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Introduction

Cross multiplication is a mathematical technique used to solve equations that involve fractions. It is a simple method that involves multiplying both sides of an equation by the same value to eliminate fractions and simplify the equation. In this article, we will explore what cross multiplication is, how it works, and some examples of how it can be used.

What is Cross Multiplication?

Cross multiplication is a method used to solve equations that involve fractions. It is a technique that can be used to simplify complex equations and solve for an unknown variable. The basic idea behind cross multiplication is that we can eliminate fractions by multiplying both sides of an equation by the same value.

For example, consider the equation:

To solve for x, we can use cross multiplication. We can multiply both sides of the equation by 6, the denominator of the fraction on the right-hand side of the equation. This gives us:

Simplifying the left-hand side of the equation gives us:

So the solution to the equation is:

This is an example of how cross multiplication can be used to solve equations that involve fractions.

How Does Cross Multiplication Work?

The basic idea behind cross multiplication is to eliminate fractions by multiplying both sides of an equation by the same value. When we multiply fractions, we multiply the numerators together and the denominators together. For example, if we have:

We can simplify this expression by multiplying the numerators together and the denominators together:

This tells us that if we have an equation that involves fractions, we can eliminate the fractions by multiplying both sides of the equation by the product of the denominators.

For example, if we have the equation:

We can eliminate the fractions by multiplying both sides of the equation by bd:

Simplifying the left-hand side of the equation gives us:

And simplifying the right-hand side of the equation gives us:

These two expressions are equivalent and represent the same equation. This is the basic idea behind cross multiplication.

Examples of Cross Multiplication

Let’s look at some examples of how cross multiplication can be used to solve equations.

Example 1:

To solve for x, we can use cross multiplication. We can multiply both sides of the equation by 9, the denominator of the fraction on the right-hand side of the equation. This gives us:

Simplifying the left-hand side of the equation gives us:

So the solution to the equation is:

Example 2:

To solve for x, we can use cross multiplication. We can multiply both sides of the equation by x, the denominator of the fraction on the right-hand side of the equation. This gives us:

Simplifying the left-hand side of the equation gives us:

So the solution to the equation is:

Multiplying both sides of the equation by 4 gives us:

Dividing both sides of the equation by 3 gives us:

Definition
Cross multiplication is a mathematical operation used to compare two or more fractions. It involves multiplying the numerator of one fraction by the denominator of the other fraction and vice versa. The result is two new fractions that are equivalent to the original fractions. These new fractions can then be compared or combined in various ways, depending on the specific problem.

Cross Multiplication Examples

Example 1: Comparing Fractions
Suppose we want to compare the fractions 1/2 and 2/3. To do this, we can cross multiply as follows:

1/2 = x/3

2x = 3

x = 3/2

Therefore, we can see that 1/2 is less than 2/3, as 3/2 is less than 2.

Example 2: Solving Equations
Suppose we want to solve the equation 3/x + 2/5 = 1/2. To do this, we can cross multiply as follows:

3/x + 2/5 = 1/2

6x + 2x = 15

8x = 15

x = 15/8

Therefore, we can see that x = 15/8 is the solution to the equation.

Example 3: Solving Proportions
Suppose we have the following proportion:

a/b = c/d

To solve for a, we can cross multiply as follows:

ad = bc

a = bc/d

Similarly, to solve for b, we can cross multiply as follows:

bd = ac

b = ad/c

Example 4: Comparing Mixed Numbers
Suppose we want to compare the mixed numbers 3 1/2 and 4 1/3. To do this, we first convert them to improper fractions:

3 1/2 = 7/2

4 1/3 = 13/3

Then, we cross multiply as follows:

7/2 = x/3

13x = 21

x = 21/13

Therefore, we can see that 3 1/2 is less than 4 1/3, as 21/13 is less than 2.

Example 5: Finding Unknown Variables
Suppose we have the following equation:

2/x + 3/y = 4

We want to solve for x and y. To do this, we can cross multiply as follows:

2y + 3x = 4xy

We can then solve for x or y by isolating the variable:

x = (2y – 4)/(3 – 4y)

y = (4x – 3)/(2 – 3x)

Cross Multiplication in Algebra

In algebra, cross multiplication is often used to solve linear equations involving fractions. For example, consider the equation:

3/4x + 1/2 = 5/8

To solve for x, we can cross multiply as follows:

3/4x + 1/2 = 5/8

24(3/4x + 1/2) = 24(5/8)

18x + 12 = 15

18x = 3

x = 1/6

Therefore, we can see that x = 1/6 is the solution to the equation.

Quiz

• What is cross multiplication?

Answer: Cross multiplication is a method used to solve proportions by multiplying the numerator of one fraction by the denominator of another fraction and vice versa.

• What is the formula for cross multiplication?

Answer: The formula for cross multiplication is: a/b = c/d can be written as ad = bc.

• Can cross multiplication be used to solve equations other than proportions?

Answer: No, cross multiplication can only be used to solve proportions.

• What is the first step in using cross multiplication to solve a proportion?

Answer: The first step is to write the proportion as two fractions with an equal sign in between.

• What is the second step in using cross multiplication to solve a proportion?

Answer: The second step is to cross multiply the fractions by multiplying the numerator of one fraction by the denominator of the other fraction.

• What is the third step in using cross multiplication to solve a proportion?

Answer: The third step is to simplify the resulting equation by combining like terms and solving for the variable.

• Can cross multiplication be used with decimals and mixed numbers?

Answer: Yes, cross multiplication can be used with decimals and mixed numbers.

• What is the purpose of cross multiplication?

Answer: The purpose of cross multiplication is to solve proportions and find the value of an unknown variable.

• Can cross multiplication be used to compare fractions?

Answer: Yes, cross multiplication can be used to compare fractions.

• Is cross multiplication the only method for solving proportions?

Answer: No, there are other methods for solving proportions such as using equivalent ratios or using a proportionality constant. However, cross multiplication is a commonly used method.

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