# Detailed Solutions to Algebra Questions on Cross Multiplication

Detailed solutions to the algebra questions on Cross Multiplication are presented.

 Solve the equations a) $\dfrac{x}{6} = \dfrac{3}{2}$ b) $\dfrac{1}{3x} = \dfrac{2}{24}$ c) $\dfrac{3}{2} = \dfrac{12}{4x}$ d) $\dfrac{4}{6} = \dfrac{x}{9}$ e) $2 = \dfrac{x}{14}$ f) $\dfrac{2}{x+2} = \dfrac{1}{7}$ Solution a) Use cross multiplication to rewrite equation as follows. 2 x = 3 � 6 Simplify. 2 x = 18 Divide the two sides of the equation by 2. 2 x / 2 = 18 / 2 Simplify to solve for x. x = 9 b) Cross multiply the denominators and numerators and rewrite the given equation as follows. 1 � 24 = 3 x � 2 Simplify. 24 = 6 x Divide both sides by the coefficient of x which is 6. x = 24 / 6 = 4 c) Use cross multiplication to rewrite the given equation without denominators as follows. 3 � 4 x = 2 � 12 Simplify. 12 x = 24 Divide both sides by 12 to find x. x = 24 / 12 = 2 d) Cross multiply the denominators and numerators and rewrite the given equation as follows. 4 � 9 = 6 � x Simplify and solve for x. x = 36 / 6 = 6 e) Change the 2 in the equation by the fraction $\dfrac{2}{1}$ and rewrite equations as follows. $\dfrac{2}{1} = \dfrac{x}{14}$ Cross multiply. 2 � 14 = 1 � x Simplify to find x. 28 = x f) Cross multiply the denominators and numerators and rewrite the given equation as follows. 2 � 7 = (x + 2) � 1 Simplify and solve for x. 14 = x + 2 x = 14 - 2 = 12 Which of the following pairs of fractions are equivalent (equal)? a) $\dfrac{5}{6}$ and $\dfrac{15}{18}$ b) $\dfrac{5}{3}$ and $\dfrac{20}{13}$ c) $\dfrac{25}{35}$ and $\dfrac{5}{7}$ d) $\dfrac{23}{7}$ and $\dfrac{46}{17}$ Solution Important Definition to be used in the solutions below: We define the cross multiplication quantities A and B as follows: A is the product of the numerator of the first fraction by the denominator of the second fraction B is the product of the denominator of the first fraction by the numerator of the second fraction a) Find the cross multiplication quantities A and B for the two fractions in part a) above. A = 5 � 18 = 90 B = 6 � 15 = 90 Compare A and B. They are equal. Hence the two fractions are equal and we can write $\dfrac{5}{6} = \dfrac{15}{18}$ b) Cross multiply the two fractions to find A and B. A = 5 � 13 = 65 B = 3 � 20 = 60 Compare A and B. They are not equal. Hence the two fractions are not equal. c) Find A and B by cross multiplication of the two fractions. A = 25 � 7 = 175 B = 35 � 5 = 175 Compare A and B. They are equal. Hence the two fractions are equal. $\dfrac{25}{35} = \dfrac{5}{7}$ d) Cross multiply the two fractions and find A and B. A = 23 � 17 = 391 B = 7 � 46 = 322 Compare A and B: They are not equal. Hence the two fractions are not equal.