Class 7 Maths Ch 3: A Peek Beyond the Point – understand decimals (tenths, hundredths), compare and operate with them, see real‑life uses and history, with notes, solved sums, extra questions and quiz for CBSE Exam
Complete Chapter 3 guide: why smaller units like tenths and hundredths are needed (measuring screws, pencils, body parts), reading and writing decimals as fractions and on number lines, comparing and ordering decimal numbers, adding and subtracting decimals through place value, decimal sequences and estimation, common decimal mistakes in real life (time, units, medicine, money) and a short history of decimal notation, plus solved examples, practice questions and puzzles for CBSE Class 7 Maths
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A Peek Beyond The Point
Class 7 Mathematics Chapter 3 | Complete Guide | Decimals, Tenths, Hundredths, Place Value 2025
Chapter at a Glance – A Peek Beyond The Point
This chapter introduces smaller units for precise measurements, tenths and hundredths, decimal notation, place value extension, unit conversions (length, weight, money), comparing decimals, and addition/subtraction of decimals.
Main Topics Covered
- Need for smaller units: Measuring screws accurately.
- Tenths: \( \frac{1}{10} \), reading and writing mixed numbers like \( 2 \frac{7}{10} \).
- Hundredths: \( \frac{1}{100} \), more precise measurements.
- Decimal place value: Extending Indian system with decimal point.
- Unit conversions: cm to mm, m; g to kg; paise to rupees.
- Locating/comparing decimals on number line.
- Addition/subtraction of decimals using place value.
- History and disasters related to decimals.
Key Takeaways for Exams
Tenths
1 unit = 10 tenths; \( 2 \frac{7}{10} = 2.7 \).
Hundredths
1 tenth = 10 hundredths; \( 4 \frac{4}{10} \frac{5}{100} = 4.45 \).
Decimal Notation
Separator for whole and fractional parts.
Conversions
1 cm = 10 mm = 0.01 m; 1 g = 0.001 kg.
Key Concepts & Rules – A Peek Beyond The Point
Important definitions, rules for tenths/hundredths, decimals, and conversions for quick reference.
Key Rules
- Tenths: Divide unit into 10 equal parts; \( 10 \times \frac{1}{10} = 1 \).
- Hundredths: Divide tenth into 10; \( 100 \times \frac{1}{100} = 1 \).
- Decimal Point: Separates whole from fractional; e.g., 70.5 = 70 + 0.5.
- Reading: 7.05 as "seven point zero five".
- Conversions: 1 mm = 0.1 cm; 1 cm = 0.01 m; 1 g = 0.001 kg; 1 paisa = 0.01 rupee.
- Comparing: Align decimal points, compare digits left to right.
- Add/Subtract: Align decimals, add/subtract as whole numbers.
Conversion Examples
| From | To | Example |
|---|---|---|
| mm to cm | Divide by 10 | 12 mm = 1.2 cm |
| cm to m | Divide by 100 | 15 cm = 0.15 m |
| g to kg | Divide by 1000 | 254 g = 0.254 kg |
| paise to rupee | Divide by 100 | 75 paise = 0.75 rupee |
Golden Lines for Exams
“Split units into 10 for precision; Align decimals for operations.”
Concept Cards – Quick Explanations
Need for Smaller Units
Precise measurements require dividing units; e.g., screws differ by tenths cm.
Exam Tip: Explain why divide into 10 parts.
Tenth Part
\( 3 \frac{4}{10} = 3 + 4 \times \frac{1}{10} = \frac{34}{10} \).
Hundredth Part
\( 4 \frac{4}{10} \frac{5}{100} = 4 + \frac{45}{100} = 4.45 \).
Decimal Place Value
Extends to right: Tenths, hundredths; 70.5 = 7 tens + 5 tenths.
Unit Conversions
Length: 1 m = 100 cm = 1000 mm; Weight: 1 kg = 1000 g.
Comparing Decimals
Use number line or place value; 6.456 < 6.465.
Add/Subtract Decimals
2.7 + 3.5 = 6.2; Align decimals.
Examples + Solutions
Example 1: Screw Lengths
Lengths: \( 2 \frac{7}{10} \) cm and \( 3 \frac{2}{10} \) cm.
Solution: Read as two and seven-tenths, three and two-tenths.
Example 2: Arm Length
Lower: \( 2 \frac{7}{10} \), Upper: \( 3 \frac{6}{10} \).
Solution: \( 5 \frac{13}{10} = 6 \frac{3}{10} \).
Example 3: Honeybee Length
Head: \( 2 \frac{3}{10} \), Thorax: \( 5 \frac{4}{10} \), Abdomen: \( 7 \frac{5}{10} \).
Solution: \( 15 \frac{12}{10} = 16 \frac{2}{10} \).
Example 4: Shylaja's Finger
Hand: \( 12 \frac{4}{10} \), Palm: \( 6 \frac{7}{10} \).
Solution: \( 5 \frac{7}{10} \).
Example 5: Fish Difference
Danio: \( 2 \frac{4}{10} \), Goby: \( \frac{9}{10} \).
Solution: \( 1 \frac{5}{10} \) cm.
Example 6: Folded Paper
Original: \( 8 \frac{9}{10} \), Folded: \( 4 \frac{4}{10} \frac{5}{100} \).
Example 7: Wire Length
\( 1 \frac{1}{10} \frac{4}{100} = 1 \frac{14}{100} = \frac{114}{100} \).
Example 8: Sum of Decimals
\( 15 \frac{3}{10} \frac{4}{100} + 2 \frac{6}{10} \frac{8}{100} = 18 \frac{2}{100} \).
Example 9: Difference
\( 25 \frac{9}{10} - 6 \frac{4}{10} \frac{7}{100} = 19 \frac{4}{10} \frac{3}{100} \).
Example 10: 23 Tens
\( 23 \times 10 = 230 \).
Example 11: mm to cm
5 mm = 0.5 cm; 12 mm = 1.2 cm.
Example 12: cm to m
10 cm = 0.1 m; 15 cm = 0.15 m.
Example 13: g to kg
5 g = 0.005 kg; 254 g = 0.254 kg.
Example 14: Closest to 1
0.9, 1.01, 1.1, 1.11 → 1.01 is closest.
Example 15: Addition
2.7 + 3.5 = 6.2 m.
Figure it Out Solutions (All Solved)
Page 2: Screws & Objects
1. Which scale for accuracy?
The one divided into tenths; Allows precise measurement.
2. Meaning of \( 2 \frac{7}{10} \) cm
2 cm + 7 tenths cm.
3. Why divide unit?
For exact measures when objects differ slightly.
4. Measure pen, sharpener, etc.
Varies; e.g., Pen: \( 12 \frac{5}{10} \) cm.
5. Objects in picture
Pencil: \( 3 \frac{4}{10} \); Eraser: \( 2 \frac{5}{10} \); etc. (estimate from fig).
Page 4: Lengths & Order
1. Write lengths two ways
USB: \( 4 \frac{8}{10} = \frac{48}{10} \); Finger: \( 1 \frac{2}{10} = \frac{12}{10} \); Leaf: \( 9 \frac{5}{10} = \frac{95}{10} \).
2. Increasing order
(h) \( \frac{4}{10} \), (a) \( \frac{9}{10} \), (b) \( 1 \frac{7}{10} \), (g) \( 6 \frac{7}{10} \), (f) \( 7 \frac{6}{10} \), (e) \( 10 \frac{5}{10} \), (d) \( 13 \frac{1}{10} \), (c) \( \frac{130}{10} \).
Page 5: Arm Length
1. Total arm
\( 6 \frac{3}{10} \) units.
Page 6: Honeybee & Finger
1. Honeybee total
\( 15 \frac{12}{10} = 16 \frac{2}{10} \).
2. Finger length
\( 5 \frac{7}{10} \).
3. Fish difference
\( 1 \frac{5}{10} \) cm; Compare to finger (varies).
Page 7: Sequences
1. Extend patterns
(a) +0.3 each: 4.9, 5.2, 5.5, 5.8
(b) +0.5 each: 9.7, 10.2, 10.7, 11.2
(c) +1.1: 9.8, 10.9, 12, 13.1
(d) -0.4: 4.9, 4.5, 4.1, 3.7
(e) -0.5: 12, 11.5, 11, 10.5
(f) -1.1: 8.2, 7.1, 6, 4.9
Page 8: Folded Paper
1. Length now?
\( 4 \frac{4}{10} \frac{5}{100} = 4 \frac{45}{100} \).
2. Hundredths in one-tenth?
10; Yes, 4 units + 45 hundredths.
Page 9: Markings
1. Fill lengths
0.1, 0.2, ..., 1.0; 0.11, 0.12, etc.
2. Wire ways
Three ways equal.
3. Write measurements
4.95, 5.15, ..., 15.65 (estimate from scales).
Page 10: Longest/Shortest
1. Identify in groups
(a) Short: \( \frac{3}{100} \), Long: \( \frac{33}{100} \)
(b) Short: \( \frac{3}{10} \), Long: \( \frac{30}{10} \)
(c) Short: \( \frac{4}{10} \), Long: \( \frac{54}{100} \)
(d) Short: \( 3 \frac{6}{10} \), Long: \( 6 \frac{3}{10} \)
(e) Short: \( \frac{2}{100} \), Long: \( 1 \frac{8}{100} \)
(f) Short: \( \frac{5}{100} \), Long: \( 7 \frac{5}{10} \)
(g) Short: \( \frac{15}{100} \), Long: \( 6 \frac{5}{10} \)
Page 11: Sum
1. Sum of 15.34 + 2.68
18.02
Page 12: Methods Different?
1. Are methods different?
No, both use place value grouping.
Page 13: Difference
1. Difference in hundredths
1943/100
2. Similarities with 653-268
Place value borrowing.
3. Figure it Out sums/diffs
(a) \( 3 \frac{7}{10} \)
(b) \( 4 \frac{3}{100} \)
(c) \( 7 \frac{7}{100} \)
(d) \( 3 \frac{3}{100} \)
(e) \( 3 \frac{3}{100} \)
(f) \( -0 \frac{2}{100} \)
Page 15: How Big?
1. Questions on parts
(a) 1000
(b) 100
(c) 10
(d) 10
(e) 1000
Page 18: Write & Read
1. Quantities in decimal
(a) 2.35
(b) 10.5
(c) 4.06
(d) 101.01
(e) 0.98
(f) 0.05
(g) 0.1
(h) 2.0477
2. 234 tenths
23.4
3. 234 hundredths, 105 tenths
(a) 2.34 (b) 10.5
Page 19: mm to cm
1. Fill blanks
7 cm, 9 mm, 13.4 cm, 2036 mm
Page 20: cm to m
1. Fill blanks
0.36 m, 0.5 m, 89 cm, 0.04 m, 3.25 m, 207 cm
Page 22: g to kg
1. Fill blanks
0.05 kg, 0.36 kg, 50 p, 0.99 rupee, 2.5 rupee
Page 25: Number Line
1. Divisions between 1 and 1.1
1.01, 1.02, ..., 1.09
2. Identify letters
A: 1.2, B: 1.5, C: 1.8 (estimate).
3. Zero Dilemma
0.2 = 0.20 = 0.200; Trailing zeros don't change value.
4. Same values
4.5 = 4.50 = 04.50
5. Magnified number lines
(a) 4.185; (b) ? = 3.426 (estimate).
6. Labels a,b,c
a: 7.2, b: 7.5, c: 7.8
7. Other number lines
Varies; e.g., a: 6.3, b: 6.6
Page 27: Comparing
1. Greater: 6.456 or 6.465
6.465 (hundredths: 6 > 5)
2. Greater decimals
(a) 1.32
(b) 13.800
(c) 1.090
3. Closest to 1
1.01
4. Closest to 4
3.65
5. Closest to 1
1.08
6. Close to 25 using 4,1,8,2,5
24.851 or similar.
Page 29: Addition/Subtraction
1. Cloth total
6.2 m
2. Difference
0.8 m
3. Sums
(a) 7.9 (b) 26.8 (c) 7.41 (d) 18.11 (e) 39.1 (f) 1.534 (g) 0.78 (h) 6.723
4. Differences
(a) 3.3 (b) 9.2 (c) 5.9 (d) 0.802 (e) 16.95 (f) 16.405 (g) 0.81 (h) 5.749
Page 30: Sequences
1. Extend
6.4, 6.8, 7.2, 7.6
2. Mental extensions
(a) 4.55, 4.6, 4.65
(b) 27.25, 27.75, 28.25
(c) 10.89, 11, 11.11
(d) 21, 23.5, 26
(e) 11.2, 12.1, 13
(f) 4.85, 4.80, 4.75
(g) 10.95, 10.45, 9.95
(h) 26, 22.5, 19
Page 31: Estimating
1. Sonu's claim
True for two decimals; Sum > whole parts, < whole + 2.
Page 34: Figure it Out
1. Fractions to decimals
(a) 0.05 (b) 0.016 (c) 1.2 (d) 0.254
2. Decimals to fractions
(a) 3 tenths + 4 hundredths (b) 1 + 2 hundredths (c) 8 tenths (d) 3 tenths + 6 hundredths + 2 thousandths
3. Number line letters
A: 0.2, B: 0.5, C: 0.8, D: 1.1, etc. (estimate).
4. Descending order
(a) 11.10, 11.01, 1.101, 1.011, 1.01
(b) 2.768, 2.698, 2.675, 2.567, 2.499
(c) 4.678 g, 4.666 g, 4.656 g, 4.600 g, 4.595 g
(d) 33.331 m, 33.313 m, 33.31 m, 33.133 m, 33.13 m
5. Using digits
(a) 40.816 (b) 104.68
6. More digits greater?
No; 0.9 > 0.89.
7. Total weight
1.3 kg
8. Milk last three days
12.96 L
9. Weight change
Lost 1.25 kg
10. Extend pattern
7.08, 7.07
11. mm in km
1,000,000 mm
12. Insurance fee
₹45,000
13. Greater
(a) \( \frac{1}{10} \) (b) One-hundredth (c) 90 hundredths
14. Decimal forms
(a) 88.10 (b) 12.12 (c) 111.11 (d) 275.25
15. Closest to 10.5
9.876 + 1.5430 = 11.4190 (adjust digits).
16. Fractions to decimals
(a) 0.5 (b) 1.5 (c) 0.25 (d) 0.75 (e) 0.2 (f) 0.8
Extra Practice Questions (Exam-Ready) – Chapter 3 A Peek Beyond The Point
50+ Questions • Categorized by Marks • With Detailed Solutions • Difficulty Tags
1-Mark Questions (Very Short Answer)
1. What is a tenth?
2. Write \( 3 \frac{4}{10} \) in improper fraction.
3. 1 cm = ? mm
4. Read 7.05
5. 0.2 = ? hundredths
2-Mark Questions (Short Answer)
6. Explain need for smaller units with screw example.
7. Add \( 2 \frac{3}{10} + 4 \frac{5}{10} \)
8. Convert 15 cm to m.
9. Compare 4.56 and 4.65
10. Subtract 3.2 - 1.45
3-Mark Questions (Reasoning / Explanation)
11. Why base 10 for decimals?
12. Honeybee total length calculation.
13. Locate 1.4 on number line.
14. Convert 254 g to kg using place value.
15. Why 0.2 = 0.20?
4–5 Mark Questions (Application / Word Problems)
16. Shylaja hand problem.
17. Folded paper length explanation.
18. Add 15.34 + 2.68 in two methods.
19. Insurance 1 lakh at 0.45 rupee each.
20. Create situation for 2.4 cm - 0.9 cm.
Common Mistakes & How to Avoid
Mistake 1: Forgetting Decimal Alignment
Add 2.7 + 3.5 as 27 + 35 = 62 (wrong).
Avoid: Always align decimal points.
Mistake 2: Sign Change in Subtraction
12.4 - 6.7 = 6.7 (wrong borrowing).
Avoid: Borrow from whole if needed.
Mistake 3: Conversion Units
10 mm = 1 m (wrong).
Avoid: Remember 1 cm = 10 mm, 1 m = 100 cm.
Mistake 4: Comparing Digits
0.9 > 0.89 (wrong).
Avoid: Compare place values left to right.
Mistake 5: Trailing Zeros
Think 0.20 > 0.2.
Avoid: They are equal.
Quick Revision One-Pager & Mind Map
| Topic | Key Points |
|---|---|
| Tenths/Hundredths | 1 = 10/10 = 100/100; Mixed to improper. |
| Decimal Notation | Point separates; Reading without "and". |
| Conversions | Length/Weight/Money: Divide by 10/100/1000. |
| Operations | Align decimals; Place value add/subtract. |
Mind Map
Central: Decimals
- Smaller Units: Tenths, Hundredths.
- Place Value: Extend right of point.
- Conversions: mm-cm-m, g-kg, paise-rupee.
- Compare/Operations: Number line, align.
Interactive Quiz (15 Questions)
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