# Cross Multiplying with Fractions

To cross-multiply fractions, we multiply the numerator of the first fraction by the denominator of the second fraction and vice versa.

In the algebraic equation ${\dfrac{a}{b}=\dfrac{c}{d}}$, the cross multiplication is given by the formula ${a\times d=b\times c}$

Thus, by cross-multiplying ${\dfrac{3}{5}=\dfrac{6}{10}}$, we get ${3\times 10=5\times 6}$ (which implies 30 = 30)

Cross-multiplication is also used to solve the value of the unknown variables in any equation.

## With One Variable

If we have one variable on each side, we solve the equation in the following way.

Let us solve ${\dfrac{6}{5}=\dfrac{x}{20}}$ for x

Cross-Multiplying

${6\times 20=5\times x}$

⇒ ${120=5x}$

Dividing both sides by 5

⇒ ${\dfrac{5x}{5}=\dfrac{120}{5}}$

⇒ ${x=24}$

Thus, the solution is x = 24

Problem – Cross Multiplying With NEGATIVE FRACTION

Solve: ${\dfrac{x}{12}=-\dfrac{84}{48}}$

Solution:

Given, ${\dfrac{x}{12}=-\dfrac{84}{48}}$
By cross-multiplying, we get
${x\times 48=12\times \left( -84\right)}$
⇒ ${48x=-1008}$
On dividing both sides by 48,
⇒ ${\dfrac{48x}{48}=-\dfrac{1008}{48}}$
⇒ ${x=-21}$
Thus, the solution is x = -21

## With Two Variables

If the same variable is present on both sides of the equation, then we cross-multiply to find the variable.

Let us solve ${\dfrac{x}{9}=\dfrac{4}{x}}$ for x

Cross-Multiplying

${x\times x=9\times 4}$

⇒ ${x^{2}=36}$

Finding Square Roots of Both Sides

⇒ ${x=\sqrt{36}}$

⇒ ${x=6}$

Thus, the solution is x = 6

## Comparing

### Unlike Fractions

To compare unlike fractions, we convert their denominators to a common value by multiplying the two original denominators together, ensuring both fractions have the same denominator.

For example,

Let us compare ${\dfrac{3}{8}}$ and ${\dfrac{4}{5}}$ by cross multiplication.

Here, the denominators are 8 and 5

Thus, the denominators of both fractions become ${8\times 5=40}$

Now, cross -multiplying to get the numerators,

${3\times 5=15}$ and ${8\times 4=32}$

Thus, the fractions become ${\dfrac{15}{40}}$ and ${\dfrac{32}{40}}$

Now, comparing the fractions,

${\dfrac{15}{40}}$ < ${\dfrac{32}{40}}$

⇒ ${\dfrac{3}{8}}$ < ${\dfrac{4}{5}}$

When the resulting fractions are the same, they are equivalent fractions.

### Ratios

If two ratios are equal (where b, d ≠ 0), then the cross-multiplication product is also equal.

On comparing the ratios:

• ${\dfrac{a}{b}=\dfrac{c}{d}}$, if ${a\times d=b\times c}$
• ${\dfrac{a}{b}<\dfrac{c}{d}}$, if ${a\times d<b\times c}$
• ${\dfrac{a}{b}>\dfrac{c}{d}}$, if ${a\times d>b\times c}$

Now, let us compare 2:4 and 5:10

Here, the fractions are ${\dfrac{2}{4}}$ and ${\dfrac{5}{10}}$

${2\times 10=20}$ and ${4\times 5=20}$

Since ${2\times 4=5\times 10}$

Thus, the ratios 2:4 and 5:10 are equivalent.

If 12 cookies cost $60, what would be the cost of 5 cookies? Solution: Given, The cost of 12 cookies = 60 The cost of 1 cookie =${\dfrac{60}{12}}$…..(i) Let the cost of 5 such cookies be x The cost of 1 cookies =${\dfrac{x}{5}}$…..(ii) From (i) and (ii),${\dfrac{60}{12}=\dfrac{x}{5}}$⇒${60\times 5=12\times x}$⇒${300=12x}$On dividing both sides by 12, ⇒${\dfrac{12x}{12}=\dfrac{300}{12}}$⇒${x=25}$Thus, 5 cookies cost$25