Basic Operations with Fractions and Decimals

Understanding how to add, subtract, multiply, and divide both fractions and decimals is essential for everyday calculations and more advanced math. 

Operations with Fractions

Adding Fractions

• Rule: Find a common denominator, convert each fraction, then add numerators.

Example: $$\frac{2}{3} + \frac{1}{4} \;=\;\frac{2\cdot4}{3\cdot4} + \frac{1\cdot3}{4\cdot3} =\frac{8}{12} + \frac{3}{12} = \frac{11}{12}.$$

Subtracting Fractions

• Rule: Same as addition, but subtract numerators.

Example: $$\frac{5}{6} - \frac{1}{3} = \frac{5}{6} - \frac{2}{6} = \frac{3}{6} = \frac{1}{2}.$$

Multiplying Fractions

• Rule: Multiply numerators together and denominators together.

Example: $$\frac{3}{5} \times \frac{4}{7} = \frac{3\cdot4}{5\cdot7} = \frac{12}{35}.$$

Dividing Fractions

• Rule: Multiply by the reciprocal of the divisor.

Example: $$\frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6}.$$

 

Operations with Decimals

Adding Decimals

• Rule: Line up decimal points, add as usual, then place decimal point in sum.

Example: $$3.45 + 2.7 = 6.15.$$

 

Subtracting Decimals

• Rule: Line up decimal points, subtract as usual, then place decimal.

Example: $$5.00 - 1.275 = 3.725.$$

 

Multiplying Decimals

• Rule: Multiply as integers, then count total decimal places in factors and place in product.

Example: $$0.6 \times 0.25 = 6 \times 25 = 150 \;\Rightarrow\; 0.150 = 0.15\quad(\text{three total decimal places}).$$

 

Dividing Decimals

• Rule: Move decimal in divisor to make it whole, shift decimal in dividend same places, then divide.

Example: $$4.5 \div 0.15 \;\Rightarrow\; 4.50 \div 0.15 = 450 \div 15 = 30.$$