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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