Chapter Overview

We already know how to read, compare, add and subtract decimals. Now we extend those ideas to multiplication and division.

1. Powers of Ten

Examples:

Multiplication reverses the movement:

2. Multiplying Decimals

For two decimals:

1. multiply as whole numbers;
2. count total decimal places;
3. place the decimal in the product.

3. Decimal Quotients

Division can continue beyond the ones place.

For example:

4. Decimal Dividend

5. Decimal Divisor

Make the divisor a whole number by multiplying both numbers by the same power of 10.

6. Terminating and Continuing Decimals

Some divisions end:

Some continue:

7. Place Value and Powers of Ten

Multiplying or dividing by 10, 100 or 1000 changes the place value of every digit.

It is more accurate to think of the digits shifting place value rather than imagining the decimal point physically moving.

8. Multiplying Two Decimals

A practical method is:

1. multiply as if the numbers were whole numbers;
2. count the total number of decimal places in both factors;
3. place that many decimal places in the product.

9. Estimating Decimal Products

Estimate before calculating. For example:

is close to:

So an answer such as 81.9 would clearly be unreasonable.

10. Dividing by a Decimal

To divide by a decimal, multiply both the dividend and divisor by the same power of 10 until the divisor becomes a whole number.

For example:

Multiply both by 10:

So:

11. Decimal Quotients That Continue

Some divisions end after a finite number of decimal places, while others continue. At this level, the important skill is to perform the division accurately and report the number of decimal places required by the question.

For example:

terminates, while some other divisions continue beyond the displayed digits.

12. Applications

Decimal multiplication and division are used in:

• cost calculations;
• distance and speed;
• mass and length conversions;
• unit pricing;
• area measurements;
• sharing quantities into equal decimal-sized parts.

13. Complete Decimal-operation Notes

Multiplying by 10, 100 and 1000

Think in place values:

Every multiplication by 10 shifts each digit one place to a higher place value.

Dividing by 10, 100 and 1000

Each division by 10 shifts each digit one place to a smaller place value.

Decimal × Whole Number

Multiply as with whole numbers, then use place value to place the decimal.

Example:

This can also be understood as repeated addition:

Decimal × Decimal

Estimate first. For $4.8\times2.2$, the result should be close to $5\times2=10$.

Exact calculation:

There are two decimal places in total:

The estimate confirms that the decimal placement is reasonable.

Whole Number ÷ Whole Number with Decimal Answer

Sometimes the quotient is not a whole number.

Continue long division by writing zeroes after the decimal point in the dividend as needed.

Decimal ÷ Whole Number

Example:

The decimal point in the quotient aligns according to place value as division proceeds.

Decimal Divisor

Example:

Multiply both by 100:

So:

Checking Decimal Division

If:

check by multiplication:

Applications

If 2.5 kg of fruit costs ₹180, cost per kg is:

So the unit price is ₹72 per kg.

14. More Decimal Examples

Decimal Place-value Check

Compare the size of the answer with the original numbers. If $0.4\times0.3$ is being calculated, both factors are less than 1, so the product must be less than both:

This simple observation can detect misplaced decimal points.

Multiplying by Numbers Less Than 1

Multiplication does not always make a number larger. For positive numbers:

because multiplying by one half means taking half of the number.

Dividing by Numbers Less Than 1

Division by a positive number smaller than 1 makes the quotient larger:

because there are 16 halves in 8 wholes.

Common Mistakes

Quick Revision

Practice Questions

Additional Practice

Chapter Summary