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 terms of the proportion by the denominators 2 and 8.
   \( \color{Red} {2\times8}\,\dfrac{x}{2}=\color{Red} {2\times 8}\,\dfrac{4}{8} \)
   Simplify.
   \( {\color{Red} {\cancel{2}\times8}}\,\dfrac{x}{\cancel{2}}={\color{Red} {2\times \cancel{8}}}\,\dfrac{4}{\cancel{8}} \)
   Simplify.
   \( 8x = 8 \)
   Divide both sides of the equation by the factor of x and simplify.
   \( \dfrac{8x}{8}= \dfrac{8}{8} \)
   \( x = 1 \)
   
2. Find \( p \) if \( \dfrac{3}{p} = \dfrac{1}{5} \)
   Solution
   Multiply both terms of the proportion by the denominators p and 5.
   \( \color{Red} {5\times p}\,\dfrac{3}{p}=\color{Red} {5\times p}\,\dfrac{1}{5} \)
   Simplify.
   \( {\color{Red} {5\times \cancel{p}}}\,\dfrac{3}{\cancel{p}}={\color{Red} {\cancel{5}\times p}\,\dfrac{1}{\cancel{5}} } \)
   Simplify.
   \( 15 = p \)
   
3. If \( \dfrac{31}{5} = \dfrac{w}{15}\), then what is the value of \( w \)?
   Solution
   Multiply both terms of the proportion by the denominators 5 and 15.
   \( \color{Red} {5\times 15}\,\dfrac{31}{5}=\color{Red} {5\times 15}\,\dfrac{w}{15} \)
   Simplify.
   \( {\color{Red} {\cancel{5}\times 5}}\,\dfrac{31}{\cancel{5}}={\color{Red} {5\times \cancel{15}}\,\dfrac{w}{\cancel{15}} } \)
   but do not multiply numbers.
   \( 5 \times 31 = 5 \times w \)
   Divide both sides by 5
   \( \dfrac{5 \times 31}{5} = \dfrac{5 \times w}{5} \)
   and 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.
   \( \color{Red} {3\times 6}\,\dfrac{2k}{3}=\color{Red} {3\times 6}\,\dfrac{20}{6} \)
   Simplify.
   \( \color{Red} {\cancel{3}\times 6}\,\dfrac{2k}{\cancel{3}}=\color{Red} {3\times \cancel{6}}\,\dfrac{20}{\cancel{6}} \)
   Simplify.
   \( 12k = 60 \)
   Divide both sides by the factor of k (12).
   \( \dfrac{12k}{12} = \dfrac{60}{12} \)
   and simplify.
   \( k = 5 \)
   
5. Solve the proportion \( \dfrac{3}{7} = \dfrac{y}{0} \) if possible.
   Solution
   No solution because the division by zero is not allowed in mathematics.
   
6. Solve the proportion \( \dfrac{1}{4} = \dfrac{0}{x} \) if possible.
   Solution
   The left side 1/4 is not equal to zero. The right side is either zero or undefined for x = 0. Hence the above proprotion does not have a solution.
   
7. If \( \dfrac{m}{4} = \dfrac{3}{12} \), then what is the value of m?.
   Solution
   Note here that the fraction on the right side 3 / 12 may be reduced to 1 / 4 by dividing both numerator and denominator by 3 as follows.
   \( \dfrac{3}{12} = \dfrac{3 \div 3}{12\div 3} = \dfrac{1}{4} \)
   Hence the given proportion may be written as.
   \( \dfrac{m}{4} = \dfrac{1}{4} \)
   The two fractions have the same denominator and therefore their numerators must be equal. Hence
   \( m = 1 \)
   
8. Find \(t \) if \( \dfrac{6}{14} = \dfrac{2t}{14} \).
   Solution
   Note the two fractions making the proportion given above have equal denominator and therefore their numerator must be equal. Hence
   \( 6 = 2 t \)
   Divide by the coefficient of t (2)
   \( 6 / 2 = 2 t / 2 \)
   and simplify
   \( 3 = t \)