Jaka EduTech
Jaka EduTech
Parent Sign In

© 2026 Jaka EduTech

Jaka EduTech

📖 Mathematics

Std 7
Books Book Guide
1 2 3 4 5 6 7 8
Chapters
1 Large Numbers Around Us 2 Arithmetic Expressions 3 A Peek Beyond the Point 4 Expressions Using Letter-Numbers 5 Parallel and Intersecting Lines 6 Number Play 7 A Tale of Three Intersecting Lines 8 Working with Fractions
3
Chapter 3
Skill: 30%

A Peek Beyond the Point

A Peek Beyond the Point

Chapter at a Glance

This chapter introduces the concept of decimals as an extension of the base-10 place-value system. By exploring the need for smaller units in real-world measurements—such as comparing screws, measuring paper folds, and tracking insect lengths—the text details the fractional units of tenths ($1/10$), hundredths ($1/100$), and thousandths ($1/1000$). The chapter details representation on the number line, place value notation (using the decimal point), conversion metrics for metric measurements and currency, and decimal arithmetic. It also highlights historical milestones in decimal notation and explains common "non-decimal" notation traps (such as overs in cricket or decimal-based time formats).

Key Definitions & Terminology

  • Decimal Point: A separator dot (‘.’) used in the place value system to separate the whole number part from the fractional parts (tenths, hundredths, thousandths).
  • Tenth: A unit divided into 10 equal parts; represented as $1/10$ or $0.1$.
  • Hundredth: A tenth divided into 10 equal parts, or a unit divided into 100 equal parts; represented as $1/100$ or $0.01$.
  • Thousandth: A hundredth divided into 10 equal parts, or a unit divided into 1000 equal parts; represented as $1/1000$ or $0.001$.
  • Trailing Zeros: Zeros added to the rightmost end of a decimal number that do not change its value (e.g., $0.2 = 0.20 = 0.200$).
  • Decimal System: A numeral system based on ten, originating from the Latin decem and cognate to the Sanskrit daśha.

Formulas, Rules & Properties

  • Positional Expansion of Decimals:
    $$N = ... + (d_1 \times 10) + (d_0 \times 1) + \left(t \times \frac{1}{10}\right) + \left(h \times \frac{1}{100}\right) + \left(m \times \frac{1}{1000}\right) + ...$$
  • Decimal Comparison Rule: Compare digits column-by-column starting from the highest place value (leftmost). The number with the larger digit at the first mismatch column is greater, regardless of the remaining digits to the right.
  • Conversion Metric Formulas:
  • Length:
    • $1\text{ mm} = 0.1\text{ cm} = 0.001\text{ m}$
    • $1\text{ cm} = 0.01\text{ m}$
    • $1\text{ m} = 100\text{ cm} = 1000\text{ mm}$
    • $1\text{ km} = 1000\text{ m} = 1,000,000\text{ mm}$
  • Weight:
    • $1\text{ g} = 0.001\text{ kg}$
    • $1\text{ mg} = 0.001\text{ g} = 0.000001\text{ kg}$
  • Currency:
    • $1\text{ Paisa} = 0.01\text{ Rupees}$
  • Fractional Reference Constants:
  • $\frac{1}{2} = 0.5$, $\frac{3}{2} = 1.5$
  • $\frac{1}{4} = 0.25$, $\frac{3}{4} = 0.75$
  • $\frac{1}{5} = 0.2$, $\frac{4}{5} = 0.8$

Core Concepts & Topics

  • The Place Value Grid Extension:
  • The columns extend infinitely to the right of the units place, dividing by 10 at each step (Tenths, Hundredths, Thousandths, Ten-thousandths, etc.).
  • Number Line Representation & Magnification:
  • Decimals are located by dividing intervals between integers into 10 segments.
  • To locate thousandths, intervals are magnified step-by-step (e.g., locating $4.185$ by magnifying between $4.1$ and $4.2$, then $4.18$ and $4.19$).
  • Decimal Arithmetic (Regrouping & Borrowing):
  • Addition: Summing same-column values; regrouping 10 hundredths to 1 tenth, or 10 tenths to 1 unit.
  • Subtraction: Borrowing 1 unit from the left column and converting it to 10 tenths, or 1 tenth to 10 hundredths (e.g., $12 \frac{4}{10} - 6 \frac{7}{10} = 11 \frac{14}{10} - 6 \frac{7}{10} = 5 \frac{7}{10}$).
  • Sonu's Range Estimation Rule:
  • The sum of two decimal numbers will be greater than the sum of their whole number parts, and less than 2 more than the sum of their whole parts.
  • Deceptive Decimal Notations:
  • Time: $4.5\text{ hours}$ post noon means 4 hours and 30 minutes ($0.5 \times 60$), not 4 hours 50 minutes.
  • Cricket: $5.5\text{ overs}$ means 5 overs and 5 balls, not 5 and a half overs (since 1 over $= 6$ balls).
  • Dimensions: $2.5\text{ ft}$ means 2 ft and 6 inches ($0.5 \times 12$), not 2 ft 5 inches.

Worked Examples

  • Sonu's Arm Length (Regrouping Addition):
  • Problem: Sonu's lower arm is $2 \frac{7}{10}$ units, and his upper arm is $3 \frac{6}{10}$ units. What is the total length?
  • Solution:
    $$2 \frac{7}{10} + 3 \frac{6}{10} = (2 + 3) + \frac{7+6}{10} = 5 + \frac{13}{10} = 5 + 1 \frac{3}{10} = 6 \frac{3}{10}\text{ units}$$
  • Honeybee Length:
  • Problem: Total length of a honeybee with head $2 \frac{3}{10}$ units, thorax $5 \frac{4}{10}$ units, and abdomen $7 \frac{5}{10}$ units.
  • Solution:
    $$2 \frac{3}{10} + 5 \frac{4}{10} + 7 \frac{5}{10} = 14 + \frac{12}{10} = 15 \frac{2}{10}\text{ units}$$
  • Finger Length (Borrowing Subtraction):
  • Problem: Find middle finger length if total hand is $12 \frac{4}{10}$ units and palm is $6 \frac{7}{10}$ units.
  • Solution:
    $$12 \frac{4}{10} - 6 \frac{7}{10} = 11 \frac{14}{10} - 6 \frac{7}{10} = 5 \frac{7}{10}\text{ units}$$
  • Tenths and Hundredths Conversions:
  • $234\text{ tenths} = 234 / 10 = 23.4$.
  • $234\text{ hundredths} = 234 / 100 = 2.34$.
  • $105\text{ tenths} = 10.5$.
  • Arranging Fractional Expressions:
  • Problem: Arrange in increasing order: $4/10$, $41/10$, $4 \frac{1}{10}$, $41 \frac{1}{10}$.
  • Solution: Converts to: $0.4$, $4.1$, $4.1$, $41.1$. Order: $4/10 < 41/10 = 4 \frac{1}{10} < 41 \frac{1}{10}$.
  • Closest Decimal Riddles:
  • Problem 1: Using digits 1, 4, 0, 8, 6 exactly once, make the decimal closest to 30.
  • Solution: By checking bounds: $40.168$ is $10.168$ away; $18.640$ is $11.36$ away. The closest is $40.168$.
  • Problem 2: Using same digits, make the smallest decimal between 100 and 1000.
  • Solution: $104.68$.
  • Vegetables Purchase Weight:
  • Problem: Calculate total weight of 0.25 kg beans, 0.3 kg carrots, 0.5 kg potatoes, 0.2 kg capsicum, and 0.05 kg ginger.
  • Solution:
    $$0.25 + 0.30 + 0.50 + 0.20 + 0.05 = 1.30\text{ kg}$$
  • Pinto's Dairy Supply:
  • Problem: Pinto supplies 3.79 L, 4.2 L, and 4.25 L of milk on the first 3 days. Total supply over 6 days is 25 L. Find the last 3 days' supply.
  • Solution: Last 3 days $= 25 - (3.79 + 4.20 + 4.25) = 25 - 12.24 = 12.76\text{ L}$.
  • Tinku's Weight Change:
  • Problem: January weight is 35.75 kg, February is 34.50 kg. Find weight change.
  • Solution: $35.75 - 34.50 = 1.25\text{ kg}$ lost.
  • Indian Railways Travel Insurance:
  • Problem: Optional travel insurance costs 45 paise per passenger. If 1 lakh people opt, find the total fee in rupees.
  • Solution:
    $$\text{Fee} = 1,00,000 \times 45\text{ p} = 4,500,000\text{ p} = \text{₹}45,000$$
  • Comparing Fractional Units:
  • $\frac{10}{1000}$ vs $\frac{1}{10} \implies 0.01 < 0.1$ (so $\frac{1}{10}$ is larger).
  • One-hundredth ($0.01$) vs 90 thousandths ($0.09$) $\implies 90$ thousandths is larger.
  • One-thousandth ($0.001$) vs 90 hundredths ($0.9$) $\implies 90$ hundredths is larger.
  • Weird Decimal Addition Conversions:
  • 12 tens and 12 tenths: $12 \times 10 + 12 \times 0.1 = 120 + 1.2 = 121.2$.
  • 10 tens, 10 ones, 10 tenths, 10 hundredths: $100 + 10 + 1.0 + 0.10 = 111.1$.
  • 25 tens, 25 ones, 25 tenths, 25 hundredths: $250 + 25 + 2.5 + 0.25 = 277.75$.

Practical Activities & Experiments

  • Measuring Screws: Positioning two screws on a ruler to measure the millimeter differences, reading them as fractional decimals ($2 \frac{7}{10}$ cm).
  • Folding Paper Divisions: Folding a sheet of paper measuring $8 \frac{9}{10}$ units in half to observe it ends at $4$ units, $4$ tenths, $5$ hundredths ($4.45$ units).
  • Decimal Sequence Challenge: Mental games to identify pattern increments/decrements (e.g. extending $5.5, 6.4, 6.39, 7.29, 7.28, 8.18, 8.17...$ by alternating adding 0.9 and subtracting 0.01 to get $9.07, 9.06$).
  • Coin Price Auditing: Discovering old coins (paise denominations) and interviewing adults about 1970s prices (dosa for 50 paise, banana for 20 paise, rice for ₹2.45/kg).
Previous Chapter 3 of 8 Next