Questions: Arithmetic on algebraic fractions
Before attempting these questions, it is highly recommended that you read Guide: Arithmetic on algebraic fractions.
Q1
Calculate the following additions and subtractions by first finding a common denominator. You should give your answer in its simplest form. You may assume that that all denominators are non-zero, so you do not need to write restrictions.
1.1. \(\displaystyle \quad \dfrac{2}{x} + \dfrac{5}{x}\)
1.2. \(\displaystyle \quad \dfrac{x+1}{3} + \dfrac{x-2}{3}\)
1.3. \(\displaystyle \quad \dfrac{4}{y} - \dfrac{1}{y}\)
1.4. \(\displaystyle \quad \dfrac{3}{4x} + \dfrac{1}{4x}\)
1.5. \(\displaystyle \quad \dfrac{b-3}{5} - \dfrac{b+2}{5}\)
1.6. \(\displaystyle \quad \dfrac{1}{x} + \dfrac{1}{2x}\)
1.7. \(\displaystyle \quad \dfrac{3}{5y} - \dfrac{2}{15y}\)
1.8. \(\displaystyle \quad \dfrac{x}{3} + \dfrac{x}{6}\)
1.9. \(\displaystyle \quad \dfrac{2}{x} + \dfrac{3}{x+1}\)
1.10. \(\displaystyle \quad \dfrac{t+4}{t-1} - \dfrac{t-2}{t-1}\)
1.11. \(\displaystyle \quad \dfrac{5}{x-3} + \dfrac{2}{2x-6}\)
1.12. \(\displaystyle \quad \dfrac{2}{x+2} - \dfrac{1}{x-2}\)
1.13. \(\displaystyle \quad \dfrac{c+1}{c^2} + \dfrac{2}{c}\)
1.14. \(\displaystyle \quad \dfrac{3x}{x^2-9} + \dfrac{2x}{x+3}\)
1.15. \(\displaystyle \quad \dfrac{x-1}{x+2} - \dfrac{2x+3}{x^2+4x+4}\)
Q2
Calculate the following multiplications. Write your answer in its simplest form. Assume that all denominators are non-zero.
2.1. \(\displaystyle \quad \dfrac{x}{3} \cdot \dfrac{2}{5}\)
2.2. \(\displaystyle \quad \dfrac{3a}{4} \cdot \dfrac{8}{a}\)
2.3. \(\displaystyle \quad \dfrac{x+1}{2} \cdot \dfrac{x}{3}\)
2.4. \(\displaystyle \quad \dfrac{5y}{2x} \cdot \dfrac{3x}{10y}\)
2.5. \(\displaystyle \quad \dfrac{x-3}{x} \cdot \dfrac{x}{4}\)
2.6. \(\displaystyle \quad -\dfrac{2x}{5} \cdot \dfrac{15}{x^2}\)
2.7. \(\displaystyle \quad \dfrac{m^2-9}{m+3} \cdot \dfrac{m}{2m-6}\)
2.8. \(\displaystyle \quad \dfrac{x+2}{x-1} \cdot \dfrac{x-1}{x+3}\)
2.9. \(\displaystyle \quad \dfrac{3x}{x^2+2x} \cdot \dfrac{x+2}{4}\)
2.10. \(\displaystyle \quad \dfrac{x^2+5x+6}{x+1} \cdot \dfrac{1}{x+2}\)
2.11. \(\displaystyle \quad -\dfrac{z-4}{2z} \cdot \dfrac{3z}{z-4}\)
2.12. \(\displaystyle \quad \dfrac{2x}{x^2-4} \cdot \dfrac{x+2}{3}\)
Q3
Calculate the following divisions. Write your answer in its simplest form. Assume that all denominators are non-zero.
3.1. \(\displaystyle \quad \dfrac{t}{2} \div \dfrac{t}{5}\)
3.2. \(\displaystyle \quad \dfrac{3}{x} \div \dfrac{1}{2x}\)
3.3. \(\displaystyle \quad \dfrac{x+1}{4} \div \dfrac{x+1}{2}\)
3.4. \(\displaystyle \quad \dfrac{2x}{3y} \div \dfrac{4}{9y}\)
3.5. \(\displaystyle \quad \dfrac{x-2}{x} \div \dfrac{3}{4}\)
3.6. \(\displaystyle \quad \dfrac{v^2-1}{v+1} \div v\)
3.7. \(\displaystyle \quad -\dfrac{3x}{5} \div \dfrac{x}{10}\)
3.8. \(\displaystyle \quad \dfrac{x^2+3x}{x} \div \dfrac{x+3}{2}\)
3.9. \(\displaystyle \quad \dfrac{x}{x-4} \div \dfrac{2x}{x-4}\)
3.10. \(\displaystyle \quad \dfrac{x^2-9}{x^2-3x} \div \dfrac{x-3}{x}\)
3.11. \(\displaystyle \quad \dfrac{2n}{n+2} \div \dfrac{n}{n+2}\)
3.12. \(\displaystyle \quad \dfrac{x^2-4}{2x} \div \dfrac{x-2}{3}\)
Q4
Simplify the following compound algebraic fractions. Write your answer in its simplest form. Assume that all denominators are non-zero.
4.1. \(\displaystyle \quad \dfrac{\dfrac{1}{x} + \dfrac{1}{y}}{\dfrac{1}{x}}\)
4.2. \(\displaystyle \quad \dfrac{-\dfrac{2x}{3}}{\dfrac{x+2}{9}}\)
4.3. \(\displaystyle \quad \dfrac{\dfrac{x}{x-1}}{\dfrac{2x}{x-1}}\)
4.4. \(\displaystyle \quad \dfrac{\dfrac{1}{t} + \dfrac{2}{t^2}}{\dfrac{3}{t}}\)
4.5. \(\displaystyle \quad \dfrac{\dfrac{x+1}{2x}}{-\dfrac{x-1}{4x^2}}\)
After attempting the questions above, please click this link to find the answers.
Version history and licensing
v1.0: initial version created 12/25 by Donald Campbell as part of a University of St Andrews VIP project.