Class 7 Maths (Part II) Chapter 4 : Another Peek Beyond the Point | Decimal Operations, Long Division with Decimals, Multiplication Rules & Real-Life Problems

Chapter at a Glance – Another Peek Beyond the Point

This chapter extends fractions to decimals, focusing on multiplication and division operations with decimals using real-life examples like shopping and travel.

Main Topics Covered

Recap of decimals and place value
Converting fractions to decimals
Dividing by 10, 100, 1000, etc.
Decimal multiplication with examples
Product of decimals and natural numbers
Rule for placing decimal point in products
When product is greater/less than numbers
Decimal division with examples
Moving decimal point in division
Converting fractions to decimals using equivalents
Division using place value

Key Takeaways for Exams

Decimal Place Value

Tenths, hundredths, thousandths.

Multiplication Rule

Ignore decimals, then place based on total decimal places.

Division by Powers of 10

Move decimal left by number of zeros.

Fraction to Decimal

Equivalent fractions with denominator 10, 100, etc.

Product Relations

Greater/less based on numbers >1 or <1.

Division Place Value

For non-terminating, use long division.

Key Rules & Properties – Decimals

Important rules for multiplying and dividing decimals.

Decimal Multiplication Rule

Multiply ignoring decimals, then place decimal point in product with total decimal places from factors.

Examples	Multiplier Places	Multiplicand Places	Product Places
9.5 × 5 = 47.5	1	0	1
12.5 × 7.5 = 93.75	1	1	2
5.96 × 24.8 = 147.808	2	1	3

Decimal Division by Powers of 10

Move decimal point left by number of zeros in divisor.

Decimal	÷ 10	÷ 100	÷ 1000	÷ 10000
18.7	1.87	0.187	0.0187	0.00187
21.1	2.11	0.211	0.0211	0.00211
0.13	0.013	0.0013	0.00013	0.000013

Properties

Product > Both: Both >1.
Product < Both: Both between 0 and 1.
Mixed: Product between the two.
Fraction to Decimal: Equivalent with denominator 10^n.
Division Reciprocal: Divide by multiplying reciprocal.

Concept Cards – Quick Explanations

Decimals Recap

Extension of place value for fractions like 1/10, 1/100.

Place Value

Tens, ones, tenths, hundredths, etc.

Fraction to Decimal

Sum of unit fractions, e.g., 254/1000 = 0.254.

Division by 10^n

Move decimal left n places.

Multiplication

As fractions: numerators ×, denominators ×.

Product Placement

Total decimal places in factors.

Product Relations

Depends on if factors >1 or <1.

Division

Move decimal right in divisor, adjust dividend.

Equivalent Fractions

For decimal conversion, denominator 10^n.

Place Value Division

For non-terminating quotients.

Examples + Solutions

Example 1: Cost of 5 Pens at ₹9.5 Each

Solution: 9.5 × 5 = 47.5. Cost is ₹47.5.

Example 2: Distance with 7.5 Litres at 12.5 km/L

Solution: 12.5 × 7.5 = 93.75 km.

Example 3: Weekly Walking Distance 827 m Each Way, 6 Days

Solution: 0.827 × 2 = 1.654 km/day, ×6 = 9.924 km/week.

Example 4: Area of Rectangle 5.7 cm × 13.3 cm

Solution: 5.7 × 13.3 = 75.81 sq cm.

Example 5: Product of 5.8 and 1.24

Solution: 58 × 124 = 7192, decimal places 3, so 7.192.

Example 6: Divide 3.9 m into 10 Pieces

Solution: 3.9 ÷ 10 = 0.39 m each.

Example 7: Share 29 m Ribbon Between 2

Solution: 29 ÷ 2 = 14.5 m each.

Figure it Out Solutions (All Solved)

1. Products in Tenths, Hundredths

(a) 6 × 4 tenths

24 tenths = 2.4

(b) 7 × 0.3

2.1

(c) 9 × 5 hundredths

45 hundredths = 0.45

2. Find Products

(a) 27.34 × 6

164.04

(b) 4.23 × 3.7

15.651

(c) 0.432 × 0.23

0.09936

3. Cloth for 3 Shirts at 1.65 m Each

3 Shirts

1.65 × 3 = 4.95 m

4. Cost of 4 Notebooks ₹15.50 Each, 3 Erasers ₹2.75 Each

Total Cost

15.50 × 4 = 62, 2.75 × 3 = 8.25, Total ₹70.25

5. Height of 36 Coins at 1.45 mm Each

Total Height

1.45 × 36 = 52.2 mm = 5.22 cm

6. Price of 2.250 kg Oranges at ₹56.50/kg

Price

56.50 × 2.25 = 127.125. Same as 56.5 × 2.250.

7. Profit on 50 Notebooks ₹23.6 Cost, ₹30 Sell

Profit

Profit per = 6.4, Total = 6.4 × 50 = 320

8. Products from 18 × 12 = 216

(a) 18 × 1.2

21.6

(b) 18 × 0.12

2.16

(c) 1.8 × 1.2

2.16

(d) 0.18 × 0.12

0.0216

(e) 0.018 × 0.012

0.000216

(f) 1.8 × 12

21.6

Products <1: (d),(e)

(d),(e)

9. Products <1 Without Calculation

Options

(b) 0.7 × 0.6, (d) 0.07 × 0.06

10. Multiply by 10,100,1000

Table Completion

5.7 ×10=57, ×100=570, ×1000=5700 etc.

Extra Practice Questions (Exam-Ready) – Chapter 4

25+ Questions • Categorized by Marks • With Detailed Solutions • Difficulty Tags

1-Mark Questions (Very Short Answer)

1. 27.53 place values.

2. 254/1000 as decimal.

3. 123 ÷ 10.

4. Decimal places in product.

5. Division by 100 moves decimal.

2-Mark Questions (Short Answer)

6. 9.5 × 5.

7. 12.5 × 7.5.

8. 3.9 ÷ 10.

9. Fraction 1/2 as decimal.

10. 29 ÷ 4 as decimal.

3-Mark Questions (Reasoning)

11. Explain multiplication rule for 5.96 × 24.8.

12. When product < both numbers.

13. Convert 1/2 to decimal using equivalent.

14. Division by 10 moves decimal left because?

15. Product of decimal and natural number natural?

4–5 Mark Questions (Application)

16. Cloth for 3 shirts 1.65 m each.

17. Height 36 coins 1.45 mm each in cm.

18. Price 2.25 kg oranges ₹56.50/kg.

19. Weekly walk 0.827 km each way, 6 days.

20. Ribbon 3.9 m into 100 pieces.

Challenge Questions (6+ Marks)

21. Explain product relations with examples.

22. Convert 29/4 to decimal.

23. Profit on 50 notebooks cost 23.6 sell 30.

24. 0.432 × 0.23.

25. 3.9 ÷ 100 in cm and mm.

Common Mistakes & How to Avoid

Mistake 1: Wrong Decimal Placement in Product

Counting places incorrectly.

Avoid: Sum places from both numbers.

Mistake 2: Forgetting to Move Decimal in Division

Not shifting left for powers of 10.

Avoid: Count zeros and move left.

Mistake 3: Incorrect Fraction Equivalent

Wrong multiplier for denominator.

Avoid: Find factor to make 10^n.

Mistake 4: Assuming Product Always Greater

Ignoring cases <1.

Avoid: Check if numbers >1 or <1.

Mistake 5: Not Adding Zeros When Needed

In division or multiplication.

Avoid: Pad with zeros for places.

Mistake 6: Confusing Multiplication and Division Rules

Mixing shifts.

Avoid: Multiply: ignore then place; Divide: shift left.

History & Fun Facts

Ancient Origins

Decimals originated in ancient China and India, with the decimal point introduced by John Napier in 1616.

The place value system was developed in India around 500 AD, enabling decimal fractions.

Real-Life Applications

Money: Currency in decimals.
Measurements: Metrics like km, m, cm.
Science: Precise calculations.
Computing: Floating point numbers.

Fun Facts

Pi is infinite decimal: 3.14159...
1/3 = 0.333... repeating.
Decimals allow infinite precision in theory.
Binary fractions in computers approximate decimals.
Simon Stevin popularized decimals in 1585.

Did You Know?

Indian mathematicians used decimals in astronomy centuries before Europe.

Quick Revision One-Pager

Multiplication and Division Rules

Operation	Rule
Multiplication	Multiply whole, place decimal with sum of places.
Division by 10^n	Move decimal left n places.

Quick Rules

✓ Decimals: Fractions with denominator 10^n
✓ Multiplication: As fractions numer/denom
✓ Product Placement: Total decimal places
✓ Division: Reciprocal multiply
✓ Equivalent: Multiply num/denom for 10^n
✓ Relations: >1 or <1 affect product size

Mind Map

Central: Decimals Ops

Basics: Place value, conversion
Multiplication:
- Rule, examples
- Relations
Division: Shift, equivalents
Applications: Real-life calc

Exam Tips

Before Solving

Count decimal places

During Solving

Ignore decimals first

After Solving

Check placement

Time-Savers

Use fraction equiv

Interactive Quiz – 15 Questions

Test Your Decimals Knowledge!