How do you do cross multiplication step by step?

In Mathematics, we cross multiply the values in an equation, to find the unknown values. The cross Multiplication method is mostly used to find the unknown variable in an equation. By cross multiplying the given expression or fractions, we multiply the numerator and denominator.

Table of Contents Show

How to Cross Multiply?
Cross Multiply with Variables on Both Sides
Cross Multiply to Compare Fractions
Problems and Solutions
Practice Questions
Definition of Cross Multiply
Steps to Cross Multiply
Comparing Fractions by Cross Multiply
Comparing Ratios by Cross Multiply
Cross Multiply With One Variable
Cross Multiply With Two Variables
Related Topics
FAQs on Cross Multiply
What are the Steps to Cross Multiply?
Do You Cross Multiply When Multiplying Fractions?
What is Cross Multiply Used for?
Can You Cross Multiply Ratios?
How Do You Cross Multiply?

If a/b = c/d is an expression, then the formula of cross multiply can be given as:

a/b = c/d

a × d = b × c

Note: Cross multiplication is not applicable if any of the denominators (b and d) is equal to zero.

We can cross multiply two fractions to compare them and find which is the bigger one. Also, when we add or subtract unlike fractions, cross multiplication is widely used. Let us learn here the cross multiplication to solve equations with unknown quantities.

Also, check:

How to Cross Multiply?

Direct multiplication of two or more fractions is an easy method. We multiply the numerators and denominators together, respectively. For example, ⅔ and ⅘ are two fractions. Thus, the product of both fractions is:

⅔ x ⅘ = (2 x 4)/(3 x 5) = 8/15

But what if we cross multiply the two fractions?

When we cross multiply in a given expression, it means the numerator on the left-hand side of the equation is multiplied by the denominator of the right-hand side. In the same way, the denominator of the left-hand side is multiplied by the numerator of the right-hand side.

After cross multiplication, the product is evaluated on both sides and then the variable is determined.

Step 1: Multiply the numerator of LHS to the denominator of RHS
Step 2: Multiply the denominator of LHS to the numerator of RHS
Step 3: Equate the two products and solve for the unknown quantity or variable

Let us see examples on how to cross multiply.

Examples

Example 1. Solve ⅔ = x/5

⅔ = x/5

By cross multiplying, we get;

2 × 5 = 3 × x

10 = 3x

Dividing both sides of the equation by 3, we get;

10/3 = x

Or x = 10/3

Example 2: Solve: 9/p = 3/2

Clearly we need to find the value of p by cross multiplication.

Therefore,

Cross multiply 9/p and 3/2 to get;

9 × 2 = p × 3

18 = 3p

Dividing both sides of the equation by 3, we get;

18/3 = p

p = 6

Cross Multiply with Variables on Both Sides

If we have the same variable, present on both sides of the equation, then cross multiply to find the variable. Let us understand with examples.

Example:

Solve X/5 = 5/X.

X/5 = 5/X

Cross multiply to get;

X × X = 5 × 5

X2 = 25

X = √25

X = +5 or -5

Cross Multiply to Compare Fractions

To check and compare whether two fractions are equivalent, we can use the cross multiplication method. There are two methods to compare fractions. Let us learn both methods.

Method 1

Suppose, if we say, ⅔ is equivalent to 4/9, then the cross multiplication of both the fractions should result in equal values on both the sides of equal sign (=).

⅔ = 4/9

After cross-multiplication, we get;

2 × 9 = 3 × 4

18 = 12

But 18 is not equal to 12 and hence the given expression is false.

So, we can conclude that ⅔ is not equivalent to 4/9.

Method 2

If we have two fractions with different denominators, then follow the below steps to compare the fractions.

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

Example:

Compare 2/7 and ⅜

Multiply 7 and 8 to get;

7 x 8 = 56

Thus, 56 is the common denominator for the two fractions.

Now we need to find the numerators for both fractions.

Thus, cross multiply 2 by 8 and 7 by 3;

2 x 8 = 16

7 x 3 = 21

So, the two fractions are 16/56 and 21/56.

Now if we compare both fractions, we can see, 21 is greater than 16, therefore,

16/56 < 21/56

2/7 < 3/8

Problems and Solutions

Q.1: Solve (x + 3)/2 = (x +1)/1.

Solution: Given,

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

By cross multiplication, we get;

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

x + 3 = 2x + 2

Subtract x from both sides.

x + 3 – x = 2x + 2 – x

3 = x + 2

Subtract 2 from both sides.

3 – 2 = x + 2 – 2

1 = x

x = 1

Q.2: If 8 chocolates cost Rs.40. How much will 12 such chocolates cost?

Solution: Given, cost of 8 chocolates = Rs.40

Cost of 1 chocolate = 40/8 ……(i)

Suppose, 12 chocolates cost Rs.x.

So, cost of 1 chocolate = x/12 …(ii)

Thus, from equation (i) and (ii), we get;

40/8 = x/12

Now, on cross multiplication, we get;

40 × 12 = 8 × x

x = 480/8

x = 60

Hence, the cost of 12 chocolates is Rs.60.

Practice Questions

Find the value of x, if (x+1)/5 = 2/(x-1)
What is the value of y, if y/6 = 6/y?
Solve for x if 4/10 = x/15.
Find if ⅘ is equivalent to 20/25.
Compare ⅞ to 9/10 using cross multiplication method.

To find the value of an unknown quantity, we need to cross multiply the fractions in a given equation. If a/b = c/d, then ad = bc is the cross multiplication.

Cross multiply the numerators and denominators of the fractions, that are present in the either side of the equation. Then solve for the unknown variable.

Consider the negative sign, while cross multiplying the fractions. For example, ½ = -x/3 2 (-x) = 1 (3) -2x = 3

x = -3/2

If ⅝ = x/2, then by cross multiplication, we get; 5 (2) = 8 (x) 10 = 8x x = 10/8

x = 5/4

The denominator of the fraction cannot be zero, because the whole fraction will represent an undefined value.

Cross multiply is the process of multiplication of numbers in the form of a fraction. The process of multiplying the numerator of one fraction with the denominator of the other fraction on the other side of an equals to symbol is considered as cross multiply or cross multiplication. Fractions can be compared by using the process of cross multiplication.

Let us learn more about cross multiply, its definition, the process to do a cross multiply, and how to do a cross multiply with one and more variables, with the help of interactive questions, FAQs.

Definition of Cross Multiply

Cross multiply can be defined as the process of multiplying the numerator of the first fraction on one side of the equals to symbol, with the denominator of the second fraction on the other side of the equals to symbol. Similarly, the denominator of the first fraction is multiplied by the numerator of the second fraction. Cross multiply is also called cross-multiplication or butterfly method. When we want to determine one or more variables in a fraction, we use the method of cross multiply. Fractions can also be compared by using the process of cross multiply.

When two fractions p/q and r/t are multiplied by each other, the process is termed cross multiply or cross-multiplication method. For example p/q and r/t are two fractions and to cross multiply we multiply p × t and q × r as shown in the image below.

Steps to Cross Multiply

Cross multiply is the process of comparing fractions i.e if two fractions are equal to each other or which fraction is greater than the other. This process is useful when we use large fractions. Cross multiply is a simple method of multiplying numbers that are across each other placed diagonally. There are three simple steps in the process of cross multiplication, let us see what they are:

For example, cross multiply 4/6 = 2/3

Step 1: Multiply the numerator of the right-hand side fraction value with the denominator of the left-hand side fraction value.

Step 2: Multiply the denominator of the right-hand side fraction value with the numerator of the left-hand side fraction value. We can write the fraction as 2/1 as seen below as it means the same as 2.

Step 3: Once both the sides LHS and RHS are multiplied after the cross multiplication, we see that the LHS is equal to RHS. Hence the fraction is equal to each other.

Comparing Fractions by Cross Multiply

Cross multiply is used in both like and unlike fractions. When a fraction is unlike i.e. when the denominators of two fractions are not similar while using the process of cross multiplication we not only multiply the numerators to the denominators but we also multiply the denominators. For example, in the image below when we cross multiply 3 × 4 and 2 × 5 we get 12 and 10 respectively. 12 and 10 are the numerators of the 3/2 and 5/4. To make the denominator as common we multiply both the denominators as well 2 × 4 = 8. Hence the new fractions with the same denominators will be 12/8 and 10/8. Since 12 is a greater numerator, 12/8 > 10/8. Therefore, 3/2 > 5/4.

Comparing Ratios by Cross Multiply

The process of cross multiply can be used in comparing ratios and finding the value. We need to multiply the numerator of the first ratio with the denominator of the second ratio and the denominator of the first ratio with the numerator of the second ratio. The process is the same as comparing fractions. There are three simple ways of comparing i.e to find out if the ratios are equal and also to which ratio is greater than the other. This can be understood from the below image.

Cross Multiply With One Variable

A variable is an unknown number in the fraction and by the process of cross multiply, we can find the value of that single variable. The steps to cross multiply with one variable is very simple and similar to a fraction. Lets see how it is done:

Step 1: Multiply the numerator of the first fraction by the denominator of the second fraction.
Step 2: Multiply the numbers. One side will have the whole number while the other side will have a number along with a variable.
Step 3: Solve the equation by finding the value of the variable by dividing both sides of the equation with a coefficient of the variable.

Let us look at an example, 3/x = 15/5. The image below shows all the steps required to solve the equation.

Cross Multiply With Two Variables

Cross multiply of two variables is done in a similar manner as one variable. The same steps are used. Let us look at an example. x/6 = 6/x. Multiply the numerator of the right-hand side fraction value with the denominator of the left-hand side fraction value and vice-vera. The image below shows the steps clearly.

Listed below are a few topics related to cross multiply, take a look.

Multiplication
Integers
Natural Numbers

Example 1: Help Jamie in finding the value of b using the cross multiplication method. 7/b = 14/18

Solution: Using the cross multiplication method.

7/b = 14/18

7 × 18 = 14 × b

126 = 14b

b = 126/14

b = 9.

Therefore, the value of b is 9.
Example 2: Help Alexa in finding the value of x using the cross multiplication method. \(\dfrac{x+1}{8}=\dfrac{3}{x-1}\)

Solution:

\(\Rightarrow\dfrac{x+1}{8}=\dfrac{3}{x-1}\)

\(\Rightarrow\ x^{2}-1=24\)

\(\Rightarrow\ x^{2}=25\)

\(\Rightarrow\ x=\pm5\)

Therefore, \(x=\pm5\).
Example 3: Help Alexa in finding the value of b using the cross multiplication method. \(\dfrac{b}{36}=\dfrac{6}{b^{2}}\).

Solution:

\(\Rightarrow\dfrac{b}{36}=\dfrac{6}{b^{2}}\)

\(\Rightarrow b \times b^{2}=6\times36\)

\(\Rightarrow b^{3}=6\times36\)

\(\Rightarrow b=\sqrt[3]{216}\)

\(\Rightarrow b=6\)

Therefore, \(b=\pm6\).

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FAQs on Cross Multiply

Cross multiply is the process of multiplication used when fractions and ratios are required to be compared. When the numerator of one fraction on one side of the equals to sign is multiplied with the denominator of the second fraction on the other side of the equals to sign, the process is called cross multiply. In other words, the numbers that are multiplied across in a fraction and ratio is cross multiply. It is also called the cross-multiplication or buttery method.

What are the Steps to Cross Multiply?

There are three simple steps used to cross multiply, they are:

Multiply the numerator of the first fraction with the denominator of the second fraction.
Repeat the process with the denominator of the first fraction and the numerator of the second fraction.
Multiply the numbers to find if the first fraction is equal to or greater than the other fraction.

Do You Cross Multiply When Multiplying Fractions?

Yes, we use the cross multiplication method when we multiply fractions. For example, for x/y = z/q we apply cross multiplication method to get xq = yz.

What is Cross Multiply Used for?

We use the cross multiplication method or cross multiplying process to multiply fractions. The process of cross multiplying is used to compare both fractions and ratios i.e. if they are equal to or greater than or lesser than. Cross multiply is also used in finding the value of variables in a fraction for one variable or for two variables.

Can You Cross Multiply Ratios?

Yes, we can cross multiply ratios using the same steps used for cross multiplying fractions. Ratios are written in fraction form and then cross multiplication is used to find the value.

How Do You Cross Multiply?

To cross multiply any two equations, we need to multiply the numerator of the first equation on one side of the equals to sign with the denominator of the second equation on the other side of equals to. Then we need to multiply the denominator of the first equation with the numerator of the second equation. Once multiplied, we compare the equations by checking if they are equal or less than or greater than each other. With the process of cross multiply, we can also multiply the denominators of both the equations to make like fractions, which becomes simple to compare.