Add • Subtract • Multiply • Divide • Mixed Operations | Integers from −20 to 20 | Answers as Reduced Improper Fractions
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Adding & Subtracting Fractions
Find a common denominator, rewrite the fractions, then add or subtract the numerators.
Keep the common denominator and reduce the result.
Whole numbers can be written as fractions with denominator 1 (e.g., \( 4 = \dfrac{4}{1} \)).
Multiplying Fractions
Multiply the numerators together and the denominators together.
\( \dfrac{a}{b} \times \dfrac{c}{d} = \dfrac{a \cdot c}{b \cdot d} \)
Then reduce the product to lowest terms. Whole numbers are treated as fractions with denominator 1.
Dividing Fractions
Multiply by the reciprocal of the second fraction.
\( \dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b} \times \dfrac{d}{c} \)
Example: \( 18 \div \dfrac{2}{3} = 18 \times \dfrac{3}{2} = 27 \)
Mixed Operations & Order of Operations
Follow PEMDAS / GEMDAS: Parentheses (and absolute value), Exponents, Multiply/Divide (left to right), Add/Subtract (left to right).
Always simplify inside grouping symbols first, then apply exponents, then multiplication and division.
Important: Always reduce the final answer to lowest terms. Write answers as reduced improper fractions (no mixed numbers). Integer answers may be left as integers (positive or negative).
Add the fractions (or whole numbers and fractions). Write the answer as a reduced improper fraction.
Simplify: \( \dfrac{1}{4} + \dfrac{1}{4} \)
Same denominator, so add the numerators:
$$\dfrac{1+1}{4} = \dfrac{2}{4} = \dfrac{1}{2}$$
Answer: \( \dfrac{1}{2} \)
Simplify: \( \dfrac{2}{5} + \dfrac{1}{5} \)
$$\dfrac{2+1}{5} = \dfrac{3}{5}$$
Answer: \( \dfrac{3}{5} \)
Simplify: \( 3 + \dfrac{1}{2} \)
Write 3 as an improper fraction:
$$3 = \dfrac{6}{2}$$
$$\dfrac{6}{2} + \dfrac{1}{2} = \dfrac{7}{2}$$
Answer: \( \dfrac{7}{2} \)
Simplify: \( \dfrac{1}{3} + \dfrac{1}{6} \)
Common denominator is 6:
$$\dfrac{1}{3} = \dfrac{2}{6}$$
$$\dfrac{2}{6} + \dfrac{1}{6} = \dfrac{3}{6} = \dfrac{1}{2}$$
Answer: \( \dfrac{1}{2} \)
Simplify: \( \dfrac{2}{5} + \dfrac{3}{10} \)
Common denominator is 10:
$$\dfrac{2}{5} = \dfrac{4}{10}$$
$$\dfrac{4}{10} + \dfrac{3}{10} = \dfrac{7}{10}$$
Answer: \( \dfrac{7}{10} \)
Simplify: \( \dfrac{5}{6} + \dfrac{1}{4} \)
Common denominator is 12:
$$\dfrac{5}{6} = \dfrac{10}{12}, \quad \dfrac{1}{4} = \dfrac{3}{12}$$
$$\dfrac{10}{12} + \dfrac{3}{12} = \dfrac{13}{12}$$
Answer: \( \dfrac{13}{12} \)
Simplify: \( \dfrac{-7}{8} + \dfrac{5}{6} \)
Common denominator is 24:
$$\dfrac{-7}{8} = \dfrac{-21}{24}, \quad \dfrac{5}{6} = \dfrac{20}{24}$$
$$\dfrac{-21}{24} + \dfrac{20}{24} = \dfrac{-1}{24}$$
Answer: \( \dfrac{-1}{24} \)
Simplify: \( -4 + \dfrac{11}{5} \)
Write \(-4\) as an improper fraction:
$$-4 = \dfrac{-20}{5}$$
$$\dfrac{-20}{5} + \dfrac{11}{5} = \dfrac{-9}{5}$$
Answer: \( \dfrac{-9}{5} \)
Simplify: \( \dfrac{13}{12} + \dfrac{-5}{8} \)
Common denominator is 24:
$$\dfrac{13}{12} = \dfrac{26}{24}, \quad \dfrac{-5}{8} = \dfrac{-15}{24}$$
$$\dfrac{26}{24} + \dfrac{-15}{24} = \dfrac{11}{24}$$
Answer: \( \dfrac{11}{24} \)
Subtract the fractions (or whole numbers and fractions). Write the answer as a reduced improper fraction.
Simplify: \( \dfrac{3}{4} – \dfrac{1}{4} \)
$$\dfrac{3-1}{4} = \dfrac{2}{4} = \dfrac{1}{2}$$
Answer: \( \dfrac{1}{2} \)
Simplify: \( \dfrac{5}{6} – \dfrac{1}{6} \)
$$\dfrac{5-1}{6} = \dfrac{4}{6} = \dfrac{2}{3}$$
Answer: \( \dfrac{2}{3} \)
Simplify: \( 4 – \dfrac{1}{3} \)
$$4 = \dfrac{12}{3}$$
$$\dfrac{12}{3} – \dfrac{1}{3} = \dfrac{11}{3}$$
Answer: \( \dfrac{11}{3} \)
Simplify: \( \dfrac{5}{6} – \dfrac{1}{3} \)
$$\dfrac{1}{3} = \dfrac{2}{6}$$
$$\dfrac{5}{6} – \dfrac{2}{6} = \dfrac{3}{6} = \dfrac{1}{2}$$
Answer: \( \dfrac{1}{2} \)
Simplify: \( \dfrac{7}{8} – \dfrac{1}{4} \)
$$\dfrac{1}{4} = \dfrac{2}{8}$$
$$\dfrac{7}{8} – \dfrac{2}{8} = \dfrac{5}{8}$$
Answer: \( \dfrac{5}{8} \)
Simplify: \( \dfrac{3}{4} – \dfrac{2}{5} \)
Common denominator is 20:
$$\dfrac{3}{4} = \dfrac{15}{20}, \quad \dfrac{2}{5} = \dfrac{8}{20}$$
$$\dfrac{15}{20} – \dfrac{8}{20} = \dfrac{7}{20}$$
Answer: \( \dfrac{7}{20} \)
Simplify: \( \dfrac{-5}{6} – \dfrac{3}{4} \)
Common denominator is 12:
$$\dfrac{-5}{6} = \dfrac{-10}{12}, \quad \dfrac{3}{4} = \dfrac{9}{12}$$
$$\dfrac{-10}{12} – \dfrac{9}{12} = \dfrac{-19}{12}$$
Answer: \( \dfrac{-19}{12} \)
Simplify: \( \dfrac{7}{8} – \dfrac{-2}{3} \)
Subtracting a negative is the same as adding:
Common denominator is 24:
$$\dfrac{7}{8} = \dfrac{21}{24}, \quad \dfrac{-2}{3} = \dfrac{-16}{24}$$
$$\dfrac{21}{24} – \dfrac{-16}{24} = \dfrac{21}{24} + \dfrac{16}{24} = \dfrac{37}{24}$$
Answer: \( \dfrac{37}{24} \)
Simplify: \( -6 – \dfrac{5}{4} \)
$$-6 = \dfrac{-24}{4}$$
$$\dfrac{-24}{4} – \dfrac{5}{4} = \dfrac{-29}{4}$$
Answer: \( \dfrac{-29}{4} \)
Multiply the fractions (or whole numbers and fractions). Write the answer as a reduced improper fraction.
Simplify: \( \dfrac{1}{2} \times \dfrac{1}{3} \)
$$\dfrac{1 \times 1}{2 \times 3} = \dfrac{1}{6}$$
Answer: \( \dfrac{1}{6} \)
Simplify: \( \dfrac{2}{5} \times 3 \)
$$3 = \dfrac{3}{1}$$
$$\dfrac{2}{5} \times \dfrac{3}{1} = \dfrac{6}{5}$$
Answer: \( \dfrac{6}{5} \)
Simplify: \( \dfrac{3}{4} \times \dfrac{2}{3} \)
$$\dfrac{3 \times 2}{4 \times 3} = \dfrac{6}{12} = \dfrac{1}{2}$$
Answer: \( \dfrac{1}{2} \)
Simplify: \( \dfrac{2}{3} \times \dfrac{3}{4} \)
$$\dfrac{2 \times 3}{3 \times 4} = \dfrac{6}{12} = \dfrac{1}{2}$$
Answer: \( \dfrac{1}{2} \)
Simplify: \( \dfrac{5}{6} \times \dfrac{3}{5} \)
$$\dfrac{5 \times 3}{6 \times 5} = \dfrac{15}{30} = \dfrac{1}{2}$$
Answer: \( \dfrac{1}{2} \)
Simplify: \( 4 \times \dfrac{2}{7} \)
$$4 = \dfrac{4}{1}$$
$$\dfrac{4}{1} \times \dfrac{2}{7} = \dfrac{8}{7}$$
Answer: \( \dfrac{8}{7} \)
Simplify: \( \dfrac{-3}{4} \times \dfrac{5}{6} \)
$$\dfrac{-3 \times 5}{4 \times 6} = \dfrac{-15}{24} = \dfrac{-5}{8}$$
Answer: \( \dfrac{-5}{8} \)
Simplify: \( \dfrac{7}{8} \times \dfrac{-4}{5} \)
$$\dfrac{7 \times (-4)}{8 \times 5} = \dfrac{-28}{40} = \dfrac{-7}{10}$$
Answer: \( \dfrac{-7}{10} \)
Simplify: \( -5 \times \dfrac{6}{7} \)
$$-5 = \dfrac{-5}{1}$$
$$\dfrac{-5}{1} \times \dfrac{6}{7} = \dfrac{-30}{7}$$
Answer: \( \dfrac{-30}{7} \)
Divide the fractions (or whole numbers and fractions). Write the answer as a reduced improper fraction.
Simplify: \( \dfrac{1}{2} \div \dfrac{1}{4} \)
Multiply by the reciprocal:
$$\dfrac{1}{2} \times \dfrac{4}{1} = \dfrac{4}{2} = 2$$
Answer: \( 2 \)
Simplify: \( \dfrac{3}{4} \div \dfrac{1}{2} \)
$$\dfrac{3}{4} \times \dfrac{2}{1} = \dfrac{6}{4} = \dfrac{3}{2}$$
Answer: \( \dfrac{3}{2} \)
Simplify: \( 18 \div \dfrac{2}{3} \)
Write 18 as a fraction and multiply by the reciprocal:
$$18 \div \dfrac{2}{3} = 18 \times \dfrac{3}{2} = \dfrac{18}{1} \times \dfrac{3}{2} = \dfrac{54}{2} = 27$$
Answer: \( 27 \)
Simplify: \( \dfrac{2}{3} \div \dfrac{1}{6} \)
$$\dfrac{2}{3} \times \dfrac{6}{1} = \dfrac{12}{3} = 4$$
Answer: \( 4 \)
Simplify: \( \dfrac{5}{6} \div \dfrac{2}{3} \)
$$\dfrac{5}{6} \times \dfrac{3}{2} = \dfrac{15}{12} = \dfrac{5}{4}$$
Answer: \( \dfrac{5}{4} \)
Simplify: \( 4 \div \dfrac{3}{5} \)
$$4 \times \dfrac{5}{3} = \dfrac{20}{3}$$
Answer: \( \dfrac{20}{3} \)
Simplify: \( \dfrac{-3}{4} \div \dfrac{2}{5} \)
Multiply by the reciprocal:
$$\dfrac{-3}{4} \times \dfrac{5}{2} = \dfrac{-15}{8}$$
Answer: \( \dfrac{-15}{8} \)
Simplify: \( \dfrac{7}{8} \div \dfrac{-3}{4} \)
$$\dfrac{7}{8} \times \dfrac{-4}{3} = \dfrac{-28}{24} = \dfrac{-7}{6}$$
Answer: \( \dfrac{-7}{6} \)
Simplify: \( -9 \div \dfrac{3}{5} \)
$$-9 \times \dfrac{5}{3} = \dfrac{-45}{3} = -15$$
Answer: \( -15 \)
These problems combine fraction operations and require the order of operations (PEMDAS). Simplify completely and write the answer as a reduced improper fraction.
Simplify: \( \left( \dfrac{3}{4} – \dfrac{1}{2} \right)^2 + \dfrac{1}{8} \)
Inside parentheses first:
$$\dfrac{3}{4} – \dfrac{1}{2} = \dfrac{3}{4} – \dfrac{2}{4} = \dfrac{1}{4}$$
Apply the exponent:
$$\left( \dfrac{1}{4} \right)^2 = \dfrac{1}{16}$$
Add:
$$\dfrac{1}{16} + \dfrac{1}{8} = \dfrac{1}{16} + \dfrac{2}{16} = \dfrac{3}{16}$$
Answer: \( \dfrac{3}{16} \)
Simplify: \( \dfrac{2}{3} \times \left( \dfrac{1}{2} + \dfrac{1}{4} \right) \)
Inside parentheses:
$$\dfrac{1}{2} + \dfrac{1}{4} = \dfrac{2}{4} + \dfrac{1}{4} = \dfrac{3}{4}$$
Multiply:
$$\dfrac{2}{3} \times \dfrac{3}{4} = \dfrac{6}{12} = \dfrac{1}{2}$$
Answer: \( \dfrac{1}{2} \)
Simplify: \( \left( \dfrac{5}{6} \div \dfrac{1}{3} \right) – \dfrac{1}{2} \)
Division inside parentheses (multiply by reciprocal):
$$\dfrac{5}{6} \div \dfrac{1}{3} = \dfrac{5}{6} \times \dfrac{3}{1} = \dfrac{15}{6} = \dfrac{5}{2}$$
Subtract:
$$\dfrac{5}{2} – \dfrac{1}{2} = \dfrac{4}{2} = 2$$
Answer: \( 2 \)
Simplify: \( \left( \dfrac{2}{5} + \dfrac{1}{10} \right)^2 \)
Inside parentheses:
$$\dfrac{2}{5} + \dfrac{1}{10} = \dfrac{4}{10} + \dfrac{1}{10} = \dfrac{5}{10} = \dfrac{1}{2}$$
Apply the exponent:
$$\left( \dfrac{1}{2} \right)^2 = \dfrac{1}{4}$$
Answer: \( \dfrac{1}{4} \)
Simplify: \( 3 \times \left( \dfrac{1}{2} – \dfrac{1}{6} \right) + \dfrac{1}{3} \)
Inside parentheses:
$$\dfrac{1}{2} – \dfrac{1}{6} = \dfrac{3}{6} – \dfrac{1}{6} = \dfrac{2}{6} = \dfrac{1}{3}$$
Multiply:
$$3 \times \dfrac{1}{3} = 1$$
Add:
$$1 + \dfrac{1}{3} = \dfrac{3}{3} + \dfrac{1}{3} = \dfrac{4}{3}$$
Answer: \( \dfrac{4}{3} \)
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