Working with Decimals

Understanding Decimals

Decimals are another way to represent parts of a whole.

Place value chart:

| Ones | . | Tenths | Hundredths | Thousandths | |------|---|--------|------------|-------------| | 3 | . | 4 | 7 | 5 |

So $3.475$ means: 3 ones + 4 tenths + 7 hundredths + 5 thousandths.

Adding and Subtracting Decimals

Rule: Line up the decimal points, then add or subtract normally.

Example: $3.45 + 2.7$

Add a zero: $3.45 + 2.70 = 6.15$

$3.45 + 2.70 = 6.15$

Multiplying Decimals

Steps:

1. Multiply as if they're whole numbers (ignore the decimals)
2. Count the total number of decimal places in both numbers
3. Put that many decimal places in the answer

Example: $2.5 \times 0.3$

| Step | Work | |------|------| | Multiply without decimals | $25 \times 3 = 75$ | | Count decimal places | $2.5$ has 1, $0.3$ has 1 → total: 2 | | Place the decimal | $0.75$ |

$2.5 \times 0.3 = 0.75$

Dividing Decimals

Steps:

1. Move the decimal in the divisor to make it a whole number
2. Move the decimal in the dividend the same number of places
3. Divide normally
4. Place the decimal point in the quotient

Example: $6.5 \div 0.5$

Move both decimals one place right: $65 \div 5 = 13$

$6.5 \div 0.5 = 13$

Converting Between Fractions and Decimals

Fraction → Decimal: Divide the numerator by the denominator

| Fraction | Division | Decimal | |----------|----------|---------| | $\frac{1}{2}$ | $1 \div 2$ | $0.5$ | | $\frac{3}{4}$ | $3 \div 4$ | $0.75$ | | $\frac{1}{5}$ | $1 \div 5$ | $0.2$ |

Decimal → Fraction: Use place value

| Decimal | Fraction | Simplified | |---------|----------|-----------| | $0.75$ | $\frac{75}{100}$ | $\frac{3}{4}$ | | $0.6$ | $\frac{6}{10}$ | $\frac{3}{5}$ | | $0.125$ | $\frac{125}{1000}$ | $\frac{1}{8}$ |

Comparing Decimals

To compare decimals, look at each place value from left to right:

• $0.5$ vs $0.35$? Since $0.50 > 0.35$, we have $0.5 > 0.35$
• $3.14$ vs $3.2$? Compare tenths: $1 < 2$, so $3.14 < 3.2$

Tip: Add zeros so both numbers have the same number of decimal places, then compare!