Another Peek Beyond the Point

Chapter Overview

This chapter extends decimal understanding to multiplication and division. Place value, estimation and connections with fractions explain the algorithms rather than treating them as mechanical rules.

This chapter belongs to Ganita Prakash, Grade 7. Focus on explaining each step, checking whether an answer is reasonable, and comparing more than one solution method.

Learning Objectives

After studying this chapter, you should be able to:

• explain and apply multiplying by powers of ten;
• explain and apply decimal multiplication;
• explain and apply decimal division;
• explain and apply measurement conversions;
• communicate the reasoning behind a solution clearly;
• check answers using estimation, substitution, or a second method.

Key Concepts

| Concept | Meaning | |---|---| | Multiplying by powers of ten | Multiplication by 10, 100 or 1000 increases every digit’s place value; division decreases it. | | Decimal multiplication | Multiply as whole numbers, then place the decimal so the product has the total number of decimal places in the factors. | | Decimal division | Make the divisor a whole number by multiplying both dividend and divisor by the same power of ten. | | Measurement conversions | Metric units use powers of ten, so decimals convert naturally between metres and centimetres or kilograms and grams. |

Detailed Explanation

Multiplying by powers of ten

Multiplication by 10, 100 or 1000 increases every digit’s place value; division decreases it.

Decimal multiplication

Multiply as whole numbers, then place the decimal so the product has the total number of decimal places in the factors.

Decimal division

Make the divisor a whole number by multiplying both dividend and divisor by the same power of ten.

Measurement conversions

Metric units use powers of ten, so decimals convert naturally between metres and centimetres or kilograms and grams.

Important Rules and Formulae

• Estimate first to check the position of the decimal point.
• Multiplying by a number smaller than 1 makes a positive number smaller.
• Changing both dividend and divisor by the same non-zero factor keeps the quotient unchanged.

Worked Examples

Example 1

Problem: Calculate 3.6 × 0.25.

Solution: 36 × 25 = 900; three decimal places in total give 0.900 = 0.9.

Example 2

Problem: Calculate 7.56 ÷ 0.6.

Solution: Multiply both numbers by 10: 75.6 ÷ 6 = 12.6.

Common Mistakes

• Counting decimal places incorrectly.
• Moving only the divisor’s decimal point in a division.
• Expecting every multiplication result to be larger.

Quick Revision

• Multiplying by powers of ten: Multiplication by 10, 100 or 1000 increases every digit’s place value; division decreases it.
• Decimal multiplication: Multiply as whole numbers, then place the decimal so the product has the total number of decimal places in the factors.
• Decimal division: Make the divisor a whole number by multiplying both dividend and divisor by the same power of ten.
• Measurement conversions: Metric units use powers of ten, so decimals convert naturally between metres and centimetres or kilograms and grams.

Practice Questions

1. Find 4.08 × 10.
2. Find 5.4 ÷ 100.
3. Calculate 2.5 × 1.2.
4. A 6.3 kg load is shared equally among 9 boxes. Mass per box?

Answers and Explanations

1. 40.8.
2. 0.054.
4. 0.7 kg.

Self-Check

Explain one rule from this chapter in your own words, create a fresh example, solve it, and verify the answer. If the explanation and verification agree, the concept is understood rather than merely memorised.