Learn – Algebra foundations – Fractions

1 Simplifications

1.1 Exercises

1.1.1 Problem set

Simplify the following fractions, if possible.

  1. \frac{3x}{xy}

  2. \frac{ab}{5a}

  3. \frac{4q}{2pq}

  4. \frac{x^2}{x}

  5. \frac{m^2}{2m}

1.1.2 Problem set

Simplify the following fractions, if possible.

  1. \frac{a^2}{2a^3}

  2. \frac{p^2}{3p^3}

  3. \frac{m^3}{2m^5}

  4. \frac{x^{50}}{6x^{53}}

  5. \frac{2a^{100}}{2a^{104}}

  6. \frac{3m^{3}n^{5}}{3m^{8}n^{10}}

  7. \frac{5x^{41}y^{15}}{5x^{25}y^{40}}

1.1.3 Problem set

Simplify the following fractions, if possible.

  1. \frac{x+3}{x}

  2. \frac{x^2+2}{x}

  3. \frac{x^2+2}{x^2}

  4. \frac{x+2}{x+1}

  5. \frac{p^2+q^2}{p+q}

  6. \frac{a^2+b^2}{a+b}

  7. \frac{x^2+3}{x^2+1}

1.1.4 Problem set

Simplify the following fractions, if possible.

  1. \frac{xy+x}{xy+y}

  2. \frac{x^2y^2+xy+x}{x^2y^2+xy+y}

  3. \frac{(m+6)(m+9)+6}{(m+9)+6}

  4. \frac{(a+2)(a+3)+2}{(a+2)(a+3)}

  5. \frac{(x+8)(x+6)+4}{(x+8)(x+6)}

  6. \frac{(p+4)(p+8)+4}{(p+4)(p+8)}

  7. \frac{x(x^2+4)}{x+4}

1.1.5 Problem set

Simplify the following fractions, if possible.

  1. \frac{(x+1)^2}{x+1}

  2. \frac{(a+1)(a+2)}{2(a+1)}

  3. \frac{(p+3)(p+5)}{5(p+3)}

  4. \frac{(a-4)(a-3)}{3(a-4)}

1.1.6 Problem set

Simplify the following fractions, if possible.

  1. \frac{(p+2)^2}{2(p+2)}

  2. \frac{(x+1)^2(x+2)^2}{2(x+2)}

  3. \frac{(a+2)^2(a+2)^2}{2(a+2)^3}

  4. \frac{(p+5)(p+5)^2}{5(p+5)^5}

1.1.7 Problem set

Which of the following are correct equations?

  1. \frac{2x+3}{x+3} = 2

  2. \frac{x+1}{x+3} = \frac{1}{3}

  3. \frac{6x}{5x} = \frac{6}{5}

  4. \frac{2x}{3x} = \left(\frac{2}{3}\right)\times x

  5. \frac{x^2+x+5}{x^2+2x+1} = \frac{x+5}{2x+1}

1.1.8 Problem set

Simplify the following fractions, if possible.

  1. \frac{2y-2x}{2}

  2. \frac{2b-2a}{2b}

  3. \frac{2q-2p}{q-p}

  4. \frac{a-b}{2a-2b}

  5. \frac{2y-2x}{x-y}

  6. \frac{p-q}{3q-3p}

1.1.9 Problem set

Simplify the following fractions, if possible.

  1. \frac{x^3+x}{x^2+1}

  2. \frac{x^3-2x^2+3x}{-x^2+2x-3}

  3. \frac{2x^3-5x-9x^2-x+x^3-12}{-x^5+3x^4+2x^3+4x^2}

2 Multiplication and division

2.1 Exercises

2.1.1 Problem set

For each of the following problems, simplify or write as a single fraction.

  1. \frac{a}{b}\times\frac{a}{b}

  2. \frac{x^2}{y^2}\times\frac{y}{x}

  3. \frac{p^3}{q^2}\times\frac{q}{p}

  4. \frac{a^2}{2b^2}\times\frac{2b}{a}

  5. \frac{3m^2}{n^3}\times\frac{n}{3m}

  6. \frac{5p^3}{q^5}\times\frac{q^2}{5p^5}

2.1.2 Problem set

For each of the following problems, simplify or write as a single fraction.

  1. \frac{x}{y}\times\frac{x+1}{y+1}

  2. \frac{p+q}{r+s}\times\frac{r+s}{p+q}

  3. \frac{m+3}{m+4}\times\frac{m+4}{m+3}

  4. \frac{a+1}{a}\times\frac{a+2}{a+1}\times\frac{a+3}{a+2}

  5. \frac{x+6}{x+5}\times\frac{x+5}{x+4}\times\frac{x+4}{x+3}\times\frac{x+3}{x+2}

  6. \frac{2}{a}(ab+ac)

2.1.3 Problem set

For each of the following problems, simplify or write as a single fraction.

  1. \frac{a/b}{c/d}

  2. \frac{a/b}{b/a}

  3. \frac{p/q}{p^2/q^2}

  4. \frac{1/a}{b^2/a^2}

  5. \frac{p^2/q^3}{p^3/q^2}

  6. \frac{x^3/y^4}{x^4/y^3}

2.1.4 Problem set

For each of the following problems, simplify or write as a single fraction.

  1. \frac{a/b}{(a+2)/(b+2)}

  2. \frac{(x+2)/(y+2)}{(x+2)/(y+2)}

  3. \frac{(m+2)/(n+2)}{(n+2)/(m+2)}

  4. \frac{(a+2)/(b+2)}{(a+1)/(b+1)}

3 Addition and subtraction

3.1 Exercises

3.1.1 Problem set

For each of the following problems, simplify or write as a single fraction.

  1. \frac{1}{a}+\frac{1}{b}

  2. \frac{3}{x}+\frac{4}{y}

  3. \frac{2}{p}-\frac{7}{q}

  4. \frac{2}{a}-\frac{a}{2}

  5. \frac{b}{a}+\frac{a}{b}

  6. \frac{y}{2x}-\frac{2x}{y}

3.1.2 Problem set

For each of the following problems, simplify or write as a single fraction.

  1. \frac{3}{2p}-\frac{7}{2q}

  2. \frac{3}{4a}-\frac{5}{2b}

  3. \frac{1}{p}+\frac{1}{p^2}

  4. \frac{1}{2x}+\frac{1}{x^2}

  5. \frac{1}{6a^2}+\frac{1}{4a^3}

3.1.3 Problem set

For each of the following problems, simplify or write as a single fraction.

  1. \frac{3}{8x^3}-\frac{5}{6x^2}

  2. \frac{3}{4a^2} - \frac{5}{6ab^2}

  3. \frac{7}{3p^2q} - \frac{5}{6pq^2}

  4. \frac{1}{p} + \frac{2}{p+1}

  5. \frac{4}{a+2} + \frac{3}{a}

  6. \frac{1}{x} + \frac{1}{2(x+1)}

  7. \frac{1}{2x} - \frac{1}{x+2}

3.1.4 Problem set

For each of the following problems, simplify or write as a single fraction.

  1. \frac{1}{a+1}-\frac{1}{a}

  2. \frac{2}{x+4} - \frac{1}{x}

  3. \frac{a}{a^2+1}-\frac{1}{a}

  4. \frac{1}{a^2} + \frac{1}{a(b+c)}

  5. \frac{x^2+1}{x^2}- \frac{x-1}{x}

  6. \frac{1}{p^2+2}-\frac{1}{p^2}

3.1.5 Problem set

For each of the following problems, simplify or write as a single fraction.

  1. \frac{1}{x^2} - \frac{1}{x(x+2)}

  2. \frac{p-1}{p(p-2)} - \frac{1}{p^2}

  3. \frac{2}{(x+1)^2} - \frac{1}{2x(x+1)}

  4. \frac{3}{(a+2)^2} - \frac{1}{a(a+2)}

  5. \frac{a}{b+c} + \frac{b+c}{a}

4 Combinations of operations

4.1 Exercises

4.1.1 Problem set

For each of the following problems, simplify or write as a single fraction.

  1. \frac{a}{b/c} + \frac{a/b}{c}

  2. \frac{\frac{b}{2}+\frac{c}{2}}{4(b+c)}

  3. \frac{(b+c)/a}{1/(ab+ac)}

  4. \frac{a/(b+c)}{1/(ab+ac)}

  5. a(b-c)\frac{1}{\frac{b}{a}-\frac{c}{a}}

4.1.2 Problem set

For each of the following problems, simplify.

  1. \frac{\frac{x^2}{y}+\frac{x}{y^2}}{\frac{x^2}{y^2}+\frac{x}{y}}

  2. \frac{xy+y(x^2+xy)}{xy+x^2y^2}

4.1.3 Problem set

Which of the following are correct equations?

  1. \frac{a/b}{c} = \frac{a}{b/c}

  2. \frac{a}{b} + c = \frac{a+c}{b}

  3. x\times\left(\frac{2}{3}\right) = \frac{x\times2}{3}

5 Miscellaneous

5.1 Exercises

5.1.1 Problem set

  1. \frac{10}{x} = \frac{15}{y}. What fraction of y is x?
  2. What can you say about the relationship (equal to, less than, greather than or nothing can be said) between \frac{b+a}{c+a} and \frac{b}{c}?
  3. Solve for x: \frac{1/x}{4/x^2} = 1
  4. Solve for x: \frac{1/x}{4/x^2} - \frac{1}{2/x} = 1