Cross Multiplication in Proportions
The cross multiplication property for two equivalent fractions states that the product of the numerator of the first fraction and the denominator of the second fraction equals the product of the numerator of the second fraction and the denominator of the first fraction. $$ \frac{a}{b} = \frac{c}{d} \Longleftrightarrow a \cdot d = b \cdot c $$
In a proportion, this property asserts that the product of the means equals the product of the extremes.
$$ a:b = c:d \Longleftrightarrow a \cdot d = b \cdot c $$
What is it used for?
Cross multiplication is used to determine whether a proportion is valid or whether two fractions are equivalent.
A practical example
Example 1
Consider the following fractions:
$$ \frac{3}{9} \ , \ \frac{5}{15} $$
To determine whether they are equivalent, express them as an equality. Equivalent fractions represent the same value.
$$ \frac{3}{9} = \frac{5}{15} $$
Now compute the cross products:
$$ 3 \cdot 15 = 5 \cdot 9 $$
$$ 45 = 45 $$
Since the cross products are equal, the fractions are equivalent.
Example 2
Consider the following fractions:
$$ \frac{4}{7} \ , \ \frac{6}{10} $$
To determine whether they are equivalent, write:
$$ \frac{4}{7} = \frac{6}{10} $$
Then compute the cross products:
$$ 4 \cdot 10 = 7 \cdot 6 $$
$$ 40 = 42 $$
Here the equality does not hold because the cross products differ.
Therefore, the fractions are NOT equivalent.
Example 3
Consider the proportion:
$$ 2:8 = 3:12 $$
To verify whether the proportion is valid, compute the cross products of the extremes and the means:
$$ 2 \cdot 12 = 3 \cdot 8 $$
$$ 24 = 24 $$
Since the cross products are equal, the proportion is valid.
Example 4
Consider the proportion:
$$ 8:9 = 3:4 $$
To verify whether the proportion is valid, compute the cross products:
$$ 8 \cdot 4 = 9 \cdot 3 $$
$$ 32 = 27 $$
In this case, the equality does not hold because the cross products differ. Therefore, it is NOT a valid proportion.
Proof
Consider any two equivalent fractions:
$$ \frac{a}{b} \ , \ \frac{c}{d} $$
Since they are equivalent, they represent the same value:
$$ \frac{a}{b} = \frac{c}{d} $$
Using the property of equivalent fractions, multiply both the numerator and the denominator of the first fraction by d:
$$ \frac{a}{b} \cdot \frac{d}{d} = \frac{c}{d} $$ $$ \frac{ad}{bd} = \frac{c}{d} $$
Apply the same property again by multiplying both the numerator and the denominator of the second fraction by b:
$$ \frac{ad}{bd} = \frac{c}{d} \cdot \frac{b}{b} $$ $$ \frac{ad}{bd} = \frac{bc}{bd} $$
Using the invariant property of equations, multiply both sides of the equation by bd:
$$ \frac{ad}{bd} \cdot bd = \frac{bc}{bd} \cdot bd $$
Now simplify:
$$ \require{cancel} \frac{ad}{\cancel{bd}} \cdot \cancel{bd} = \frac{bc}{\cancel{bd}} \cdot \cancel{bd} $$
This yields the cross multiplication identity:
$$ ad = bc $$
Note. To establish the cross multiplication property for a proportion, recall that a proportion is the equality between two equivalent fractions $$ a:b = c:d \Longleftrightarrow \frac{a}{b} = \frac{c}{d} $$ Once this equivalence is recognized, the proof follows exactly the same steps.
And so on.
