Detailed Solutions to Questions on Cross Multiplication

Detailed solutions to the questions on Cross Multiplication are presented.

  1. 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




  2. 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.


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Updated: 20 January 2017 (A Dendane)

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