Detailed Solutions to Algebra Questions on Cross Multiplication

Detailed solutions to the algebra 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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