Cross-Multiplication Method – Formula, Derivation, Examples

In mathematics, we cross multiply the terms in an equation to determine the unknown values. The method of cross multiplication is mostly employed to identify the unknown variable in an equation. In the cross multiplication method, we multiply the numerator of the first fraction and the denominator of the second fraction, the numerator of the second fraction, and the denominator of the first fraction of the given expression or fraction. The cross multiplication method is a way to solve an equation that involves an unknown variable as part of two fractions that are set equal to each other. By using cross multiplication, we can compare two fractions to find which is the bigger one and the smaller one. When two fractions form an equation, the products of the numerator on one side and the denominator on the other side are equal. This equality property of quantities in product form is referred to as the cross-multiplication method.

Let a/b = c/d be an expression. Now, the formula for the cross multiplication is given as follows:

Note: The cross multiplication method is not applicable if any of the denominators are equal to zero, i.e., b and d = 0.

How to solve Cross-Multiplication?

Follow the steps given below when employing the method of cross multiplication.

Step 1: First, multiply the numerator on the left-hand side of the fraction with the denominator on the right-hand side of the fraction.

Step 2: Next, multiply the numerator on the right-hand side of the fraction with the denominator on the left-hand side of the fraction.

Step 3: Now, equate both the products.

Step 4: Finally, solve for the variable.

Example: Solve: (m + 1)/3 = 4/9.

Solution:

Given: (m + 1)/3 = 4/9.

By cross multiplying, we get;

⇒ (m + 1) × 9 = 4 × 3

⇒ 9m + 9 = 12

⇒ 9m = 12 – 9 = 3

Now, divide both sides of the equation by 9.

⇒ 9m/9 = 3/9

⇒ m = 1/3

Hence, the value of “m” is 1/3.

Cross-Multiplication with Variables on Both Sides

Let us suppose that we have the same variable present on both sides of the equation. Now, we can apply the cross multiplication method to determine the variable. Let us go through some examples to understand better.

Example: Solve: (x – 1)/2 = 8/(x – 1).

Solution:

Given: (x – 1)/2 = 8/(x – 1) 

By cross multiplying, we get;

(x – 1) × (x – 1) = 2 × 8

⇒ (x – 1)² = 16

⇒ (x – 1) = √16 = ±4

⇒ (x – 1) = +4

⇒ x = 4 + 1 = 5

(or)

⇒ (x – 1) = –4

⇒ x = –4 + 1 = –3

Hence, the value of x is –3 or 5.

Cross-Multiplication for Comparing Fractions

By using cross multiplication, we can compare two fractions to find which is the bigger one and the smaller one, or check whether the two fractions are equivalent or not. We have two methods to compare fractions.

Method 1:

For example, let us assume that 5/8 is equivalent to 10/24. 

Note: Two fractions are said to be equivalent if the cross multiplication of both the fractions results in equal values. That is, the product on the left-hand side (LHS) must be equal to the product on the right-hand side (RHS).

5/8 = 10/24

By cross multiplying, we get;

5 × 24 = 10 × 8

120 = 80

Here, the product on LHS i.e., 120 is not equal to the product on RHS, i.e., 80. So, our statement is false.

Hence, we can conclude that 5/8 is not equivalent to 10/24.

Method 2:

Follow the steps given below to compare the fractions if the two fractions have different denominators:

• Step 1: First, multiply the denominators of both fractions to get the same denominator.
• Step 2: Now, to get the numerator of the first fraction, multiply the numerator of one fraction with the denominator of the other.
• Step 3: Again, to get the numerator of the second fraction, multiply the denominator of the first fraction by the numerator of the other fraction.
• Step 4: Now, compare the two new fractions.

Example: Compare 2/3 and 4/9.

Solution:

Multiply the denominators, i.e., 3 and 9 to get;

3 × 9 = 27

So, 27 is the common denominator for the two fractions.

Now, we need to find the numerators for both fractions.

So, by cross-multiplication:

2 × 9 = 18

4 × 3 = 12

So, the two fractions are 18/27 and 12/27.

Now, if we compare both fractions, we can observe, that 18 is greater than 12, hence,

12/27 < 18/27 

⇒ 4/9 < 2/3.

So, 4/9 is less than 2/3.

Solved Examples on Quadratic Equations

Example 1: Solve 13/5 = a/2.

Solution:

Given: 13/5 = a/2

By cross multiplying, we get;

⇒ 13 × 2 = a × 5

⇒ 26 = 5a

Now, divide both sides of the equation by 5.

⇒ 26/5 = 5a/5

⇒ a = 26/5

Hence, the value of “a” is 26/5.

Example 2: Compare 4/7 and 3/4.

Solution:

Multiply the denominators, i.e., 7 and 4 to get;

7 × 4 = 28

So, 28 is the common denominator for the two fractions.

Now, we need to find the numerators for both fractions.

So, by cross-multiplication:

4 × 4 = 16

7 × 3 = 21

So, the two fractions are 16/28and 21/28.

Now, if we compare both fractions, we can observe, that 21 is greater than 16, hence,

16/56 < 21/56

⇒ 4/7 < 3/4.

So, 4/7 is less than 3/4.

Example 3: Solve: 3/(x+6) = 5/8.

Solution:

Given: 3/(x+6) = 5/8

By cross-multiplication, we get

3 × 8 = 5 × (x + 6)

⇒ 24 = 5x + 30

⇒ 5x = 24 – 30 = –6

Now, divide both sides of the equation by 5.

⇒ 5x/5 = –6/5

⇒ x = –6/5

Hence, the value of “x” is –6/5.

Example 4: If the cost of 10 mangoes is Rs. 350, then how much will two dozen such mangoes cost?

Solution:

Given that the cost of 10 mangoes = ₨ 350

So, the cost of 1 mango = 350/10  ————— (1)

Let the cost of two dozens of mangoes, i.e., 24 mangoes be “x”.

So, the cost of 1 mango = x/24————— (1)

From equations (1) and (2)

350/10 = x/24

By cross multiplication, we get;

350 × 24= 10 × x

⇒ x = (350 × 24)/10

⇒ x = 840

Hence, the cost of two dozen of mangoes is ₨ 840.

Example 5: Find whether 6/7 is equivalent to 36/42 or not.

Solution:

Two fractions are said to be equivalent if the cross multiplication of both the fractions results in equal values.

6/7 = 36/42

By applying cross-multiplication, we get;

6 × 42 = 36 × 7

252 = 252

Since both values are equal, both fractions are equivalent.

Hence, 6/7 is equivalent to 36/42.

FAQs on Cross-Multiplication

Question 1: What is meant by cross-multiplication?

Answer:

When two fractions form an equation, the products of the numerator on one side and the denominator on the other side are equal. This equality property of quantities in product form is referred to as the cross-multiplication method. The cross multiplication method is not applicable if any of the denominators are equal to zero, i.e., b and d ≠ 0.

If a/b = c/d, then ad = bc. 

Question 2: How to find a variable using cross multiplication?

Answer:

Multiply the numerator of the first fraction and the denominator of the second fraction, the numerator of the second fraction, and the denominator of the first fraction of the given expression or fraction, and equate both the products. Finally, solve for the variable.

Question 3: Can the denominators of the fractions be equal to zero?

Answer: 

The cross multiplication method is not applicable if any of the denominators are equal to zero as the fraction’s value will become an undefined value if the denominator of a fraction is equal to zero.

Question 4: How to find equivalent fractions using the cross-multiplication method?

Answer:

Two fractions are said to be equivalent if the cross multiplication of both the fractions results in equal values. That is, the product on the left-hand side (LHS) must be equal to the product on the right-hand side (RHS).

Question 5: How to compare two fractions with unlike denominators using the cross-multiplication method?

Answer:

Follow the steps given below to compare the fractions if the two fractions have different denominators

Step 1: First, multiply the denominators of both fractions to get the same denominator.
Step 2: Now, to get the numerator of the first fraction, multiply the numerator of one fraction with the denominator of the other.
Step 3: Again, to get the numerator of the second fraction, multiply the denominator of the first fraction by the numerator of the other fraction.
Step 4: Now, compare the two new fractions.