Detailed Solutions to Questions on Cross Multiplication
Detailed solutions to the 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.


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Updated: 20 January 2017 (A Dendane)