Key Points

To multiply decimals, first multiply the numbers as if they were whole numbers, ignoring the decimal points. Then, count the total number of decimal places in the original numbers and place the decimal point in the product so it has that many decimal places.

The number of decimal places in the product of two decimals is the sum of the number of decimal places in the multiplier and the multiplicand. For example, $1.25 \times 0.3$ will have $2+1=3$ decimal places.

To multiply a decimal by $10, 100, 1000$, etc., move the decimal point to the right by the same number of places as there are zeros in the multiplier. For example, $2.345 \times 100 = 234.5$.

To divide a decimal by $10, 100, 1000$, etc., move the decimal point to the left by the same number of places as there are zeros in the divisor. For example, $45.6 \div 10 = 4.56$.

To divide by a decimal, first convert the divisor into a whole number by multiplying it by a power of 10 ($10, 100$, etc.). Then, multiply the dividend by the same power of 10 and perform the division. For example, $4.68 \div 1.3 = (4.68 \times 10) \div (1.3 \times 10) = 46.8 \div 13$.

Use the long division method. Place the decimal point in the quotient directly above the decimal point in the dividend. Continue the division by adding zeros to the right of the dividend if needed.

To convert a fraction to a decimal, divide the numerator by the denominator using long division. Alternatively, if the denominator has only prime factors of 2 and 5, you can find an equivalent fraction with a denominator of $10, 100, 1000$, etc. For example, $\frac{3}{4} = \frac{3 \times 25}{4 \times 25} = \frac{75}{100} = 0.75$.

When multiplying two decimals, if both numbers are greater than 1, the product is greater than both numbers (e.g., $2 \times 3 = 6$). If both numbers are between 0 and 1, the product is smaller than both numbers (e.g., $0.2 \times 0.3 = 0.06$).

When dividing, if the divisor is greater than 1, the quotient is smaller than the dividend (e.g., $10 \div 2 = 5$). If the divisor is between 0 and 1, the quotient is greater than the dividend (e.g., $10 \div 0.5 = 20$).

Some divisions, like $10 \div 3$, result in a decimal that never ends and has a repeating pattern of digits. These are called non-terminating recurring decimals. For example, $10 \div 3 = 3.333...$.