Detailed Solutions to Questions on Math Proportions
Detailed solutions the questions on proportions are presented with full explanations.

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.
\( \dfrac{8x}{8}= \dfrac{8}{8} \)
\( x = 1 \)

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.
\( 15 = p \)

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.
\( 31 = w \)

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.
\( k = 5 \)

Solve the proportion \( \dfrac{3}{7} = \dfrac{y}{0} \) if possible.
Solution
No solution because the division by zero is not allowed in mathematics.

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.

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
\( m = 1 \)

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
\( 3 = t \)


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