Cross Multiply

In Mathematics, we cross multiply the values in an equation, to find the unknown values. 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.

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 in 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 the 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 equation by 3, we get;

18/3 = p

Or

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 = +25 or -25

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 the 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 to 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

Or

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

Or

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

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

Frequently Asked Questions on Cross Multiply

What do you mean by cross multiply?

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.

How to cross multiply to find a variable?

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.

How to cross multiply with negative fraction?

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

What is the value of x, if ⅝ = x/2?

If ⅝ = x/2, then by cross multiplication, we get;
5 (2) = 8 (x)
10 = 8x
x = 10/8
x = 5/4

Can we have denominators of the fraction equal to zero?

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

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