Solution:
Step 1: Like decimals (3 decimal places): 3.050, 3.500, 3.005, 3.550, 3.505.\nStep 2: Comparing decimal parts: 3.005 < 3.050 < 3.500 < 3.505 < 3.550.\nAscending: 3.005 < 3.05 < 3.50 < 3.505 < 3.55.\nDescending: 3.55 > 3.505 > 3.50 > 3.05 > 3.005.
Solved Model 2
Model Solution
Evaluate: (0.144 × 0.081) / (0.009 × 0.012 × 0.6).
Step-by-Step Model Solution:
Numerator decimal places = 3 + 3 = 6. Denominator places = 3 + 3 + 1 = 7.\nMultiply numerator and denominator by 10^7: (144 × 81 × 10) / (9 × 12 × 6).\nCancel factors: 81/9 = 9; 144/12 = 12; 12/6 = 2.\nResult = 2 × 9 × 10 = 180.
Solved Model 3
Model Solution
A merchant bought 45.5 kg rice at ₹ 84.50/kg and 35.25 kg wheat at ₹ 32.40/kg. Handed ₹ 5000 note. Find balance.
Step-by-Step Model Solution:
Rice: 45.5 × 84.50 = ₹ 3844.75. Wheat: 35.25 × 32.40 = ₹ 1142.10.\nTotal bill = 3844.75 + 1142.10 = ₹ 4986.85.\nBalance returned = 5000.00 - 4986.85 = ₹ 13.15.
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
Which is greater: 0.8 or 0.789? Show like decimal conversion.
Model Descriptive Solution:
0.8 = 0.800. 0.800 > 0.789 because tenths digit 8 > 7. 0.8 is greater.
Question 2
5 Marks
Convert 7/8 and 13/25 into decimals without long division.
Model Descriptive Solution:
7/8 = (7×125)/(8×125) = 875/1000 = 0.875. 13/25 = (13×4)/(25×4) = 52/100 = 0.52.
Question 3
5 Marks
Evaluate: 18.5 × 0.04 × 0.002. State decimal place count.
Model Descriptive Solution:
185 × 4 × 2 = 1480. Decimal places = 1 + 2 + 3 = 6. Result = 0.001480 = 0.00148.
Question 4
5 Marks
Evaluate: 0.0102 ÷ 0.017.
Model Descriptive Solution:
(0.0102 × 1000) / (0.017 × 1000) = 10.2 / 17 = 0.6.
Question 5
5 Marks
Express in metres using decimals: (a) 15 m 45 cm, (b) 9 cm, (c) 845 mm.
Model Descriptive Solution:
(a) 15.45 m; (b) 0.09 m; (c) 0.845 m.
Question 6
10 Marks
Simplify using a² - b² identity: (2.3 × 2.3 - 0.7 × 0.7) / (2.3 - 0.7).
Model Descriptive Solution:
(a - b)(a + b) / (a - b) = a + b = 2.3 + 0.7 = 3.
Question 7
10 Marks
Milkman had 75.5 L. Sold 24.75 L and 31.8 L. Remaining distributed equally to 5 families. Find share.
Model Descriptive Solution:
Sold = 24.75 + 31.80 = 56.55 L. Remaining = 75.50 - 56.55 = 18.95 L. Share = 18.95 ÷ 5 = 3.79 litres.
Question 8
10 Marks
Carton with 15 books weighs 18.45 kg. Empty carton weighs 1.2 kg. Find weight of each book in kg and g.
Model Descriptive Solution:
Net weight of 15 books = 18.45 - 1.20 = 17.25 kg. Each book = 17.25 ÷ 15 = 1.15 kg = 1150 grams.
Question 9
10 Marks
Evaluate following BODMAS: (4.8 × 1.5 - 3.2 ÷ 0.8) + (0.05 × 40).
Model Descriptive Solution:
(7.2 - 4) + 2 = 3.2 + 2 = 5.2.
Question 10
10 Marks
Tailor needs 2.35 m per shirt. Roll has 45 m. How many shirts can be stitched and what length remains?
Model Descriptive Solution:
45 ÷ 2.35 = 4500 ÷ 235 = 19 complete shirts. Cloth used = 19 × 2.35 = 44.65 m. Remaining cloth = 0.35 m (35 cm).
High-Yield FAQs & Revision Notes
Does appending trailing zeroes to a decimal alter its numerical value?
No. 0.5 = 0.50 = 0.500. Trailing zeroes do not change value.
What is the procedure for comparing two decimal numbers?
Compare whole parts first; if equal, convert to like decimals and compare digit by digit from tenths place.
How is a decimal divided by another decimal number?
Multiply both dividend and divisor by 10^k to make divisor an integer, then divide.
How do you determine the position of the decimal point when multiplying two decimals?
Count total decimal places in factors, multiply as integers, and count that total from right.
What fraction will terminate into a finite decimal?
A fraction in lowest terms p/q terminates iff q = 2^m × 5^n.
How are units of money converted using decimals in RIMC descriptive problems?
1 paisa = ₹ 0.01; divide paise by 100 to get decimal rupees.
How are metric lengths converted into decimals of a metre?
1 cm = 0.01 m, 1 mm = 0.001 m. E.g. 15 m 45 cm = 15.45 m.
How is weight converted into decimals of a kilogram?
1 g = 0.001 kg. E.g. 4 kg 250 g = 4.25 kg.
Why must decimal points be aligned vertically in addition and subtraction?
To ensure digits of identical place values are added or subtracted together.
How does this module map to the official RIMC syllabus?
Directly covers Serial No. 21 (Decimals) and Serial No. 38 (Operations & applications) of Chapter 8.
How does this topic relate to the official RIMC syllabus?
This module directly covers Serial No. 21, 38 of the broader syllabus published in Chapter 8 of the official RIMC admission brochure, ensuring 100% syllabus alignment.