Cross Multiplication: Steps, Comparisons, and Solved Examples

Cross multiplication is a fundamental mathematical technique used to compare and solve proportions or equations involving fractions. It is a simple yet powerful method that allows us to find unknown values by manipulating the ratios between different quantities. 

• Cross multiplication is the process of multiplying the numerator of one fraction by the denominator of another fraction to generate an equal equation. 
• By doing so, isolation of the unknown variable and determining its value can be possible. 
• This technique is particularly useful in scenarios where there are two fractions or ratios and need to determine if they are equal or find the missing value. 
• Cross multiplication provides a straightforward approach to solving such problems efficiently.

Key Terms: Numerator, Denominator, Fraction, Ratio, Variable, Missing Value, Equations

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What is Cross Multiplication?

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Cross multiplication is a mathematical operation used to solve equations or proportions involving fractions. It is a method that allows to compare the ratios of different quantities and find unknown values. Cross multiplication involves multiplying the numerator of one fraction by the denominator of another fraction, and vice versa. 

• This process is applied when there are two fractions or ratios and need to determine if they are equal or find the missing value.
• To perform cross multiplication, an equation is set up with two fractions on either side, separated by an equal sign. 
• The numerator of the first fraction is multiplied by the denominator of the second fraction, while the numerator of the second fraction is multiplied by the denominator of the first fraction. 
• The resulting equation equates these two products.
• By cross-multiplying, we create an equivalent equation that helps to isolate and solve for the unknown variable. 
• The equation can be rearranged to find the value of the unknown by dividing both sides or performing any necessary arithmetic operations.

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Cross Multiplication Illustration

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Consider two fractions or ratios:

a/b = c/d

Cross multiplication is performed by multiplying the numerator of the first fraction (a) by the denominator of the second fraction (d), and the numerator of the second fraction (c) by the denominator of the first fraction (b). This gives us the equation:

a * d = c * b

This equation is known as the cross multiplication equation. By cross-multiplying, an equivalent equation is created that helps to find the unknown value or determine if the two fractions are equal.

Then rearranging the equation to solve for the unknown variable. To find the value of 'a', both sides of the equation are divided by 'd'

a = (c * b) / d

Similarly, To find the value of 'b', both sides of the equation are divided by 'c':

b = (a * d) / c

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Steps to Cross Multiply

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Suppose we have the following equation:

2/3 = 4/6

Steps to cross multiply and determine if the two fractions are equal:

Step 1: Write down the given equation:

2/3 = 4/6

Step 2: Cross-multiply the first fraction's numerator (2) by the denominator of the second fraction (6), and the second fraction's numerator (4) by the denominator of the first fraction (3):

(2 * 6) = (4 * 3)

Step 3: Simplify both sides of the equation:

12 = 12

Step 4: Since both sides of the equation are equal (12 = 12), it can be concluded that the two fractions are equal.

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Comparing Fractions by Cross Multiply

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Comparing fractions can be easily done using cross multiplication, a powerful technique that allows one to determine which fraction is greater or if they are equal. Cross multiplication provides a straightforward method to compare fractions by examining the products of their numerators and denominators.

Steps to compare fractions using cross multiplication:

Step 1: Write down the two fractions to compare.

Let's consider the fractions 3/4 and 5/8.

Step 2: Cross-multiply the first fraction's numerator by the denominator of the second fraction, and the second fraction's numerator by the denominator of the first fraction.

(3 * 8) and (5 * 4)

Step 3: Simplify both products.

3 * 8 = 24

5 * 4 = 20

Step 4: Compare the products obtained in Step 3.

Since 24 is greater than 20, we conclude that 3/4 is greater than 5/8.

By using cross multiplication, it can be seen that the product of the numerator and denominator of the first fraction (3/4) is larger than the product of the numerator and denominator of the second fraction (5/8). Therefore, it can be determined that 3/4 is greater than 5/8.

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Comparing Ratios by Cross Multiply

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Comparing ratios can be effectively done using cross multiplication, a technique that allows us to determine the relationship between two sets of ratios. Cross multiplication provides a convenient method to compare ratios by examining the products of their corresponding terms.

Steps to compare ratios using cross multiplication:

Step 1: Write down the two ratios you want to compare.

Let's consider the ratios 2:5 and 3:8.

Step 2: Cross multiply by multiplying the numerator of the first ratio by the denominator of the second ratio and the numerator of the second ratio by the denominator of the first ratio.

(2 * 8) and (3 * 5)

Step 3: Simplify both products.

2 * 8 = 16

3 * 5 = 15

Step 4: Compare the products obtained in Step 3.

Since 16 is greater than 15, it can be concluded that the ratio 2:5 is greater than the ratio 3:8.

By using cross multiplication, it can be seen that the product of the numerator and denominator of the first ratio (2:5) is larger than the product of the numerator and denominator of the second ratio (3:8). Therefore, the determined ratio 2:5 is greater than the ratio 3:8.

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Cross Multiply With One Variable

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When cross-multiplying with one variable, we are usually solving for the unknown variable in an equation or proportion. The following are the stages involved in the calculation:

Step 1: Write down the equation or proportion with one variable. For example

2/3 = x/5

Step 2: Cross multiplies by multiplying the numerator of the first fraction (2) by the denominator of the second fraction (5) and the numerator of the second fraction (x) by the denominator of the first fraction (3):

(2 * 5) = (x * 3)

Step 3: Simplify both sides of the equation:

10 = 3x

Step 4: Isolate the variable by dividing both sides of the equation by the coefficient of the variable (3 in this case):

10/3 = x

Step 5: Simplify the division to obtain the value of the variable:

x = 10/3

Step 6: If necessary, convert the fraction to a mixed number or decimal:

x ≈ 3.33

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Cross Multiply With Multiple Variables

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Cross multiplication with variables on both sides typically involves solving equations where variables appear in both the numerators and denominators. The steps for performing cross-multiplication with variables on both sides are as follows:

Step 1: Write down the equation with variables on both sides. For example:

(2x + 3) / (4x - 5) = (6x + 1) / (8x + 2)

Step 2: Cross multiply by multiplying the numerator of the first fraction (2x + 3) by the denominator of the second fraction (8x + 2), and the numerator of the second fraction (6x + 1) by the denominator of the first fraction (4x - 5):

(2x + 3)(8x + 2) = (6x + 1)(4x - 5)

Step 3: Simplify both sides of the equation by expanding the products.

(16x² + 4x + 24x + 6) = (24x² - 30x + 4x - 5)

Step 4: Combine like terms on both sides.

16x² + 28x + 6 = 24x² - 26x - 5

Step 5: Bring all terms to one side of the equation to form a quadratic equation.

0 = 24x² - 16x² - 28x + 26x - 5 - 6

0 = 8x² - 2x - 11

Step 6: Simplify the quadratic equation further.

In this case, the quadratic equation cannot be easily factored, so by the quadratic formula:

x = (-b ± √(b² - 4ac)) / (2a)

Plugging in the values, a = 8, b = -30, c = -11:

x = (-(-30) ± √((-30)² - 4 * 8 * (-11))) / (2 * 8)

Simplifying further:

x = (30 ± √(900 + 352)) / 16

x = (30 ± √1252) / 16

Step 8: Simplify the square root if possible.

x = (30 ± √(4 * 313)) / 16

x = (30 ± 2√313) / 16

Therefore, the solutions to the equation (2x + 3)/(4x - 5) = (6x + 1)/(8x + 2) are:

x = (30 + 2√313) / 16

x = (30 - 2√313) / 16

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Cross Multiply to Solve Equations Involving Multiple Variable

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Cross multiplication with multiple variables involves solving equations or systems of equations that contain more than one unknown. The steps for performing cross multiplication with multiple variables are as follows:

Step 1: Write down the equations or system of equations with multiple variables. For example:

2x + 3y = 10

4x - 2y = 6

Step 2: Choose one equation and isolate one variable in terms of the other. Let's isolate 'x' in terms of 'y' in the first equation:

2x = 10 - 3y

x = (10 - 3y) / 2

Step 3: Substitute the expression for the isolated variable into the other equation(s). Substituting the expression for 'x' into the second equation gives:

4((10 - 3y) / 2) - 2y = 6

Step 4: Simplify the equation(s) by performing the necessary arithmetic operations.

Simplifying the equation obtained in Step 3 gives: 20 - 6y - 2y = 6

Step 5: Combine like terms and isolate the remaining variable(s).

Simplifying the equation further gives: 20 - 8y = 6

-8y = 6 - 20

-8y = -14

y = -14 / -8

y = 7/4

Step 6: Substitute the value of the solved variable(s) back into one of the original equations to find the value of the remaining variable(s). Using the first equation and substituting 'y' gives:

2x + 3(7/4) = 10

2x + 21/4 = 10

2x = 10 - 2¼

2x = 40/4 - 2¼

2x = 19/4

x = (19/4) / 2

x = 19/8

Step 7: Verify the answer by replacing the variable values into the original equations and checking that both equations are satisfied.

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Solved Examples

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Example 1: Solve the equation using cross multiplication: (4x + 3)/(7x - 2) = (2x - 1)/(3x + 4)

Solution: Cross multiply the fractions:

(4x + 3)(3x + 4) = (7x - 2)(2x - 1)

Expand and simplify both sides:

12x² + 16x + 9x + 12 = 14x² - 7x - 4x + 2

Combine like terms:

12x² + 25x + 12 = 14x² - 11x + 2

Move all terms to one side to obtain a quadratic equation:

12x² - 14x² + 25x + 11x + 12 - 2 = 0

-2x² + 36x + 10 = 0

Solve the quadratic equation using factoring, completing the square, or the quadratic formula. In this case, let's use the quadratic formula:

x = (-b ± √(b² - 4ac)) / (2a)

Substituting the values:

x = (-36 ± √(36² - 4(-2)(10))) / (2(-2))

x = (-36 ± √(1296 + 80)) / (-4)

x = (-36 ± √1376) / (-4)

Therefore, the solutions to the quadratic equation are:

x ≈ -2.37 or x ≈ 9.37

Example 2: Solve the equation using cross multiplication: (5x - 2)/(3x + 1) = (2x + 3)/(4x - 5)

Solution: Cross multiply the fractions:

(5x - 2)(4x - 5) = (3x + 1)(2x + 3)

Expand and simplify both sides:

20x² - 25x - 8x + 10 = 6x² + 9x + 2x + 3

Combine like terms:

20x² - 33x + 10 = 6x² + 11x + 3

Move all terms to one side to obtain a quadratic equation:

20x² - 6x² - 33x - 11x + 10 - 3 = 0

14x² - 44x + 7 = 0

Solve the quadratic equation using factoring, completing the square, or the quadratic formula. In this case, let's use the quadratic formula:

x = (-b ± √(b² - 4ac)) / (2a)

Substituting the values:

x = (-(-44) ± √((-44)² - 4(14)(7))) / (2(14))

x = (44 ± √(1936 - 392)) / 28

x = (44 ± √1544) / 28

Therefore, the solutions to the quadratic equation are:

x ≈ 3.29 or x ≈ 0.18

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Things to Remember

• Cross multiplication is a technique used to compare fractions, solve proportions, and manipulate equations involving fractions and ratios.
• It is primarily used to determine the relative sizes of fractions or ratios by comparing the products of their corresponding terms.
• The process involves multiplying the numerator of one fraction by the denominator of the other fraction and vice versa.
• Cross multiplication is particularly useful when comparing fractions or ratios to determine which is greater or if they are equal.
• It is also used to solve proportions by setting up an equation with two ratios and finding the value of the unknown variable.
• In equations involving fractions, cross multiplication can be used to manipulate the equation and isolate the variable.
• By cross-multiplying and simplifying the equation, the unknown variable can be found.
• Cross multiplication is not applicable to other mathematical operations like addition, subtraction, multiplication, or division, as it is specific to fractions and ratios.