Cross Multiplication: Definition, Steps, and Examples

Cross multiplication: Mathematics is all about logic, which can be understood by analysing patterns and applying shortcuts. One of the most useful method for solving math problems involving fractions, ratios, and proportions is cross multiplication. 

Even in real-life scenarios, while comparing prices across quantities or adjusting ingredient ratios in cooking, this multiplication method becomes handy. This blog explains cross-multiplication techniques and the scope of applications to help you handle typical calculations in practical life and make informed decisions. 

Read More: Unit Fraction

What is Cross Multiplication?

Cross multiplication is a technique used in mathematical calculations to solve equations involving two fractions. It is a simple process of multiplying the first fraction's numerator by the second's denominator and vice versa. The method eliminates the denominators of fractions and helps compare the two numbers directly. 

For example, a/b and c/d are two fractions and the relationship between them is given by the following equation:

a/b = c/d

Then, we can cross-multiply the numerators and set them equal as follows:

a x d = b x c

When to Use Cross Multiplication?

Cross multiplication forms the basis for solving word problems in algebra, geometry, and arithmetic. You must understand when to use it for calculation. Cross-multiplication becomes useful in the following cases:

• When solving an equation with two equal fractions.

•  When comparing two ratios, to check if they are equivalent.

• When dealing with practical scenarios involving ratios and proportions.

• When working with unknown values in fraction-based equations. 

Read More:  Division of Fractions

How to Do Cross Multiplication?

Learning how to cross-multiply helps solve many mathematical problems by simplifying equations involving complex fractions. Here, we will discuss how cross-multiplication is used to solve different mathematical problems with some cross multiplication examples. 

Cross Multiplication to Find the Value of a Variable

If a condition is developed where two fractions are set equal and any numerator or denominator has an unknown value, we use cross multiplication to solve it.

For example, 5/8 = p/24, where p is a variable.

Multiply the first fraction's numerator by the second's denominator and vice versa, and set them equal.

It gives the expression as follows:

5 x 24 = p x 8

Or, 8p = 120

Now, by solving the equation for p, we get:

p = 120/8 = 15

Cross Multiplication to Verify if Two Ratios are Equal

Cross multiplication workds best if two ratios are given, and we must verify whether they are equivalent fractions.

For example, we use the cross-multiplication method to find whether the fractions 3/7 and 9/21 are equivalent.

By cross multiplying, we get:

3 x 21 = 9 x 7

Or, 63 = 63

Therefore, 3/7 and 9/21 are equivalent fractions.

Read More:  Prime Factorization Method

Cross Multiplication when Adding or Subtracting Fractions

Cross multiplication is a method used to add or subtract fractions that have different denominators. You use this when you want to avoid finding the Least Common Denominator (LCD).

For example, to add two fractions, 4/5 and 7/8, we will proceed as follows:

Step 1: Cross multiply 4 with 8 and 7 with 5 to get 4 x 8 = 32 and 7 x 5 =35

Step 2: We add 32 and 35 to get 32 + 35 = 67

Step 3: We multiply the denominators 5 and 8 to get 5 x 8 = 40

Step 4: We will take 67 as a numerator and 40 as a denominator to form a new fraction which is the sum of the given fractions.

So, we can say 4/5 + 7/8 = 67/40

Similarly, if we subtract 7/8 from 4/5, we get the answer as follows:

7/8 – 4/5

= (7x5 - 4x8)/8x5

= (35-32)/40

= 3/40

Read More: Improper Fractions

Tips to Remember While Cross-Multiplying

Cross multiplication is simple, but to perform it accurately, you must keep in mind a few tips as follows:

• Always check that the two fractions or ratios are set equal.

• Multiply the numerator with the other denominator and set equality.

• Isolate the variable, if any, after cross-multiplying to find its value.

• Simplify, if possible, before cross multiplying to avoid handling large numbers. 

Also read: Find the Sum - Definition, Formulas and Examples

Why Cross Multiplication Works?

Cross multiplication is based on the principle of equality between two fractions.

Suppose there are two fractions such that m/n = p/q

We multiply both sides by n × q and get:

(m/n) x n x q = (p/q) x n x q

After simplifying, we get:

m x q = p x n

The result is the same as cross multiplication. This method clears the denominators and shows that the equality of fractions leads to equal cross-products.

Cross Multiplication Examples

1. Solve for ‘a’ if 21/a = 63/72

Solution:

21/a = 63/72

Or, 63 x a = 21 x 72

Or, a = (21 x 72)/63

Or, a = (7 x 24)/7

Or, a = 24

2. Are the fractions 7/13 and 9/16 equivalent?

Solution:

Using cross multiplication, we get 7 x 16 = 112 and 9 x 13 = 117

Since these values are unequal, 7/13 and 9/16 are not equivalent. 

3. If a car travels 120 km in 3 hours, how long will it take to travel 200 km at the same speed? Solve using the cross multiplication method.

Solution:

With the same speed, the ratio of distance to speed will be the same in both cases.

So, if the time taken to travel 200 km is 't' hours, we can say 120/3 = 200/t

Now, by cross multiplying, we get:

120 x t = 3 x 200

Or, t = (3 x 200)/120

Or, t = 5

Answer: The time taken to travel 200 km is 5 hours.

Cross-multiplication is a must-know skill for performing basic arithmetic and algebraic calculations. Once you understand cross multiplication, you can easily solve various math problems, including solving proportions, working with word problems, or simply checking the equality of two ratios.

Also read: Fractions

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