Decimals - Multiplication and Division of Decimals

Multiplying Decimals by Whole Numbers: Multiply the numbers as if they were whole numbers, ignoring the decimal point. After finding the product, place the decimal point so that the product has the same number of decimal places as the original decimal factor. For example, in $4.2 imes 3$, there is one decimal place, so the product $12.6$ also has one decimal place.

Multiplying a Decimal by another Decimal: First, multiply the numbers as whole numbers. Count the total number of decimal places in both the multiplier and the multiplicand. Starting from the right of the product, move the decimal point to the left by that total number of places. If you have $0.2$ (1 place) and $0.15$ (2 places), your product must have $1 + 2 = 3$ decimal places.

Multiplication by $10, 100, \text{ and } 1000$: When multiplying by powers of $10$, the decimal point shifts to the right. Visualise the decimal point 'jumping' over digits to the right; the number of jumps equals the number of zeros in the multiplier. For instance, $5.67 imes 100 = 567$.

Division of Decimals by Whole Numbers: Perform the division just as you would with whole numbers. The most important step is to place the decimal point in the quotient directly above the decimal point in the dividend. This creates a vertical alignment where the points are stacked on top of each other.

Division by $10, 100, \text{ and } 1000$: When dividing by powers of $10$, the decimal point shifts to the left. The number of places it moves is equal to the number of zeros in the divisor. This makes the number smaller. For example, $45.8 \div 100 = 0.458$.

Dividing a Decimal by another Decimal: To divide by a decimal, convert the divisor into a whole number by moving its decimal point to the right until it is at the end. You must move the decimal point in the dividend the same number of places to the right. This is like 'sliding' the decimal point in both numbers simultaneously to maintain the ratio.

Annexing Zeros in Division: If you cannot finish a division because there is a remainder, you can add (annex) zeros to the right of the dividend's decimal part. This allows you to continue the division process until the remainder becomes zero. Visualise this as extending the number with invisible placeholders to complete the calculation.