Detailed Solutions to Algebra Questions on Cross Multiplication

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

  1. Solution

    a) Use cross multiplication to rewrite the equation:

    \[ 2x = 3 \times 6 \] Simplify: \[ 2x = 18 \] Divide both sides by 2: \[ \frac{2x}{2} = \frac{18}{2} \implies x = 9 \]

    b) Cross multiply denominators and numerators:

    \[ 1 \times 24 = 3x \times 2 \] Simplify: \[ 24 = 6x \] Divide both sides by 6: \[ x = \frac{24}{6} = 4 \]

    c) Use cross multiplication:

    \[ 3 \times 4x = 2 \times 12 \] Simplify: \[ 12x = 24 \] Divide both sides by 12: \[ x = \frac{24}{12} = 2 \]

    d) Cross multiply:

    \[ 4 \times 9 = 6 \times x \] Simplify and solve for \(x\): \[ 36 = 6x \implies x = \frac{36}{6} = 6 \]

    e) Express 2 as \(\frac{2}{1}\) and cross multiply:

    \[ \frac{2}{1} = \frac{x}{14} \] Cross multiply: \[ 2 \times 14 = 1 \times x \] Simplify: \[ 28 = x \]

    f) Cross multiply:

    \[ 2 \times 7 = (x + 2) \times 1 \] Simplify and solve for \(x\): \[ 14 = x + 2 \implies x = 14 - 2 = 12 \]

  2. Solution

    Definition: For two fractions, define cross multiplication quantities:

    a) Calculate \(A\) and \(B\) for \(\frac{5}{6}\) and \(\frac{15}{18}\):

    \[ A = 5 \times 18 = 90 \] \[ B = 6 \times 15 = 90 \] Since \(A = B\), the fractions are equal: \[ \frac{5}{6} = \frac{15}{18} \]

    b) Calculate \(A\) and \(B\) for \(\frac{5}{3}\) and \(\frac{20}{13}\):

    \[ A = 5 \times 13 = 65 \] \[ B = 3 \times 20 = 60 \] Since \(A \neq B\), the fractions are not equal.

    c) Calculate \(A\) and \(B\) for \(\frac{25}{35}\) and \(\frac{5}{7}\):

    \[ A = 25 \times 7 = 175 \] \[ B = 35 \times 5 = 175 \] Since \(A = B\), the fractions are equal: \[ \frac{25}{35} = \frac{5}{7} \]

    d) Calculate \(A\) and \(B\) for \(\frac{23}{7}\) and \(\frac{46}{17}\):

    \[ A = 23 \times 17 = 391 \] \[ B = 7 \times 46 = 322 \] Since \(A \neq B\), the fractions are not equal.

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