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# Cross Multiplication – Techniques & Examples

Before we can discuss about the cross-multiplication process, let’s remind ourselves about the parts of a fraction. A fraction normally is a number that is written in the form a/b where a and b are integers and b is a non-zero.

The number at the top in a fraction is known as the numerator while the number at the bottom is known as the denominator. The numerator and denominator are separated by a slash line or division bar.

For instance, 4/5, 2/7, 1/3, 1/4 etc. are all examples of fractions. It is also important to note that, a rational expression similarly takes the form of a fraction a/b, where a and b are algebraic expressions.

**Examples** of rational expressions are; (x +5)/3, 2/x- 8, 3x/5 etc.

## What is Cross Multiplication?

In mathematics, cross multiplication is a process where a variable in an equation is determined by cross multiplying two fractions or expressions. Cross multiplication can be applied also compare fractions by multiplying the numerator of each fraction by the other’s denominator.

## How to Cross Multiply?

To perform cross multiplication, the numerator of the first fraction is multiplied with the denominator of the second fraction. Similarly, the denominator of the first fraction is multiplied by the numerator of the second fraction.

The two products are equated and the value of the variable is determined. In order to master how to do cross multiplication, let’s examine the following cross multiplication cases:

### How to cross multiply with a variable?

*Example 1*

Given, 9/x = 3/2

__Solution__

To find the value of x, we apply the cross-multiplication process where;

- Multiply the numerator of the first fraction by the denominator of the second fraction;

9* 2 =18

- Similarly multiply the denominator of the first fraction by the numerator of the second fraction;

x * 3 =3x

- Now equate the two products and divide both side of the equation by 3;

3x = 18

x =6

*Example 2*

Solve x/5 = 4/2

__Solution__

Apply the same procedures for cross multiplication;

- x * 2 = 2x
- 5 * 4 = 20

Now equate the two products;

2x = 20

x = 10

Cross Multiplying with two of the Same Variable

*Example 3*

(x + 3)/2 = (x +1)/1

__Solution__

In this case, the numerator of the first and second fraction are x +3 and x + 1 respectively.

Now, apply cross multiplication by multiplying the numerator of the first fraction by the denominator of the second fraction;

- (x + 3) * 1 = x + 3

Multiply dominator of 1^{ST }fraction by numerator of 2^{ND} fraction;

- 2 * (x + 1) = 2x + 2

Equate the two products and combine the like terms

*4x + 12 = 2x + 2.*

Isolate the variable x by adding -2x to both sides of the equation;

- 4x -2x +12 = 2x -2x + 2

= 2x + 12 = 2

Now add -12 to both sides,

- 2x + 12 -12 = 2 -12

2x = -10

x = -5

*Example 4*

Solve 8/ (x – 2) = 4/x

__Solution__

Cross multiply;

- 8 * x = 8x
- (x- 2) * 4= 4x – 8

Equate the two products and combine the like terms;

8x = 4x -8

Isolate the variable x;

- Add -4x to both sides of the equation;

8x – 4x = 8

4x = 8

x = 2

*Example 5*

Solve for x 2x/3 + x/2 = 5/6

__Solution__

In this case, we multiply each term by the LCM. The LCM of 3, 2 and 6 is 6, Therefore, the equation will be;

- (2x/3)6 + (x/2)6 = (5/6)6

= 4x + 3x = 5

Combine the like terms and divide both sides by 7;

7x = 5

x = 5/7

*Example 6*

Solve for x 4/10 = x/15

__Solution__

Cross multiply and equate the products;

4 * 15 = 10 * x

Divide both sides of the equation by 10;

x = 60/10

= 6

*Practice Questions*

- Solve the following:
- (x + 5)/x = (2x + 10)/3
- -6x + 2 = 12x/3
- -x/9 = -9/x
- To prepare a lemonade, 3 liters of water are mixed with 4 liters of lemon juice. How many liters of water can be mixed with 8 liters of lemon juice?
- An 8-meter flag post cast a shadow of 15 meters on the ground. How tall is an electric post which cast a shadow of 30 meters in the same condition?
- A fire engine has a capacity of holding 3000 gallons of water. If its nozzle can deliver 80 gallons of water per minute. Calculate:
- How many gallons of water can be delivered in 10 minutes?
- How long will it take for the tank to be empty?
- 4 gallons of paint can cover 800 square feet of a floor. Calculate the quantity of paint need to cover 200 square feet?
- A number when divided by 2, the result is equal to the 3 more than the number whole divided by 5. What is the number?
- The reciprocal of a positive rational number is 2 times the number itself. Determine the number.
- The ratio of w to x is equal to the ratio of y to z. If x = 2w and y = 3w, express z in terms of w.

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