Cross Multiply

Cross multiplication is a common method of multiplying numbers in fraction form. Cross multiplication involves multiplying the numerators of one fraction by the denominator of the other across an equation. This method can be easily applied when solving linear equations with two variables. For example:

\(a_1 x + b_1 y + c_1 = 0 \)

\(a_2 x + b_2 y + c_2 = 0 \)

If cross-multiplication is performed correctly, we can quickly obtain the values of x and y by simplifying the equation.

Cross multiplication is primarily used with fractions to compare them or check their equality. This method can help solve equations when dealing with complex fractions. Given below are the different steps involved in this process:

For example: \(\frac{5}{8} = \frac{10}{16} \)

Step 1: To begin, multiply the fraction's numerator on the right by the fraction's denominator on the left. Multiplying \(10 \times 8 = 80 \).

Step 2:  Now, we multiply the fraction's denominator on the right by the fraction's numerator on the left. We can represent \(\frac{10}{16} \) as \(\frac{5}{8} \) (since they are equivalent fractions).

Step 3: After the cross multiplication, always compare the LHS and RHS.
If they are equal, we can conclude that the fractions are equivalent. Here, in this 80 = 80, the fractions are equivalent.

The cross-multiplication formula is used for solving linear equations with two variables, as given below:

\(a_1 x + b_1 y + c_1 = 0 \)

\(a_2 x + b_2 y + c_2 = 0 \)

The cross-multiplication formula used to solve linear equations is:
 

\({x \over b_1c_2 \space –\space b_2c_1} = {y \over c_1a_2 \space– \space c_2a_1} = {1 \over b_2a_1 \space–\space b_1a_2}\)

How to derive cross-multiplication formula?

In linear equations with two variables, we derive the cross-multiplication formula by eliminating one variable, often by making the coefficients of that variable equal.

For example: Let’s consider two linear equations as

\(a_1 x + b_1 y + c_1 = 0 \)

\(a_2 x + b_2 y + c_2 = 0 \)

We solve these equations by making the coefficients of y equal in both equations:

• We first multiply all terms of equation (1) by b₂ and all terms of equation (2) by b₁:
  
  \(b_2 \times (a_1 x + b_1 y + c_1) = 0 \)
  
  ⇒ \(b_2 a_1 x + b_2 b_1 y + c_1 \)....(3)
  
  \(b_1 \times (a_2 x + b_2 y + c_2) = 0 \)
  
  ⇒ \(b_1 a_2 x + b_1 b_2 y + b_1 c_2 \)…(4)
   
• Then, subtract equation (4) from equation (3), 
  
  \((b_2 a_1 - b_1 a_2)x + (b_2 c_1 - b_1 c_2) = 0\)
   
• Isolating x,
  
  \(x = \frac{b_1 c_2 - b_2 c_1}{b_2 a_1 - b_1 a_2}\)…(a)
  
  where \((b_2 a_1 - b_1 a_2) \neq 0\)
   
• Similarly, solve (1) and (2) for y:
  
  \(y = \frac{c_1 a_2 - c_2 a_1}{b_2 a_1 - b_1 a_2}\)…(b)
  
  where \((b_2 a_1 - b_1 a_2) \neq 0\)
   
• We now combine (a) and (b),
  
  \({x \over b_1c_2 \space –\space b_2c_1} = {y \over c_1a_2 \space– \space c_2a_1} = {1 \over b_2a_1 \space–\space b_1a_2}\)
   

Thus, we derived the required cross-multiplication formula.

To compare two fractions using cross-multiplication, start by writing the fractions you want to compare, like \(\frac{a}{b} \ and \ \frac{c}{d}.\) Then, cross-multiply by multiplying the numerator of the first fraction (a) with the denominator of the second fraction (d),and multiplying the numerator of the second fraction (c) with the denominator of the first fraction (b). This gives you two products: a × d and b × c.

After that, compare the two products:

If a × d > b × c, then \(\frac{a}{b}\) is greater than \(\frac{c}{d}\) 

If a × d < b × c, then \(\frac{a}{b}\) is smaller than \(\frac{c}{d}\)

If a × d = b × c, then the two fractions are equal.

This method is useful because it allows you to directly compare the fractions without needing to find a common denominator.

To compare two ratios using cross-multiplication, follow these simple steps:

Write down the two ratios you want to compare. For example, you have:
\(\frac{a}{b}\ and \ \frac{c}{d}\)
    ​
Cross-multiply: Multiply the numerator of the first ratio (a) by the denominator of the second ratio (d), and multiply the numerator of the second ratio (c) by the denominator of the first ratio (b). This gives you:
\(a×d\)
\(b×c\)

Compare the products:

If a×d>b×c, then the first ratio is larger.

If a×d<b×c, then the second ratio is larger.

If a×d=b×c, the ratios are equal.

Example: Let’s compare the ratios 34 and 56.

Cross-multiply: Multiply 3 by 6: 3×6=18

Multiply 4 by 5: 4×5=20

Compare: Since 18 is less than 20, we know that 34 is smaller than  56.

Set up the equation: You’ll start with two fractions, and one of them will have a variable (like x). For example:\(\frac{a}{b}\ =\ \frac{c}{x}\)

Cross-multiply: This means you multiply the top number of the first fraction by the bottom number of the second fraction, and multiply the top number of the second fraction by the bottom number of the first fraction. This gives you: 
                      \(   a×x=b×c\)

Solve for x: To find x, just divide both sides of the equation by a
 \(x = \frac{b \times c}{a}\)
     

Example: Let’s say you have this equation: \(\frac{3}{4} \ =\ \frac{6}{x}\)

Cross-multiply:

Multiply 3 by x to get 3x.

Multiply 4 by 6 to get 24.

So, now you 3x = 24.

Solve for x: To isolate x, divide both sides by 3:

\(x=\frac{24}{3}\)

So, x = 8.

Write down the equation: You’ll start with two fractions, each containing a variable (like x and y). For example:
\(\frac{x}{a} \ = \ \frac{y}{b}\)  

Cross-multiply: This means you multiply the top of the first fraction by the bottom of the second fraction, and the top of the second fraction by the bottom of the first fraction. You get:

\( x×b=y×a\)

Simplify: Now, you have an equation without fractions, which is easier to work with. You can solve for one variable.

Example: Let’s say you have the equation:

\(\frac{x}{3} \ = \ \frac{y}{4}\)

Cross-multiply:

Multiply x by y: \( x \times y.\)

Multiply three by 4:\( 3×4=12.\)

So now the equation looks like this:

\(x×y=12\)

Solve for one variable: If you want to find x, you can divide both sides of the equation by y

x = 12y

Now, you can solve for x once you know the value of 𝑦

Cross-multiplication is a quick and effective method for solving linear equations with two variables. Solving linear equations with two variables, we apply the cross-multiplication method as given below:

Use the cross-multiplication formula:
 
\({x \over b_1c_2 \space –\space b_2c_1} = {y \over c_1a_2 \space– \space c_2a_1} = {1 \over b_2a_1 \space–\space b_1a_2}\)

Solve the linear equations:

\(3x + 4y = 8\)

\(2x + y = 5\)
 

• First, we convert the equations into standard form:
  Since the general form of a linear equation is:
  
  \(ax + by + c = 0 \)
  
  We rewrite the given equation as:
  
  \(3x + 4y – 8 = 0\)
  
  \(2x + y – 5 = 0\)
   
• When comparing the general form: \(a_1 x + b_1 y = -c_1\) and \(a_2 x + b_2 y = -c_2\):
  
  \(a_1 = 3, \quad b_1 = 4, \quad c_1 = -8\)
  
  \(a_2 = 2, \quad b_2 = 1, \quad c_2 = -5\)
   
• Now, apply the cross-multiplication formula:
  
  \({x \over b_1c_2 \space –\space b_2c_1} = {y \over c_1a_2 \space– \space c_2a_1} = {1 \over b_2a_1 \space–\space b_1a_2}\)
   
• Substituting the values:
  
  \({x \over (4\times-5) \space –\space (1\times-8)} = {y \over (-8\times2) \space– \space (-5\times3)} = {1 \over (1\times 3) \space–\space (4\times2)}\)
   
• Simplifying the fractions:\({x \over (-20) \space –\space (-8)} = {y \over (-16) \space– \space (-15)} = {1 \over 3 \space–\space 8}\)

  
  \({x \over (-12)} = {y \over (-1)} = {1 \over (-5)}\)
   

• Isolating x:
  Comparing the first and third fractions:
  
  \({x \over (-12)} = {1 \over (-5)}\)
  
  \({x } = {{-12 \times 1} \over (-5) } = {12 \over 5}\)
   
• Isolating y:
  Compare the second and third fractions:
  
  \({y \over (-1)} = {1 \over (-5)}\)
  
  \({y } = {1 \times -1 \over (-5)} = {1 \over 5}\)
  
  ⇒ \(x = \frac{12}{5}, \quad y = \frac{1}{5}\)

Cross multiplication can help learners to find many values in real life on a daily basis. It is one of the important topics of mathematics. Here are some tips and tricks to be a master in cross multiplying.
 

• Understand the concept very well. Learn the equation and always remember it.
   
• Before cross-multiplying, always remember to check if the fraction is in its simplest form or not. If not, simplify it.
   
• After solving, estimate to see if the answer is making an equivalent.
   
• Try to solve problems by using diagonal arrow marks for easier identification.
   
• Practice more with some real-life examples such as finding cooking recipes, shopping discounts, etc.

Cross-multiplication in equations helps students determine the value of unknown numbers easily. This technique can be applied to various real-life situations. Here are a few examples of its applications:

• Cross-multiplication is used to compare the prices of different items for choosing the cheaper option when the quantity remains the same.
   
• This method helps adjust the quantities of ingredients required for a recipe without changing the proportions.
   
• Cross-multiplication is utilized in calculating the time required for a journey at a constant speed. Suppose a car travels 180 km in 3 hours, then we can calculate how far it can travel in 5 hours at the same speed using cross multiplication. Which on the other hand, will give us the distance as 300 km.
   
• We apply cross-multiplication to convert currencies accurately on the basis of their exchange rates. If 1 USD values up to 75 rupees in India, We can calculate the value of 50 USD by cross multiplying them. 
   
• While shopping a product with discount, we can calculate the amount we need to pay by cross multiplying the discounted percentage and the product's cost.