Fractions Formula

The fractions formula helps in conveniently performing the numerous operations on fractions. Compared to normal integral numbers the basic arithmetic operations for fractions, follow different rules. Fractions formulas help us to carry out basic operations with fractions easily. The basic arithmetic operation of addition or subtraction requires the denominators of the fractions to be equal. And while dividing one fraction by another fraction, the division is transformed to multiplication, by taking the reciprocal of the 2nd fraction. Let us learn more about the fraction formula and solve a few examples in this section. 

What is Fractions Formula?

Fractions are one of the most important aspects of arithmetic that we use in our daily life. Fractions represent a numerical value that is a part of the whole value and it is represented using this symbol / (called the fractional line), for example, a/b. Fractions' formulas help in framing rules to be followed while we perform the four main arithmetic operations i.e addition, subtraction, multiplication, and division. Listed below are the fractions formulas:

Formula 1

A mixed fraction has a whole number and a fraction associated with it. The mixed fraction is converted into an improper fraction by multiplying the denominator with the whole number and adding it to the numerator, to form the numerator of the improper fraction.  

\( A\dfrac{b}{c}  = \dfrac{Ac + b}{c} \)

Formula 2

The addition of like fractions is possible by the simple addition of numerators and having the same denominator for the answer. The denominator of the given fractions is equal to the denominator of the final answer.

\( \frac{a}{b}  +\frac{c}{b} = \frac{a + c}{b}\)

Formula 3

For the addition of unlike fractions, each of the fractions is multiplied with suitable constants to make the denominators of the two fractions equal. The aim is to get the denominators of the fractions as equal, before performing the addition process.

\(\frac{a}{b}  +\frac{c}{d} =\frac{a .d}{b. d}  +\frac{c . b}{d . b} = \frac{ad + bc}{bd}\)

Formula 4

Multiplication of fractions is possible by multiplying the numerators and then the denominators of both the fractions and then writing it as a single fraction. Further, this product is simplified and reduced to get the final answer.

\( \frac{a}{b} \times\frac{c}{d} = \frac{ac}{bd}\)

Formula 5

Division of fractions is transformed into a multiplication of fractions by first inverting the fraction in the denominator, and then multiplying it with the numerator fraction.

\(\dfrac{(a/b)}{(c/d)} = \frac{a}{b} \times \frac{d}{c}\)

Fraction Formulas

The other important formulas we use are listed as follows:

Cross multiply rule to check if two fractions are equivalent: if a/b = c/d , then ad = bc

Reciprocal rule to flip numerator and denominator: if a/b is the fraction, then b/a is its reciprocal.

Examples Using Fractions Formula

Example 1: Find the sum of the fractions \(\frac{4}{11} \) and  \( \frac{5}{8} \) using fractions formula.

Solution:

\(\begin{align} \frac{4}{11}  +   \frac{5}{8} &=\frac{4 \times 8}{11 \times 8}  +   \frac{5 \times 11}{8 \times 11} \\ &= \frac{32}{88}  +   \frac{55}{88} \\ &= \frac{32 + 55}{88} \\ &= \frac{87}{88} \end{align} \)

Therefore the sum of the fractions is \(\frac{87}{88}\).

Example 2: Find the value of \(\frac{24}{36} \div \frac{96}{288} \).

Solution:

\( \begin{align} \frac{24}{36} \div \frac{96}{288} & = \dfrac{\frac{24}{36}}{ \frac{96}{288} } \\ &= \frac{24}{36} \times \frac{288}{96} \\ &=  \frac{24}{36} \times \frac{36 \times 8}{24 \times 4} \\ &= 2\end{align} \)

Hence the final value is 2 using the fractions formula.

Example 3: Milk is sold at $16 per gallon. Find the cost of \(6\dfrac{2}{5}\) gallons of milk.

Solution:

Cost of one gallon of milk = $16

Therefore, the cost of \(6\dfrac{2}{5}\) gallons i.e. 32/5 gallons will be 32/5 * 16 = $102.4

Therefore, the cost of gallons of milk is $102.4