Detailed Solutions to Questions on Maths Proportions

Detailed solutions the questions on proportions are presented with full explanations.


  1. Find \( x \) if \[ \dfrac{x}{2} = \dfrac{4}{8}. \]

    Solution

    Multiply both sides of the proportion by the denominators 2 and 8: \[ \color{red}{2 \times 8} \times \dfrac{x}{2} = \color{red}{2 \times 8} \times \dfrac{4}{8}. \] Simplify by canceling common factors: \[ \dfrac{\color{red}{\cancel{2}} \times 8}{\cancel{2}} x = \dfrac{2 \times \color{red}{\cancel{8}}}{\cancel{8}} \times 4, \] which reduces to \[ 8x = 8. \] Divide both sides by 8: \[ \dfrac{8x}{8} = \dfrac{8}{8}, \] so \[ x = 1. \]

  2. Find \( p \) if \[ \dfrac{3}{p} = \dfrac{1}{5}. \]

    Solution

    Multiply both sides of the proportion by the denominators \( p \) and 5: \[ \color{red}{5 \times p} \times \dfrac{3}{p} = \color{red}{5 \times p} \times \dfrac{1}{5}. \] Simplify by canceling common factors: \[ \dfrac{5 \times \color{red}{\cancel{p}}}{\cancel{p}} \times 3 = \dfrac{\color{red}{\cancel{5}} \times p}{\cancel{5}} \times 1, \] which gives \[ 15 = p. \]

  3. Find \( w \) if \( \dfrac{31}{5} = \dfrac{w}{15} \)

    Solution

    Multiply both terms of the proportion by the denominators 5 and 15: \[ \color{Red} {5 \times 15} \times \dfrac{31}{5} = \color{Red} {5 \times 15} \times \dfrac{w}{15} \] Simplify by canceling: \[ \dfrac{\color{Red}{\cancel{5}} \times 15}{\cancel{5}} \times 31 = \dfrac{5 \times \color{Red}{\cancel{15}}}{\cancel{15}} \times w \] Do not multiply numbers: \[ 5 \times 31 = 5 \times w \] Divide both sides by 5: \[ \dfrac{5 \times 31}{5} = \dfrac{5 \times w}{5} \] Simplify: \[ 31 = w \]

  4. Find \( k \) if \( \dfrac{2k}{3} = \dfrac{20}{6} \)

    Solution

    Multiply both terms of the proportion by the denominators 3 and 6: \[ \color{Red} {3 \times 6} \times \dfrac{2k}{3} = \color{Red} {3 \times 6} \times \dfrac{20}{6} \] Simplify by canceling: \[ \dfrac{\color{Red}{\cancel{3}} \times 6}{\cancel{3}} \times 2k = \dfrac{3 \times \color{Red}{\cancel{6}}}{\cancel{6}} \times 20 \] Simplify: \[ 12k = 60 \] Divide both sides by 12: \[ \dfrac{12k}{12} = \dfrac{60}{12} \] Simplify: \[ k = 5 \]

  5. Solve the proportion \( \dfrac{3}{7} = \dfrac{y}{0} \) if possible

    Solution

    No solution because division by zero is not allowed in mathematics.

  6. Solve the proportion \( \dfrac{1}{4} = \dfrac{0}{x} \) if possible

    Solution

    The left side \( \dfrac{1}{4} \) is not equal to zero. The right side is either zero or undefined for \( x = 0 \). Hence, the proportion has no solution.

  7. If \( \dfrac{m}{4} = \dfrac{3}{12} \), then what is the value of \( m \)?

    Solution

    Reduce the right side fraction: \[ \dfrac{3}{12} = \dfrac{3 \div 3}{12 \div 3} = \dfrac{1}{4} \] Rewrite the proportion: \[ \dfrac{m}{4} = \dfrac{1}{4} \] Since denominators are equal, numerators must be equal: \[ m = 1 \]

  8. Find \( t \) if \( \dfrac{6}{14} = \dfrac{2t}{14} \)

    Solution

    The denominators are equal, so numerators must be equal: \[ 6 = 2t \] Divide both sides by 2: \[ \dfrac{6}{2} = \dfrac{2t}{2} \] Simplify: \[ 3 = t \]

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